Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS

# Tight Majorizations and Convergence Ratesof Nuclear Norm Minimization IRLS
Source: [https://arxiv.org/html/2608.23765](https://arxiv.org/html/2608.23765)
Christian Kümmerle††thanks:School of Data, Mathematical, and Statistical Sciences, Department of Computer Science & Institute of Artificial Intelligence, University of Central Florida, Oviedo, FL 32816, USA \([kuemmerle@ucf\.edu](mailto:[email protected])\)\.Tomas Masak††thanks:Institute for Statistics and Mathematics, Wirtschaftsuniversität Wien, 1020 Vienna, Austria \([tomas\.masak@wu\.ac\.at](mailto:[email protected])\)\.Dominik Stöger††thanks:Department of Mathematics, KU Eichstätt\-Ingolstadt, 85049 Ingolstadt, Germany \([dominik\.stoeger@ku\.de](mailto:[email protected])\)\.

###### Abstract

Iteratively reweighted least squares \(IRLS\) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood\. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low\-rank recovery\. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic\-mean weight operator defines a valid global quadratic majorizer\. Furthermore, we show that this weight operator is optimal within the family of power\-mean weights, clarifying why it improves over classical one\-sided reweighting schemes that use only row\- or column\-space information\. Under a Schatten\-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic\-mean weights\. For IRLS with harmonic\-mean weights, we prove a dimension\-independent, locally linear convergence rate\. We provide a counterexample showing that this dimension\-independent local rate cannot in general be obtained for IRLS algorithms using one\-sided weight operators, which predominate in the literature\. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic\-mean reweighting across square, rectangular, and adversarially initialized recovery problems\.

Keywords:iteratively reweighted least squares, low\-rank matrix recovery, majorization\-minimization, harmonic\-mean weight operator, global and local convergence\.

###### Contents

1. [1Introduction](https://arxiv.org/html/2608.23765#S1)
2. [2Related Work](https://arxiv.org/html/2608.23765#S2)
3. [3TheMatrixIRLSAlgorithm and Its Majorization Properties](https://arxiv.org/html/2608.23765#S3)1. [3\.1IRLS for Low\-Rank Recovery and Basic Properties](https://arxiv.org/html/2608.23765#S3.SS1) 2. [3\.2Majorization of Quadratic Model Implied by Harmonic\-Mean Weights](https://arxiv.org/html/2608.23765#S3.SS2) 3. [3\.3Choice of the Smoothing Parameter](https://arxiv.org/html/2608.23765#S3.SS3) 4. [3\.4Optimality of Harmonic\-Mean Weight Operator](https://arxiv.org/html/2608.23765#S3.SS4)
4. [4Linear Convergence Rates of IRLS for Nuclear Norm Minimization](https://arxiv.org/html/2608.23765#S4)1. [4\.1Preliminaries](https://arxiv.org/html/2608.23765#S4.SS1) 2. [4\.2Global Linear Convergence of IRLS for Nuclear Norm Minimization](https://arxiv.org/html/2608.23765#S4.SS2)1. [4\.2\.1Rank\-rrGround Truth𝐗⋆\\mathbf\{X\}\_\{\\star\}](https://arxiv.org/html/2608.23765#S4.SS2.SSS1) 2. [4\.2\.2Approximately Low\-Rank Ground Truth](https://arxiv.org/html/2608.23765#S4.SS2.SSS2) 3. [4\.3Fast Local Linear Rate ofMatrixIRLSwith Harmonic\-Mean Weights](https://arxiv.org/html/2608.23765#S4.SS3)
5. [5Numerical Experiments](https://arxiv.org/html/2608.23765#S5)1. [5\.1Setup](https://arxiv.org/html/2608.23765#S5.SS1) 2. [5\.2Linear Convergence Rate Factors for IRLS with Weight Operator Variants](https://arxiv.org/html/2608.23765#S5.SS2) 3. [5\.3Recovery of Rectangular Low\-Rank Matrices](https://arxiv.org/html/2608.23765#S5.SS3) 4. [5\.4Adversarial Initialization](https://arxiv.org/html/2608.23765#S5.SS4) 5. [5\.5Role of the Smoothing Parameter Update Rules](https://arxiv.org/html/2608.23765#S5.SS5)
6. [6Conclusion](https://arxiv.org/html/2608.23765#S6)
7. [AProofs of Main Results](https://arxiv.org/html/2608.23765#A1)1. [A\.1Proof of\(Harmonic\-Mean Quadratic Model Majorization\)](https://arxiv.org/html/2608.23765#A1.SS1)1. [A\.1\.1Proof of Lemmavia Iterative Pinching](https://arxiv.org/html/2608.23765#A1.SS1.SSS1) 2. [A\.2Proof of\(Optimality of Harmonic Mean Quadratic Model\)](https://arxiv.org/html/2608.23765#A1.SS2)1. [A\.2\.1Proof of\(Monotonicity of Power Mean Weight Operators\)](https://arxiv.org/html/2608.23765#A1.SS2.SSS1) 2. [A\.2\.2Proofs ofand\(Second Order Necessary Condition for Majorization\)](https://arxiv.org/html/2608.23765#A1.SS2.SSS2) 3. [A\.2\.3Proof of](https://arxiv.org/html/2608.23765#A1.SS2.SSS3) 3. [A\.3Proofs ofand\(Global Linear Convergence\)](https://arxiv.org/html/2608.23765#A1.SS3)1. [A\.3\.1Preliminaries and General Proof Strategy](https://arxiv.org/html/2608.23765#A1.SS3.SSS1) 2. [A\.3\.2Upper Bounds on Quadratic Forms Implied by Weight Operators](https://arxiv.org/html/2608.23765#A1.SS3.SSS2) 3. [A\.3\.3Proof of](https://arxiv.org/html/2608.23765#A1.SS3.SSS3) 4. [A\.3\.4Proof of](https://arxiv.org/html/2608.23765#A1.SS3.SSS4) 4. [A\.4Proofs ofand\(Local Linear Convergence and Counterexample\)](https://arxiv.org/html/2608.23765#A1.SS4)1. [A\.4\.1Tight Upper Bounds on Harmonic\-Mean Quadratic Forms](https://arxiv.org/html/2608.23765#A1.SS4.SSS1) 2. [A\.4\.2Proof of\(Dimension\-Free Fast Linear Rate ofMatrixIRLS\)](https://arxiv.org/html/2608.23765#A1.SS4.SSS2) 3. [A\.4\.3Proof of\(Counterexample for One\-Sided IRLS\)](https://arxiv.org/html/2608.23765#A1.SS4.SSS3)
8. [BComplementary Proofs](https://arxiv.org/html/2608.23765#A2)1. [B\.1Proof of\(Properties of the IRLS Quadratic Model\)](https://arxiv.org/html/2608.23765#A2.SS1) 2. [B\.2Lipschitz Gradients of Smoothed Nuclear Norm](https://arxiv.org/html/2608.23765#A2.SS2) 3. [B\.3Prior Art of Majorization Proofs for Low\-Rank IRLS Algorithms](https://arxiv.org/html/2608.23765#A2.SS3)1. [B\.3\.1Proof ofUsing Concavity Arguments](https://arxiv.org/html/2608.23765#A2.SS3.SSS1) 2. [B\.3\.2Proof ofUsing Variational Calculus](https://arxiv.org/html/2608.23765#A2.SS3.SSS2) 3. [B\.3\.3Proof ofBased on](https://arxiv.org/html/2608.23765#A2.SS3.SSS3) 4. [B\.3\.4Challenges for Harmonic\-Mean Majorization Proofs](https://arxiv.org/html/2608.23765#A2.SS3.SSS4)
9. [References](https://arxiv.org/html/2608.23765#bib)

## 1Introduction

Optimization methods that treat a matrix variable through its*spectrum*rather than through its entries have moved to the center of large\-scale machine learning\. TheMuonoptimizer\([Jordan et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib99)\)replaces the raw gradient \(or momentum\) matrix𝐆=𝐔​diag⁡\(𝝈\)​𝐕⊤\\mathbf\{G\}=\\mathbf\{U\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}^\{\\top\}by its*orthogonalized*counterpartmsign⁡\(𝐆\)=𝐔r​𝐕r⊤\\operatorname\{msign\}\(\\mathbf\{G\}\)=\\mathbf\{U\}\_\{r\}\\mathbf\{V\}\_\{r\}^\{\\top\}, computed in practice by a few Newton–Schulz iterations\([Amsel et al\., 2026](https://arxiv.org/html/2608.23765#bib.bib104)\), and has been adopted for the pre\-training of large language models at the trillion\-parameter scale\([Liu et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib105);[Kimi Team, 2025](https://arxiv.org/html/2608.23765#bib.bib106);[DeepSeek\-AI, 2026](https://arxiv.org/html/2608.23765#bib.bib107)\)\. The fact that makes steps involvingmsign⁡\(𝐆\)\\operatorname\{msign\}\(\\mathbf\{G\}\)principled using spectral geometry is a duality statement: sincemsign⁡\(𝐆\)\\operatorname\{msign\}\(\\mathbf\{G\}\)attains the maximum in‖𝐆‖∗=max‖𝚫‖≤1⁡⟨𝐆,𝚫⟩F\\left\\lVert\\mathbf\{G\}\\right\\rVert\_\{\\ast\}=\\max\_\{\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\\leq 1\}\\langle\\mathbf\{G\},\\mathbf\{\\Delta\}\\rangle\_\{F\}, so aMuonstep is exactly steepest descent with respect to the spectral norm, with the nuclear norm‖⋅‖∗\\left\\lVert\\cdot\\right\\rVert\_\{\\ast\}as the dual norm in which progress is measured\([Carlson et al\., 2015](https://arxiv.org/html/2608.23765#bib.bib101);[Bernstein and Newhouse, 2025](https://arxiv.org/html/2608.23765#bib.bib100);[Chen et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib103);[Pethick et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib102)\);𝐆↦msign⁡\(𝐆\)\\mathbf\{G\}\\mapsto\\operatorname\{msign\}\(\\mathbf\{G\}\)is one of the subgradients of the nuclear norm\.

Beyond its role as a dual norm in this context, the nuclear norm has long served as the canonical convex surrogate for matrix rank\([Fazel, 2002](https://arxiv.org/html/2608.23765#bib.bib31);[Recht et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib65)\)\. This second role is central to low\-rank recovery: from underdetermined linear measurements𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\), where𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{m\}andm≪d1​d2m\\ll d\_\{1\}d\_\{2\}, one seeks to reconstruct a matrix𝐗⋆\\mathbf\{X\}\_\{\\star\}of rankr⋆≪min⁡\(d1,d2\)r\_\{\\star\}\\ll\\min\(d\_\{1\},d\_\{2\}\), or one that is well approximated by such a matrix\. Problems for which this type of modeling arises have been prominently studied in signal processing, data science, and machine learning\. Instances include phase retrieval problems\([Candès et al\., 2013](https://arxiv.org/html/2608.23765#bib.bib30)\), the design of recommender systems\([Candès and Recht, 2009](https://arxiv.org/html/2608.23765#bib.bib67);[Koren et al\., 2009](https://arxiv.org/html/2608.23765#bib.bib26);[Chi et al\., 2019](https://arxiv.org/html/2608.23765#bib.bib68)\), blind demixing\([Ling and Strohmer, 2017](https://arxiv.org/html/2608.23765#bib.bib77);[Jung et al\., 2018](https://arxiv.org/html/2608.23765#bib.bib78)\)and the quantum state tomography\([Gross et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib24);[Tariq et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib98)\)problem\. For these problems, the benchmark convex estimator is

min𝐗∈ℝd1×d2​‖𝐗‖∗subject to𝒜⁡\(𝐗\)=𝐲\.\\underset\{\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\}\{\\min\}\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\qquad\\text\{subject to \}\\qquad\\mathcal\{A\}\(\\mathbf\{X\}\)=\\mathbf\{y\}\.\(1\)Under standard random sensing models,[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)succeeds at near\-intrinsic sample sizes\. For example,m≳r⋆​\(d1\+d2\)m\\gtrsim r\_\{\\star\}\(d\_\{1\}\+d\_\{2\}\)Gaussian rank\-one measurements suffice, optimally up to constant factors\([Cai and Zhang, 2015](https://arxiv.org/html/2608.23765#bib.bib83);[Kueng et al\., 2017](https://arxiv.org/html/2608.23765#bib.bib80);[Kabanava et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib81)\), and precise phase transitions have been studied in[Romanov and Gavish \(2018\)](https://arxiv.org/html/2608.23765#bib.bib25)\. Thus, the statistical rationale for nuclear norm minimization is mature\. However, the computational aspects of this problem remain challenging at scale:[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)admits an exact semidefinite programming formulation, which allows for polynomial time guarantees using generic SDP algorithms\([Nesterov and Nemirovskii, 1994](https://arxiv.org/html/2608.23765#bib.bib23)\)\. The non\-smoothness and spectral structure of‖𝐗‖∗\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}pose challenges for specialized solvers, which may suffer from a slow \(e\.g\., sublinear\) convergence rate and the need for repeated full singular value decompositions\.

A structurally suitable algorithmic paradigm for solving[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)is based on a framework going back to the 1930s\([Weiszfeld, 1937](https://arxiv.org/html/2608.23765#bib.bib19);[Beck and Sabach, 2015](https://arxiv.org/html/2608.23765#bib.bib18)\)known as iteratively reweighted least squares \(IRLS\), which iteratively solves a sequence of weighted least squares problems after implicitly smoothing and majorizing the non\-smooth objective by a sequence of quadratic model functions\. IRLS has been widely used for separable non\-smooth optimization in computer vision, robust statistics, or compressed sensing, just to name a few\([Ochs et al\., 2015](https://arxiv.org/html/2608.23765#bib.bib57);[Holland and Welsch, 1977](https://arxiv.org/html/2608.23765#bib.bib47);[Daubechies et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib38)\)\. IRLS is also known as half\-quadratic minimization\([Nikolova and Ng, 2005](https://arxiv.org/html/2608.23765#bib.bib3)\)and is related to the so\-called*η\\eta\-trick*\([Bach, 2019](https://arxiv.org/html/2608.23765#bib.bib1)\)in machine learning\.

While IRLS is relatively well understood for separable objectives, both its analysis and its design for spectral objectives such as[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)remain much less settled\. IRLS methods for spectral objectives such as[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)use updates of the form

𝐗\(k\)\\displaystyle\\mathbf\{X\}^\{\(k\)\}:=arg​min𝐙∈ℝd1×d2⁡⟨𝐙,W\(k\)​\(𝐙\)⟩F​subject to​𝒜​\(𝐙\)=𝐲,\\displaystyle:=\\argmin\_\{\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\}\\ \\langle\\mathbf\{Z\},W^\{\(k\)\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\;\\text\{ subject to \}\\;\\mathcal\{A\}\(\\mathbf\{Z\}\)=\\mathbf\{y\},\(2\)where the positive\-definite weight operatorW\(k\)W^\{\(k\)\}is constructed from spectral information of𝐗\(k−1\)\\mathbf\{X\}^\{\(k\-1\)\}\. The main design choice is precisely how to define this operator\. Following the initial works of[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54);[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48), the majority of papers on IRLS methods for spectral objective minimization\([Cai and Li, 2017](https://arxiv.org/html/2608.23765#bib.bib17);[Radhakrishnan et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib86);[Zhu et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib87)\)use*one\-sided*reweighting, whereW\(k\)W^\{\(k\)\}encodes only column space information of𝐗\(k−1\)\\mathbf\{X\}^\{\(k\-1\)\}and corresponds to a multiplication by a matrix from the left \(or the analogue for row space information and multiplication by a matrix from the right\), whereas another line of research\([Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36);[Kümmerle and Mayrink Verdun, 2021](https://arxiv.org/html/2608.23765#bib.bib33);[Kraemer, 2025](https://arxiv.org/html/2608.23765#bib.bib13)\)considers different weight operators that use information from both the row and column spaces of𝐗\(k−1\)\\mathbf\{X\}^\{\(k\-1\)\}\. This raises the question of whether different weight operators merely reflect different trade\-offs, or whether there is a principled optimal choice for the spectral minimization problem at hand\.

The understanding of IRLS methods with iterates[eq\.2](https://arxiv.org/html/2608.23765#S1.E2)for nuclear norm minimization is also limited by a lack of available convergence guarantees\. The best available results\([Mohan and Fazel, 2012](https://arxiv.org/html/2608.23765#bib.bib48);[Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54)\)only imply \(global\) convergence to a ground truth low\-rank matrix, in case where the nuclear norm minimization problem has a unique low\-rank solution, and subsequence convergence to limiting matrices satisfying error bounds in the case of noisy measurements and/or only appropriately low\-rank matrices to be recovered\([Cai and Li, 2017](https://arxiv.org/html/2608.23765#bib.bib17)\)\. This is somewhat at odds with the mature trajectory analysis of other low\-rank matrix recovery methodologies for which rigorous convergence guarantees involving*convergence rates*are available\([Tu et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib76);[Chen and Chi, 2018](https://arxiv.org/html/2608.23765#bib.bib2)\)\.

This paper aims to provide, on the one hand, principled answers to the IRLS design question for[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)and, on the other hand, the first convergence rate analysis of IRLS methods for this problem\. These two goals are intertwined\. In[Figure1](https://arxiv.org/html/2608.23765#S1.F1), we illustrate, for a representative rank\-two matrix recovery problem involving rank\-one measurements, a typical convergence trajectory of IRLS iterates𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}following the updates of[eq\.2](https://arxiv.org/html/2608.23765#S1.E2)simply depending on different weight operator choices\. We observe that the objective gap decreases linearly for different choices, but with a notably different linear rate, in which a so\-called harmonic\-mean weight operatorW\(k\)W^\{\(k\)\}leads to a notably improved linear rate\.

Figure 1:Smoothed nuclear norm gap𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}of IRLS iterates for harmonic, left\-sided, right\-sided, and arithmetic mean weight operators\. The rank\-22ground truth𝐗⋆∈ℝ140×140\\mathbf\{X\}\_\{\\star\}\\in\\R^\{140\\times 140\}and them=2520m=2520rank\-one measurements are generated using independent𝒩⁡\(0,1\)\\mathcal\{N\}\(0,1\)entries\.In[Figure1](https://arxiv.org/html/2608.23765#S1.F1), the objective gap is measured with respect to a family of smoothed nuclear norm objectives𝒥ε:ℝd1×d2→ℝ\\mathcal\{J\}\_\{\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rgiven by

𝒥ε​\(𝐗\):=∑i=1djε​\(σi​\(𝐗\)\),withjε​\(σ\):=\{\|σ\|,if​\|σ\|\>ε,σ22​ε\+ε2,if​\|σ\|≤ε,\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\):=\\sum\_\{i=1\}^\{d\}j\_\{\\varepsilon\}\(\\sigma\_\{i\}\(\\mathbf\{X\}\)\),\\quad\\text\{ with \}\\quad j\_\{\\varepsilon\}\(\\sigma\):=\\begin\{cases\}\|\\sigma\|,&\\text\{ if \}\|\\sigma\|\>\\varepsilon,\\\\ \\frac\{\\sigma^\{2\}\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\},&\\text\{ if \}\|\\sigma\|\\leq\\varepsilon,\\end\{cases\}\(3\)which are implicitly minimized by the IRLS updates[eq\.2](https://arxiv.org/html/2608.23765#S1.E2)—specifically, by virtue of[eq\.2](https://arxiv.org/html/2608.23765#S1.E2)minimizing a quadratic modelQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\,\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\)of𝒥εk\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}about the current iterate, together with a smoothing\-parameter update forε\\varepsilon\. An accurate convergence analysis of the resulting IRLS iterates requires that the quadratic modelQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\,\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\)*majorizes pointwise*the smoothed objective𝒥εk\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}, which has been only known for models defined by one\-sided weight operators\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Mohan and Fazel, 2012](https://arxiv.org/html/2608.23765#bib.bib48)\)\. For the harmonic\-mean IRLS method, however, no majorization result has been established, impeding its convergence analysis\.

In this paper, we show that the harmonic\-mean weight operator not only induces a valid majorizing quadratic model of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}, providing the foundation for global convergence guarantees, but also that it is optimal within a family of power\-mean weight operators\. Building on this, we provide the first convergence analysis for IRLS methods for nuclear norm minimization involving convergence rates, leading also to a sharper local linear rate for the harmonic\-mean IRLS method than for classical IRLS methods\. We summarize our contributions as follows:

- •The Harmonic\-mean Quadratic Model is a Valid Majorizer\.In[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1), we prove that the quadratic model induced by the harmonic mean weight operator globally majorizes the smoothed nuclear norm𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}\. The proof develops a new spectral comparison argument based on a Sylvester\-equation characterization and iterative pinching, thereby overcoming the noncommutativity that obstructs standard separability\-based arguments\.
- •Harmonic Mean is the Tightest Power Mean\.Takingqq\-power means of the left\- and right\-sided weights yields a Loewner\-ordered family of weight operators, and hence an ordered family of quadratic models\. In[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), we show thatq≥−1q\\geq\-1is precisely the range that guarantees global majorization of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}\. Consequently, the harmonic\-mean choiceq=−1q=\-1induces the smallest, and thus tightest, quadratic majorizer within this family\.
- •Global and Local Convergence with Linear Rates\.[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)establishes global linear convergence of IRLS for every admissible one\-sided or power\-mean weight operator and any positive\-definite initialization of the weights, provided that𝒜\\mathcal\{A\}satisfies the null space property \(NSP\) of orderrr\.[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)extends this linear decay to approximately low\-rank ground truths, up to their best rank\-rrapproximation error\. The global contraction factors retain a dependence on the matrix dimension\. By contrast, once the iterates enter a specified neighborhood of the ground truth,[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)establishes a local linear rate for harmonic\-mean IRLS whose contraction factor is independent ofd1d\_\{1\}andd2d\_\{2\}\. Complementing this positive result,[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)constructs local instances satisfying the NSP for which a one\-sided IRLS step reduces the relative error by at most orderr/dr/d, ruling out an analogous dimension\-independent local rate in general\. The separation theoretically justifies the performance gap between the IRLS variants observed in[Figure1](https://arxiv.org/html/2608.23765#S1.F1)\.[Table1](https://arxiv.org/html/2608.23765#S1.T1)places these guarantees in the context of the existing IRLS convergence theory, for both sparse vector and low\-rank matrix recovery\.

Table 1:IRLS convergence results at the convex endpointp=1p=1: the sparseℓ1\\ell\_\{1\}benchmark and the direct nuclear\-norm comparison\. The final row records the harmonic\-mean specialization of this paper\.ReferenceWeightGlobal guaranteeLocal / distinguishing resultSparse recovery \(ℓ1\\ell\_\{1\}\): benchmark[Daubechies et al\. \(2010\)](https://arxiv.org/html/2608.23765#bib.bib38)entrywiseconv\. under NSP; no ratelocal linear rate; basinO⁡\(𝐱⋆min\)O\(\\mathbf\{x\}\_\{\\star\}^\{\\operatorname\{min\}\}\)[Kümmerle et al\. \(2021\)](https://arxiv.org/html/2608.23765#bib.bib37)entrywiselinear rate,1−Cη1​d1\-\\frac\{C\}\{\\eta\_\{1\}d\}arbitrary initializationLow\-rank recovery \(nuclear norm\): direct comparison[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54);[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48)one\-sidedconv\. under NSP; no rateno local rate established[Cai and Li \(2017\)](https://arxiv.org/html/2608.23765#bib.bib17)one\-sidedconv\. under RIP; no rateapprox\. rank, noise stability[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)harm\. meann/asuperlinear of order2−p2\-p, does not apply to nuclear norm \(p=1p=1\)This paperharm\. meanlinear rate,1−Cη1​d1\-\\frac\{C\}\{\\eta\_\{1\}d\}local linear rate1−cηr1\-c\_\{\\eta\_\{r\}\}; basinO⁡\(σr​\(𝐗⋆\)/d\)O\(\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)/\\sqrt\{d\}\);tight majorizer
For visual comparison,dddenotes the ambient vector dimension in the sparse block andd=min⁡\(d1,d2\)d=\\min\(d\_\{1\},d\_\{2\}\)in the low\-rank block;η1\\eta\_\{1\}is the respective order\-one NSP constant\.

A preliminary version of the global linear convergence result for nuclear norm IRLS was presented at the 2024 IEEE 13th Sensor Array and Multichannel Signal Processing Workshop \(SAM\) and appeared in its proceedings\([Kümmerle and Stöger, 2024](https://arxiv.org/html/2608.23765#bib.bib14)\)\. This four\-page workshop paper did not contain proofs nor the majorization, optimality, or fast local\-rate results developed here\.

##### Outline\.

In[Section3](https://arxiv.org/html/2608.23765#S3), we present relevant notions of weight operators and outline the IRLS methodology for nuclear norm minimization, as well as the majorization and optimality results for the harmonic\-mean weight operator\.[Section4](https://arxiv.org/html/2608.23765#S4)contains the convergence results\. The numerical experiments of[Section5](https://arxiv.org/html/2608.23765#S5)corroborate the predicted advantage of harmonic\-mean reweighting across square, rectangular, and adversarially initialized recovery problems \(showing in particular that this advantage is not merely an artifact\), and investigate the practical trade\-offs between different smoothing\-parameter schedules\.[Section6](https://arxiv.org/html/2608.23765#S6)contains a conclusion and discussion of future research directions\. Finally, all proofs of presented main results are contained in[AppendixA](https://arxiv.org/html/2608.23765#A1), with some complementary proofs implicitly known in the literature being provided in[AppendixB](https://arxiv.org/html/2608.23765#A2)\.

##### Notation\.

Matrices are denoted by bold uppercase letters \(such as𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\) and vectors are denoted by bold lowercase letters \(𝐱∈ℝd\\mathbf\{x\}\\in\\R^\{d\}\)\. The singular values of a matrix𝐗\\mathbf\{X\}are denoted byσ1​\(𝐗\)≥σ2​\(𝐗\)≥…≥σmin⁡\(d1,d2\)​\(𝐗\)≥0\\sigma\_\{1\}\(\\mathbf\{X\}\)\\geq\\sigma\_\{2\}\(\\mathbf\{X\}\)\\geq\\ldots\\geq\\sigma\_\{\\min\(d\_\{1\},d\_\{2\}\)\}\(\\mathbf\{X\}\)\\geq 0\. We use the convention ofσi=0\\sigma\_\{i\}=0fori\>min⁡\(d1,d2\)i\>\\min\(d\_\{1\},d\_\{2\}\)\. The Frobenius inner product of two matrices𝐗,𝐘∈ℝd1×d2\\mathbf\{X\},\\mathbf\{Y\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is denoted by⟨𝐗,𝐘⟩F=tr⁡\(𝐗⊤​𝐘\)\\langle\\mathbf\{X\},\\mathbf\{Y\}\\rangle\_\{F\}=\\trace\(\\mathbf\{X\}^\{\\top\}\\mathbf\{Y\}\)\. The spectral norm of a matrix𝐗\\mathbf\{X\}is denoted by‖𝐗‖=σ1​\(𝐗\)\\left\\lVert\\mathbf\{X\}\\right\\rVert=\\sigma\_\{1\}\(\\mathbf\{X\}\)and the nuclear norm of a matrix𝐗\\mathbf\{X\}is denoted by‖𝐗‖∗=∑i=1min⁡\(d1,d2\)σi​\(𝐗\)\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}=\\sum\_\{i=1\}^\{\\min\(d\_\{1\},d\_\{2\}\)\}\\sigma\_\{i\}\(\\mathbf\{X\}\)\. We denote by𝐗∘𝐘\\mathbf\{X\}\\circ\\mathbf\{Y\}the Hadamard \(entrywise\) product of two matrices𝐗\\mathbf\{X\}and𝐘\\mathbf\{Y\}\. Whenever applicable, we use the notation ofd=min⁡\(d1,d2\)d=\\min\(d\_\{1\},d\_\{2\}\)andD=max⁡\(d1,d2\)D=\\max\(d\_\{1\},d\_\{2\}\), and\[d\]:=\{1,2,…,d\}\[d\]:=\\\{1,2,\\ldots,d\\\}\. Finally, forr∈\{0,…,d\}r\\in\\\{0,\\ldots,d\\\}and𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, we denote the error of the best rank\-rrapproximation of𝐗\\mathbf\{X\}in nuclear norm byβr​\(𝐗\)∗:=minrank⁡\(𝐙\)≤r⁡‖𝐗−𝐙‖∗=∑i=r\+1dσi​\(𝐗\)\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}:=\\min\_\{\\rank\(\\mathbf\{Z\}\)\\leq r\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{Z\}\\right\\rVert\_\{\\ast\}=\\sum\_\{i=r\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{X\}\)\. Equivalently,βr​\(𝐗\)∗\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}is the nuclear\-norm tail of the singular\-value vector of𝐗\\mathbf\{X\}after its firstrrentries\.

## 2Related Work

##### Low\-Rank Matrix Recovery\.

Popular methodologies for low\-rank matrix recovery impose a rank\-rrconstraint by construction, using iterative rank projections\([Jain et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib70)\), factorizations such as𝐗=𝐔𝐕⊤\\mathbf\{X\}=\\mathbf\{U\}\\mathbf\{V\}^\{\\top\}\([Burer and Monteiro, 2005](https://arxiv.org/html/2608.23765#bib.bib69)\), or Riemannian optimization\([Tu et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib76);[Vandereycken, 2013](https://arxiv.org/html/2608.23765#bib.bib12)\)\. See[Chi et al\. \(2019\)](https://arxiv.org/html/2608.23765#bib.bib68)for an overview\. Under a rank\-restricted isometry property, singular value projection converges globally at a linear rate from zero, whereas factorized and Riemannian gradient methods are commonly shown to converge linearly only after a certified initialization enters a local basin\. These methods are generally analyzed together with spectral initialization, which is often costly to compute and may be unstable in the presence of noise or outliers\. Global convergence from random initialization has only been proven so far for specific measurement models, e\.g\., for rank\-one phase retrieval\([Chen et al\., 2019](https://arxiv.org/html/2608.23765#bib.bib35)\), and for small random initialization\([Stöger and Soltanolkotabi, 2021](https://arxiv.org/html/2608.23765#bib.bib88);[Soltanolkotabi et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib89)\)\. Moreover, standard factorized gradient descent slows as the condition number of the ground truth increases\. More recent variants can remove condition\-number dependence from the local contraction or attain faster local rates, although end\-to\-end guarantees still depend on the initialization\([Tong et al\., 2021](https://arxiv.org/html/2608.23765#bib.bib74);[Cai et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib75);[Zilber and Nadler, 2022](https://arxiv.org/html/2608.23765#bib.bib72);[Luo et al\., 2023](https://arxiv.org/html/2608.23765#bib.bib73)\)\. All these formulations constitute nonconvex estimators and require the target rank for provable convergence\.

Nuclear norm minimization[eq\.1](https://arxiv.org/html/2608.23765#S1.E1), in contrast, is the standard convex estimator and does not fix the rank in its formulation\. Under suitable RIP or null space conditions, it achieves stable recovery at the degrees\-of\-freedom sampling orderr⁡\(d1\+d2\)r\(d\_\{1\}\+d\_\{2\}\), up to model\-dependent logarithmic factors and independently ofκ\\kappa\([Recht et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib65);[Gross, 2011](https://arxiv.org/html/2608.23765#bib.bib79);[Kueng et al\., 2017](https://arxiv.org/html/2608.23765#bib.bib80);[Kabanava et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib81)\), see also[Davenport and Romberg \(2016\)](https://arxiv.org/html/2608.23765#bib.bib64);[Fuchs et al\. \(2022\)](https://arxiv.org/html/2608.23765#bib.bib63)for an overview\. Standard first\-order schemes such as singular value thresholding may provide faster per\-iteration complexity than generic semidefinite programming solvers, but are typically analyzed with sublinear rates\([Cai et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib9)\)\. Unlike for nonconvex estimators, algorithms with provable global linear rates under near\-optimal sample complexity withoutκ\\kappa\-dependence are available for[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)or closely related unconstrained formulations, for example for the restarted accelerated primal\-dual method of[Colbrook \(2022\)](https://arxiv.org/html/2608.23765#bib.bib15)and the restarted mirror descent method of[Ding and Wang \(2026\)](https://arxiv.org/html/2608.23765#bib.bib16), under suitable sharpness conditions \(e\.g\., Frobenius\-robust NSP, cf\.[Section4\.1](https://arxiv.org/html/2608.23765#S4.SS1)\)\. The global linear rate we show for IRLS variants in[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)is likewise independent ofκ\\kappa, but neither requires a specific step size nor restart schedule\. The faster local rate of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)for harmonic meanMatrixIRLSonly depends on the NSP constantηr\\eta\_\{r\}and not directly on the dimension, unlike the certified rates of[Colbrook \(2022\)](https://arxiv.org/html/2608.23765#bib.bib15)and[Ding and Wang \(2026\)](https://arxiv.org/html/2608.23765#bib.bib16)\.

##### Low\-Rank IRLS\.

The foundational works by[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54)and[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48)proposed the first IRLS algorithms for low\-rank matrix recovery, using left\-sided reweighting for nuclear norm minimization and right\-sided reweighting for Schatten\-ppquasi\-norm minimization withp∈\(0,1\]p\\in\(0,1\]\(see[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)\), respectively\. Both works establish global convergence of the respective IRLS algorithm under the NSP assumption \(see[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)\), albeit*without any convergence rate*\. As a tool, they implicitly establish the majorization property[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)for the respective quadratic model functions \(cf\.[SectionsB\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1)and[B\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2)for details\)\.[Lai et al\. \(2013, Theorem 3\.6\)](https://arxiv.org/html/2608.23765#bib.bib49)provide an error bound for the limiting iterate of an IRLS algorithm of an unconstrained variant of[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)under an RIP assumption\. On the other hand, the local linear rate analysis that is claimed by[Lai et al\. \(2013, p\. 950\)](https://arxiv.org/html/2608.23765#bib.bib49)is not substantiated\.[Cai and Li \(2017\)](https://arxiv.org/html/2608.23765#bib.bib17)established stability guarantees for the limiting iterate of one\-sided IRLS for nuclear norm minimization subject to a residual norm inequality constraint\.

[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)proposed IRLS algorithm variants for Schatten\-ppquasi\-norm minimization withp∈\(0,1\]p\\in\(0,1\]using harmonic\-mean weight operators motivated from a perspective to treat column and row space information symmetrically, and showed local superlinear convergence of order2−p2\-pin a neighborhood of the ground truth matrix𝐗⋆\\mathbf\{X\}\_\{\\star\}under NSP assumptions for these IRLS variants; however, the relevant result\([Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), Theorem 11\)does*not apply*for the nuclear norm case ofp=1p=1, but is only meaningful in the quasi\-norm case wherep∈\(0,1\)p\\in\(0,1\)\.[Algorithm1](https://arxiv.org/html/2608.23765#alg1)studied in this paper is a variant of[Kümmerle and Sigl \(2018, Alg\. 1\)](https://arxiv.org/html/2608.23765#bib.bib36)specified top=1p=1, but using a different smoothing parameter schedule \(see[Section3\.3](https://arxiv.org/html/2608.23765#S3.SS3)\)\. Finally, the proof of the technical result in[Kümmerle and Sigl \(2018, Lemma 14\)](https://arxiv.org/html/2608.23765#bib.bib36), which would imply a majorization property of the harmonic\-mean quadratic model, is faulty, as we point out in[SectionB\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4)\. Our proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)overcomes this issue and provides the foundation for a global analysis ofMatrixIRLS\. A related line of literature\([Kümmerle and Mayrink Verdun, 2021](https://arxiv.org/html/2608.23765#bib.bib33);[Ghosh et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib11)\)focuses on the nonconvex log\-determinant minimization problem and establishes locally quadratic convergence rates for suitable geometric mean weight operator\-based IRLS variants under near\-optimal sample complexity assumptions for the respective problems of matrix completion and Euclidean distance geometry\. Recently,[Kraemer \(2025\)](https://arxiv.org/html/2608.23765#bib.bib13)established asymptotic convergence properties of IRLS methods for log\-determinant minimization and emphasizes the importance of choosing an appropriate smoothing parameter schedule, particularly at the information\-theoretic limit\.

On a related note, the framework of recursive feature machines \(RFM\), a novel methodology for sample\-efficient feature learning\([Radhakrishnan et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib85)\), has been identified to be related to IRLS methods for spectral optimization problems such as[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)if specified to linear models\([Radhakrishnan et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib86)\)\.[Radhakrishnan et al\. \(2025\)](https://arxiv.org/html/2608.23765#bib.bib86)provides a derivation of one\-sided IRLS methods for log\-determinant and nuclear norm minimization and related objectives based on the RFM framework and proposes an SVD\-free implementation of one\-sided IRLS for log\-determinant minimization\. However, the proposed SVD\-free implementation does not apply to the nuclear norm case, and no rigorous convergence analysis of low\-rank IRLS methods has emerged from this framework\.

##### IRLS for Sparse Recovery\.

IRLS algorithms for sparse vector recovery, which precede low\-rank IRLS methods, are also relevant for the context of this work\([Gorodnitsky and Rao, 1997](https://arxiv.org/html/2608.23765#bib.bib51);[Rao and Kreutz\-Delgado, 1999](https://arxiv.org/html/2608.23765#bib.bib50);[Wipf and Nagarajan, 2010](https://arxiv.org/html/2608.23765#bib.bib56)\)\. These algorithms have been well\-known as efficient solvers forℓp\\ell\_\{p\}\-quasi\-norm minimization problems for0<p≤10<p\\leq 1\.[Chartrand and Yin \(2008\)](https://arxiv.org/html/2608.23765#bib.bib52)and[Wipf and Nagarajan \(2010\)](https://arxiv.org/html/2608.23765#bib.bib56)proposed and empirically studied adaptive smoothing parameter schedules forε\\varepsilon, observing their importance for the success of the methodology\.[Daubechies et al\. \(2010\)](https://arxiv.org/html/2608.23765#bib.bib38)analyzes an adaptive scheme akin to[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)for the smoothing parameterε\\varepsilonof an IRLS algorithm forℓ1\\ell\_\{1\}\-minimization under underdetermined linear measurements and provides a global convergence guarantee for the method under a sparse NSP assumption on the measurement matrix, while also showing locally linear convergence \(independent of the ambient dimension\), the analogue of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)in this work, and locally superlinear convergence for IRLS targeted forℓp\\ell\_\{p\}\-quasi\-norm minimization with0<p<10<p<1\. Global linear convergence rates were established in[Kümmerle et al\. \(2021\)](https://arxiv.org/html/2608.23765#bib.bib37)forℓ1\\ell\_\{1\}\-minimization and in[Peng et al\. \(2022\)](https://arxiv.org/html/2608.23765#bib.bib22);[Peng et al\. \(2023\)](https://arxiv.org/html/2608.23765#bib.bib21)for related robust estimation problems\.

The fact that vectorℓp\\ell\_\{p\}\-objectives of vectors are separable significantly simplifies the design and analysis of relevant IRLS methods compared to IRLS for spectral optimization, as studied in this paper\. In particular, sparse IRLS offers fewer degrees of freedom in the design of the weight operator, with no analogue of the off\-diagonal entries of the weight operator core matrix \([Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)\)\. We note that the usage of particularly tight quadratic model designs as studied in this work is necessary to achieve convergence rate results forMatrixIRLSthat are in line with the ones for sparse recovery \(see[Table1](https://arxiv.org/html/2608.23765#S1.T1)\)\.

## 3TheMatrixIRLSAlgorithm and Its Majorization Properties

In this section, we first introduce theMatrixIRLSalgorithm, the associated weight operators, and the resulting quadratic models\. We then show that the harmonic\-mean weight operator yields a valid majorizer, before discussing its optimality within the power\-mean family\.

### 3\.1IRLS for Low\-Rank Recovery and Basic Properties

Iteratively reweighted least squares can be interpreted as a*smoothing method*\([Chen, 2012](https://arxiv.org/html/2608.23765#bib.bib32)\), where a Huber\-type smoothing\([Huber, 1964](https://arxiv.org/html/2608.23765#bib.bib40)\)of the nuclear norm is minimized via quadratic majorizing models\([Lange, 2016](https://arxiv.org/html/2608.23765#bib.bib28);[Sun et al\., 2017](https://arxiv.org/html/2608.23765#bib.bib29)\)\. In particular, IRLS mitigates the non\-smoothness of the nuclear norm objective[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)by working with*ε\\varepsilon\-smoothed nuclear norms*𝒥εk\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}of[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)\.𝒥εk\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}is in turn minimized approximately by constructing a quadratic modelQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\,\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\)about the current iterate𝐗k\\mathbf\{X\}^\{k\}\.

It can be shown that𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)is differentiable with a Lipschitz\-continuous gradient \(see[SectionB\.2](https://arxiv.org/html/2608.23765#A2.SS2)\)\. But despite the surrogate𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)being smooth, minimizing it directly remains challenging because it depends nonlinearly on the singular values of its argument\. The IRLS strategy is thus to replace𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}locally by quadratic modelsQε\(⋅∣𝐗\):ℝd1×d2→ℝQ\_\{\\varepsilon\}\\left\(\\cdot\\mid\\mathbf\{X\}\\right\):\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rgiven suitable reference points𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\(see[Equation8](https://arxiv.org/html/2608.23765#S3.E8)below\)\. IfQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\\left\(\\cdot\\mid\\mathbf\{X\}\\right\)*majorizes*𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}globally, IRLS can be interpreted as a Majorization\-Minimization \(MM\) method\([Lange, 2016](https://arxiv.org/html/2608.23765#bib.bib28);[Sun et al\., 2017](https://arxiv.org/html/2608.23765#bib.bib29)\)intertwined with smoothing\([Chen, 2012](https://arxiv.org/html/2608.23765#bib.bib32)\)\. Since𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}is a spectral function, the curvature information entering the quadratic model is naturally expressed in the singular\-vector coordinates of the reference matrix𝐗\\mathbf\{X\}\. Analogously to IRLS algorithms for separable problems\([Wipf and Nagarajan, 2010](https://arxiv.org/html/2608.23765#bib.bib56);[Daubechies et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib38)\), which encode inverse magnitude information into entrywise weights, we incorporate the singular value information into the*weight operator core matrix*, which in turn fixes the curvature of the quadratic model\. However, as we will see, the non\-diagonal entries of this matrix contain degrees of freedom which can be chosen in different ways, leading to a multitude of possible quadratic models\.

###### Definition 1\(Weight Operator Core Matrix\)\.

Let𝛔∈ℝD\\bm\{\\sigma\}\\in\\R^\{D\}be a nonincreasing vector of singular values \(padded by zeros for indices larger thandd\), and letε\>0\\varepsilon\>0be a smoothing parameter\. The associated*weight operator core matrix*𝐇𝛔,ε∈ℝd1×d2\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is defined entrywise fori∈\[d1\]i\\in\[d\_\{1\}\]andj∈\[d2\]j\\in\[d\_\{2\}\], by one of the following choices:

\(𝐇𝝈,ε\)i​j\\displaystyle\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=\[max⁡\(σi,ε\)\]−1,\\displaystyle=\[\\max\(\\sigma\_\{i\},\\varepsilon\)\]^\{\-1\},left\-sided\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54)\),\\displaystyle\\text\{left\-sided \\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{Fornasier11\}\{\\@@citephrase\{, \}\}\{\}\)\}\},\(4\)\(𝐇𝝈,ε\)i​j\\displaystyle\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=\[max⁡\(σj,ε\)\]−1,\\displaystyle=\[\\max\(\\sigma\_\{j\},\\varepsilon\)\]^\{\-1\},right\-sided\([Mohan and Fazel, 2012](https://arxiv.org/html/2608.23765#bib.bib48)\),\\displaystyle\\text\{right\-sided \\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{mohan\_fazel\}\{\\@@citephrase\{, \}\}\{\}\)\}\},\(5\)\(𝐇𝝈,ε\)i​j\\displaystyle\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=2​\[max⁡\(σi,ε\)\+max⁡\(σj,ε\)\]−1,\\displaystyle=2\[\\max\(\\sigma\_\{i\},\\varepsilon\)\+\\max\(\\sigma\_\{j\},\\varepsilon\)\]^\{\-1\},harmonic\-mean\([Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36)\)\.\\displaystyle\\text\{harmonic\-mean \\cite\[citep\]\{\(\\@@bibref\{AuthorsPhrase1Year\}\{Kummerle\-JMLR2018\}\{\\@@citephrase\{, \}\}\{\}\)\}\}\.\(6\)

The three core matrix types[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4),[5](https://arxiv.org/html/2608.23765#S3.E5)and[6](https://arxiv.org/html/2608.23765#S3.E6)differ only in the way they compute non\-diagonal entries, coinciding on the diagonal\.𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}of[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)uses the harmonic mean of[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)and[eq\.5](https://arxiv.org/html/2608.23765#S3.E5)of indices corresponding to row and column indices of the matrix, whereas the entries of[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)and[eq\.5](https://arxiv.org/html/2608.23765#S3.E5)depend only on singular values associated to the matrix’s row and column indices, respectively\. With[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1), we can define the*weight operator*W𝐗,ε:ℝd1×d2→ℝd1×d2W\_\{\\mathbf\{X\},\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}, associated to a smoothing parameterε\>0\\varepsilon\>0and a matrix iterate𝐗\\mathbf\{X\}, specifying a quadratic modelQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)of𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)about𝐗\\mathbf\{X\}\.

###### Definition 2\(,[2018](https://arxiv.org/html/2608.23765#bib.bib36);[Kümmerle and Mayrink Verdun](https://arxiv.org/html/2608.23765#bib.bib33),[2021](https://arxiv.org/html/2608.23765#bib.bib33)\)\.

Letε\>0\\varepsilon\>0and𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be a matrix with the full singular value decomposition𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}, where𝐔𝐗∈ℝd1×d1\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐕𝐗∈ℝd2×d2\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}are orthogonal anddiag⁡\(𝛔\)∈ℝd1×d2\\diag\(\\bm\{\\sigma\}\)\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is the rectangular diagonal matrix with the extended vector of singular values𝛔∈ℝD\\bm\{\\sigma\}\\in\\R^\{D\}of𝐗\\mathbf\{X\}on the diagonal\. Then the*weight operator*W𝐗,ε:ℝd1×d2→ℝd1×d2W\_\{\\mathbf\{X\},\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}is defined by

W𝐗,ε​\(𝐙\)=𝐔𝐗​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\]​𝐕𝐗⊤W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\(7\)for each𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. Here𝐇𝛔,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}is the weight operator core matrix of[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)\.

The weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is self\-adjoint\. Using[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)with any of the core matrices, we define the*quadratic model function of𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)about𝐗\\mathbf\{X\}*associated to the weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\), writtenQε\(⋅∣𝐗\):ℝd1×d2→ℝQ\_\{\\varepsilon\}\\left\(\\cdot\\mid\\mathbf\{X\}\\right\):\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R, such that for all𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\},

Qε​\(𝐙∣𝐗\):=𝒥ε​\(𝐗\)\+⟨∇𝒥ε​\(𝐗\),𝐙−𝐗⟩F\+12​⟨𝐙−𝐗,W𝐗,ε​\(𝐙−𝐗\)⟩F\.\\begin\{split\}Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)&:=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{Z\}\-\\mathbf\{X\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\.\\end\{split\}\(8\)
###### Example 1\.

To make the geometry of the smoothed nuclear norm and its quadratic surrogates \(which depend on the weighting choice, be it left\-sided, right\-sided, or harmonic\-mean\) explicit, consider a simple2×22\\times 2example chosen to be nonsymmetric so that the harmonic, left\-sided, and right\-sided weights are genuinely different\. Let

X⁡\(t\)=\(432t\),t∈ℝ,X\(t\)=\\begin\{pmatrix\}4&3\\\\ 2&t\\end\{pmatrix\},\\qquad t\\in\\mathbb\{R\},\(9\)corresponding to the situation where only the last entry of a2×22\\times 2matrix is unknown\. One can immediately see thatt=3/2t=3/2gives the only rank\-deficient candidate\. It is also straightforward to calculate that

‖X⁡\(t\)‖∗=29\+t2\+8​\|t−3/2\|\\\|X\(t\)\\\|\_\{\*\}=\\sqrt\{29\+t^\{2\}\+8\|t\-3/2\|\}and verify that on thet<3/2t<3/2branch the nuclear norm is decreasing inttwhile on thet\>3/2t\>3/2branch it is increasing\. Thus,t⋆=3/2t\_\{\\star\}=3/2is also the unique nuclear norm minimizer\.

[Figure2](https://arxiv.org/html/2608.23765#S3.F2)\(left\) displays the nuclear norm \(although it appears piecewise linear intt, it is not\) with its several smoothed versions\.[Figure2](https://arxiv.org/html/2608.23765#S3.F2)\(right\) then shows one step ofMatrixIRLSfrom the initial pointt0=1/2t\_\{0\}=1/2and the smoothing parameterε=0\.5\\varepsilon=0\.5\. The three weighting schemes introduced in[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)lead to different surrogates, of which the harmonic\-mean surrogateQ0\.5harmQ\_\{0\.5\}^\{\\mathrm\{harm\}\}is the tightest\. In the next step ofMatrixIRLS, this surrogate is minimized to obtaint1t\_\{1\},ε\\varepsilonis reduced, and a new surrogate is constructed att1t\_\{1\}, to be minimized in the next step\.

Figure 2:The nuclear norm of \([9](https://arxiv.org/html/2608.23765#S3.E9)\) from[Example1](https://arxiv.org/html/2608.23765#Thmexample1)\.*Left*: The smoothed nuclear norm𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}forε=0\.8,0\.5,0\.2\\varepsilon=0\.8,0\.5,0\.2and0\.10\.1\.*Right*: The left\-sided, right\-sided, and harmonic\-mean surrogates of𝒥0\.5\\mathcal\{J\}\_\{0\.5\}att0=1/2t\_\{0\}=1/2\. The next step of the algorithm withε\\varepsilonreduced to0\.20\.2is also indicated\.The following proposition states elementary properties of this expression for weighting schemes introduced above\. Exactness of the quadratic model at𝐙=𝐗\\mathbf\{Z\}=\\mathbf\{X\}follows from its definition, and a specific gradient matching leads to a simplification of the quadratic model\.

###### Proposition 1\(Gradient Condition and Symmetry of𝑸𝜺\\bm\{Q\}\_\{\\bm\{\\varepsilon\}\}\)\.

Letε\>0\\varepsilon\>0,𝒥ε:ℝd1×d2→ℝ\\mathcal\{J\}\_\{\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rbe defined as in[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)andQε\(⋅∣𝐗\):ℝd1×d2→ℝQ\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\):\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Ras defined in[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)via weight operatorW𝐗,ε:ℝd1×d2→ℝd1×d2W\_\{\\mathbf\{X\},\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}defined in[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)\. Then the quadratic model and the smoothed surrogate function𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)coincide at the reference point, i\.e\.,Qε​\(𝐗∣𝐗\)=𝒥ε​\(𝐗\)Q\_\{\\varepsilon\}\(\\mathbf\{X\}\\mid\\mathbf\{X\}\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)for𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. Furthermore, if the weight operator core matrix𝐇𝛔,ε∈ℝd1×d2\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}satisfies\(𝐇𝛔,ε\)i​i=max⁡\(σi,ε\)−1\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=\\max\(\\sigma\_\{i\},\\varepsilon\)^\{\-1\}for alli∈\[d\]i\\in\[d\], then, for any𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, the following gradient condition holds:

W𝐗,ε​\(𝐗\)=∇𝒥ε​\(𝐗\)for each​𝐗∈ℝd1×d2,W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)=\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\\quad\\text\{ for each \}\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\},\(10\)and the quadratic model satisfies for each𝐙,𝐗∈ℝd1×d2\\mathbf\{Z\},\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}that

Qε​\(𝐙∣𝐗\)=𝒥ε​\(𝐗\)\+12​⟨𝐙,W𝐗,ε​\(𝐙\)⟩F−12​⟨𝐗,W𝐗,ε​\(𝐗\)⟩F\.Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}\.\(11\)

The proof of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)is straightforward and provided for completeness in[SectionB\.1](https://arxiv.org/html/2608.23765#A2.SS1)\. Inspecting the weight operator core matrix of[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1), we observe that the diagonal condition\(𝐇𝝈,ε\)i​i=max⁡\(σi,ε\)−1\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=\\max\(\\sigma\_\{i\},\\varepsilon\)^\{\-1\}for alli∈\[d\]i\\in\[d\], and thus, the assertion of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1), is satisfied for both one\-sided weight operator notions[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)and[eq\.5](https://arxiv.org/html/2608.23765#S3.E5), and for the harmonic mean weight operator[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)\.

Observe that, by[eq\.11](https://arxiv.org/html/2608.23765#S3.E11), minimizingQε​\(𝐙∣𝐗\)Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)corresponds to minimizing a*reweighted*least squares objective⟨𝐙,W𝐗,ε​\(𝐙\)⟩F\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}once the reference point𝐗\\mathbf\{X\}and the smoothing parameterε\\varepsilonare fixed\. This fact gives*iteratively reweighted least squares \(IRLS\)*methods their name and motivates methods that alternate between updating the reference point𝐗\\mathbf\{X\}, the smoothing parameterε\\varepsilon, and the weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)\. Instantiating such a method with harmonic\-mean weight operators in the quadratic modelQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)yields the IRLS method of main interest in this paper, which we callMatrixIRLSand state as[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\. When discussing the convergence properties ofMatrixIRLSin[Section4](https://arxiv.org/html/2608.23765#S4), we will also refer to variants of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)which use alternative weight operators instead of harmonic\-mean weight operators—in these cases, only the update rule for the governing weight operatorW\(k\)W^\{\(k\)\}of[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)changes, whereas all other aspects of the algorithm remain unchanged\.

Algorithm 1MatrixIRLSfor Nuclear Norm MinimizationInput:Operator

𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{m\}, data vector

𝐲∈ℝm\\mathbf\{y\}\\in\\mathbb\{R\}^\{m\}, initial positive definite weight operator

W\(0\):ℝd1×d2→ℝd1×d2W^\{\(0\)\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}\(default:

W\(0\)=IdW^\{\(0\)\}=\\Id\), rank estimate

r~<d\\widetilde\{r\}<d\.

Set

ε−1=∞\\varepsilon\_\{\-1\}=\\infty\.

for

k=0,1,2,…k=0,1,2,\\ldotsdo

Solve weighted least squares problem:

𝐗\(k\)\\displaystyle\\mathbf\{X\}^\{\(k\)\}:=arg​min𝐙∈ℝd1×d2⁡⟨𝐙,W\(k\)​\(𝐙\)⟩F​subject to​𝒜​\(𝐙\)=𝐲\.\\displaystyle:=\\argmin\_\{\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\}\\ \\langle\\mathbf\{Z\},W^\{\(k\)\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\;\\text\{ subject to \}\\;\\mathcal\{A\}\(\\mathbf\{Z\}\)=\\mathbf\{y\}\.\(12\)
Update smoothing parameter

ε\\varepsilon:

εk\\displaystyle\\varepsilon\_\{k\}:=min⁡\(εk−1,βr~​\(𝐗\(k\)\)∗/d\),\\displaystyle:=\\min\\left\(\\varepsilon\_\{k\-1\},\\,\\beta\_\{\\widetilde\{r\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}/d\\right\),\(13\)where

βr~​\(𝐗\(k\)\)∗\\beta\_\{\\widetilde\{r\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}is the nuclear norm error of the best rank\-

r~\\widetilde\{r\}approximation of

𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}\.

If

εk=0\\varepsilon\_\{k\}=0, i\.e\., if

βr~​\(𝐗\(k\)\)∗=0\\beta\_\{\\widetilde\{r\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}=0,return

𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}\.

Update weight operator: Define

W\(k\+1\):=W𝐗\(k\),εkW^\{\(k\+1\)\}:=W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}using weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)from[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2), where the weight operator core matrix is defined using the*harmonic\-mean weights*[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)\.

endfor

returnSequence

\(𝐗\(k\)\)k≥0\(\\mathbf\{X\}^\{\(k\)\}\)\_\{k\\geq 0\}\.

We note that it is*not*necessary to compute the entire spectral information of𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}to implement the weight update in[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\. On the contrary, the weight operator update in[Algorithm1](https://arxiv.org/html/2608.23765#alg1)can be implemented by computing*only the first*rkr\_\{k\}singular valuesσi​\(𝐗\(k\)\)\\sigma\_\{i\}\(\\mathbf\{X\}^\{\(k\)\}\)and matrices𝐔\(k\)∈ℝd1×rk\\mathbf\{U\}^\{\(k\)\}\\in\\R^\{d\_\{1\}\\times r\_\{k\}\}and𝐕\(k\)∈ℝd2×rk\\mathbf\{V\}^\{\(k\)\}\\in\\R^\{d\_\{2\}\\times r\_\{k\}\}with leadingrkr\_\{k\}left and right singular vectors of𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}, whererk:=\|\{i∈\[d\]:σi​\(𝐗\(k\)\)\>εk\}\|r\_\{k\}:=\|\\\{i\\in\[d\]:\\sigma\_\{i\}\(\\mathbf\{X\}^\{\(k\)\}\)\>\\varepsilon\_\{k\}\\\}\|\. While not the focus of the present paper, we refer to[Kümmerle and Mayrink Verdun \(2021\)](https://arxiv.org/html/2608.23765#bib.bib33)and[Ghosh et al\. \(2024\)](https://arxiv.org/html/2608.23765#bib.bib11)for more details on the implementation of such low\-rank IRLS algorithms\. Furthermore, we note that the smoothing update[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)is different from the ones proposed in the existing literature\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36);[Kümmerle and Mayrink Verdun, 2021](https://arxiv.org/html/2608.23765#bib.bib33)\)\. The reasons for this choice are detailed in[Section3\.3](https://arxiv.org/html/2608.23765#S3.SS3)and[Section5\.5](https://arxiv.org/html/2608.23765#S5.SS5)\.

The update[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)returnsεk=0\\varepsilon\_\{k\}=0precisely if𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}has rank at mostr~\\widetilde\{r\}; the weight operator update is then undefined, which is why[Algorithm1](https://arxiv.org/html/2608.23765#alg1)returns, and the returned matrix is a feasible matrix of rank at mostr~\\widetilde\{r\}, hence equal to a ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}of rank at mostr~=r\\widetilde\{r\}=rif𝒜\\mathcal\{A\}satisfies the NSP of[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4): the difference𝐍:=𝐗\(k\)−𝐗⋆∈ker⁡\(𝒜\)\\mathbf\{N\}:=\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\in\\ker\(\\mathcal\{A\}\)has rank at most2​r2r, so that∑i\>rσi​\(𝐍\)≤∑i≤rσi​\(𝐍\)≤ηr​∑i\>rσi​\(𝐍\)\\sum\_\{i\>r\}\\sigma\_\{i\}\(\\mathbf\{N\}\)\\leq\\sum\_\{i\\leq r\}\\sigma\_\{i\}\(\\mathbf\{N\}\)\\leq\\eta\_\{r\}\\sum\_\{i\>r\}\\sigma\_\{i\}\(\\mathbf\{N\}\)by[eq\.24](https://arxiv.org/html/2608.23765#S4.E24), andηr<1\\eta\_\{r\}<1leaves only𝐍=𝟎\\mathbf\{N\}=\\mathbf\{0\}\. Accordingly, all statements about𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}andεk\\varepsilon\_\{k\}in the following refer to the iterations carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1), where a final iteration withεk=0\\varepsilon\_\{k\}=0is covered by the conventionj0​\(σ\):=\|σ\|j\_\{0\}\(\\sigma\):=\|\\sigma\|in[eq\.3](https://arxiv.org/html/2608.23765#S1.E3), i\.e\.,𝒥0=‖⋅‖∗\\mathcal\{J\}\_\{0\}=\\left\\lVert\\cdot\\right\\rVert\_\{\\ast\}, which is the pointwise limit of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}asε→0\+\\varepsilon\\to 0^\{\+\}\.

### 3\.2Majorization of Quadratic Model Implied by Harmonic\-Mean Weights

The basic properties of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)are algebraic consequences of the weight construction\. They explain why the quadratic model leads to a weighted least\-squares subproblem, but they are insufficient for an analysis of the IRLS algorithm’s properties\. The missing piece is a statement that connects the weighted least\-squares solution𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}of[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)to progress in the objective value of theε\\varepsilon\-smoothed nuclear norm objective𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\), such as

Qεk−1​\(𝐗\(k\)∣𝐗\(k−1\)\)≥𝒥εk−1​\(𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\-1\)\}\)\\geq\\mathcal\{J\}\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\(14\)for a pair of algorithm iterates𝐗\(k−1\)\\mathbf\{X\}^\{\(k\-1\)\}and𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\. Together with the basic properties from[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1), the smoothing parameter update step[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)and the fact thatε↦𝒥ε​\(𝐗\)\\varepsilon\\mapsto\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)is monotonically nondecreasing,[eq\.14](https://arxiv.org/html/2608.23765#S3.E14)implies that the two iterates𝐗\(k−1\)\\mathbf\{X\}^\{\(k\-1\)\}and𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)satisfy for allk≥1k\\geq 1that

𝒥εk​\(𝐗\(k\)\)≤𝒥εk−1​\(𝐗\(k\)\)≤Qεk−1​\(𝐗\(k\)∣𝐗\(k−1\)\)≤Qεk−1​\(𝐗\(k−1\)∣𝐗\(k−1\)\)=𝒥εk−1​\(𝐗\(k−1\)\)\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\\mathcal\{J\}\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq Q\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\-1\)\}\)\\leq Q\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\-1\)\}\\mid\\mathbf\{X\}^\{\(k\-1\)\}\)=\\mathcal\{J\}\_\{\\varepsilon\_\{k\-1\}\}\(\\mathbf\{X\}^\{\(k\-1\)\}\)\.\(15\)Therefore, the sequence\{𝒥εk​\(𝐗\(k\)\)\}k\\left\\\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\\right\\\}\_\{k\}is nonincreasing\. Ifε¯:=limk→∞εk\>0\\bar\{\\varepsilon\}:=\\lim\_\{k\\to\\infty\}\\varepsilon\_\{k\}\>0, standard IRLS arguments, under the remaining assumptions of the respective analyses, then imply that every accumulation point is stationary for the constrained minimization of𝒥ε¯\\mathcal\{J\}\_\{\\bar\{\\varepsilon\}\}, cf\.[Fornasier et al\. \(2011, Theorem 6\.11\)](https://arxiv.org/html/2608.23765#bib.bib54)and[Kümmerle and Sigl \(2018, Theorem 9\)](https://arxiv.org/html/2608.23765#bib.bib36)\. A majorization property such as[eq\.14](https://arxiv.org/html/2608.23765#S3.E14)is likewise essential for the convergence\-rate analysis in[Section4](https://arxiv.org/html/2608.23765#S4)\.

However, as we discuss in[SectionB\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4), existing proofs and analogous strategies fail to establish a corresponding statement for the quadratic model functions defined by harmonic\-mean weight operators \(and thus, for the setting of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\), which constitute in some sense a tighter approximation of the smoothed nuclear norm objectives than their one\-sided analogues\. As the main result of this paper, we establish such a*global majorization*forQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)defined by the harmonic\-mean weight operator\.

###### Theorem 1\(Global Majorization of Harmonic\-Mean Quadratic Model\)\.

Letε\>0\\varepsilon\>0, let𝒥ε:ℝd1×d2→ℝ\\mathcal\{J\}\_\{\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rbe theε\\varepsilon\-smoothed nuclear norm[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)andQε\(⋅∣𝐗\):ℝd1×d2→ℝQ\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\):\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rbe the quadratic model function of[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)defined by the harmonic\-mean weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)with core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)\. Then,Qε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)majorizes𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}globally, i\.e\.,

Qε​\(𝐙∣𝐗\)≥𝒥ε​\(𝐙\)Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\(16\)for each𝐙,𝐗∈ℝd1×d2\\mathbf\{Z\},\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.

Our novel proof approach relies on directly establishing a suitable lower bound for the weighted inner product term⟨𝐙,W𝐗,ε​\(𝐙\)⟩F\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}using the spectral properties of𝐙\\mathbf\{Z\}\. More precisely, in the harmonic\-mean case, we can show that given𝐙\\mathbf\{Z\}, the matrixW𝐗,ε​\(𝐙\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)solves a Sylvester equation\. Using pinching techniques, we can then replace this Sylvester equation iteratively by simpler ones, until we can finally obtain a closed\-form expression for the weighted inner product term\. We provide a detailed proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)in[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)\. We think that it might be of independent interest for other approximation problems involving spectral functions beyond nuclear norm\-type problems\. A specialization of the novel proof technique to the case of one\-sided weights and[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)is discussed in[SectionB\.3\.3](https://arxiv.org/html/2608.23765#A2.SS3.SSS3)\.

### 3\.3Choice of the Smoothing Parameter

It is well\-known in the IRLS literature\([Daubechies et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib38);[Aravkin et al\., 2019](https://arxiv.org/html/2608.23765#bib.bib58);[Peng et al\., 2022](https://arxiv.org/html/2608.23765#bib.bib22);[Lerman et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib10)\)that the choice of the smoothing parameterε\\varepsilonis crucial for both the theoretical analysis of an IRLS algorithm and its empirical performance\. The update rule[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)proposed in[Algorithm1](https://arxiv.org/html/2608.23765#alg1)differs from the prevailing update rules in the low\-rank IRLS literature, using the scaledℓ1\\ell\_\{1\}\-tailβr~​\(𝐗\(k\)\)∗/d=∑i=r~\+1dσi​\(𝐗\(k\)\)/d\\beta\_\{\\widetilde\{r\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}/d=\\sum\_\{i=\\widetilde\{r\}\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{X\}^\{\(k\)\}\)/dto update the smoothing parameterε\\varepsilon, which requires information about the entire spectrum of𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}, whereas other works\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36);[Kümmerle and Mayrink Verdun, 2021](https://arxiv.org/html/2608.23765#bib.bib33);[Ghosh et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib11)\)all use the\(r~\+1\)\(\\widetilde\{r\}\+1\)st singular valueσr~\+1​\(𝐗\(k\)\)\\sigma\_\{\\widetilde\{r\}\+1\}\(\\mathbf\{X\}^\{\(k\)\}\)in[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)in lieu of the scaledℓ1\\ell\_\{1\}\-tail:

εk:=min⁡\(εk−1,σr~\+1​\(𝐗\(k\)\)\)\.\\varepsilon\_\{k\}:=\\min\\left\(\\varepsilon\_\{k\-1\},\\sigma\_\{\\widetilde\{r\}\+1\}\(\\mathbf\{X\}^\{\(k\)\}\)\\right\)\.\(17\)This has certain computational advantages\.

The works of[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36),[Kümmerle and Mayrink Verdun \(2021\)](https://arxiv.org/html/2608.23765#bib.bib33), and[Ghosh et al\. \(2024\)](https://arxiv.org/html/2608.23765#bib.bib11)primarily concern nonconvex rank surrogates, such as Schatten\-ppquasi\-norms or the log\-determinant, rather than the convex nuclear norm considered here\. In the present setting, the update rule[eq\.17](https://arxiv.org/html/2608.23765#S3.E17)remains computationally attractive, but it is not covered by our global convergence\-rate analysis\. That analysis exploits the specific coupling betweenεk\\varepsilon\_\{k\}and the nuclear\-norm tail provided by[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)\. Moreover, the experiments in[Section5\.5](https://arxiv.org/html/2608.23765#S5.SS5)suggest that the scaled nuclear\-norm\-tail update recovers the nuclear\-norm minimizer over a wider range of sampling factors in the considered experimental regime\. These theoretical and empirical considerations motivate our use of[eq\.13](https://arxiv.org/html/2608.23765#S3.E13); they do not imply that this rule is optimal for every problem instance or performance criterion\. A compromise between these two update rules is the Frobenius tail based rule

εk:=min⁡\(εk−1,∑i=r~\+1dσi2​\(𝐗\(k\)\)d\)\.\\varepsilon\_\{k\}:=\\min\\left\(\\varepsilon\_\{k\-1\},\\sqrt\{\\frac\{\\sum\_\{i=\\widetilde\{r\}\+1\}^\{d\}\\sigma\_\{i\}^\{2\}\(\\mathbf\{X\}^\{\(k\)\}\)\}\{d\}\}\\right\)\.\(18\)However, it is unclear whether either of the rules[eq\.17](https://arxiv.org/html/2608.23765#S3.E17)and[eq\.18](https://arxiv.org/html/2608.23765#S3.E18)can be used to prove global convergence rates for IRLS as established in[Section4](https://arxiv.org/html/2608.23765#S4)\.

Alternative smoothing parameter update rules include a linear decrease according to a fixed schedule\([Mohan and Fazel, 2012](https://arxiv.org/html/2608.23765#bib.bib48);[Peng et al\., 2023](https://arxiv.org/html/2608.23765#bib.bib21)\), which could also be considered within[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\. While simple in principle, tuning of the per\-iteration decrease factor is challenging in practice\. Moreover, it is unclear whether convergence guarantees can be obtained with such a rule\.

In this paper, we focus on the update rule[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)for our analyses\. We further provide numerical evidence in[Section5\.5](https://arxiv.org/html/2608.23765#S5.SS5)that suggests thatMatrixIRLSwith[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)is able to find the nuclear norm minimizer for a larger range of sampling factors than the other update rules, often within a wall\-clock time comparable to IRLS methods using other update rules\.

### 3\.4Optimality of Harmonic\-Mean Weight Operator

In the case of left\- and right\-sided weight operators[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)and[eq\.5](https://arxiv.org/html/2608.23765#S3.E5), it can be verified that the weight operator action reduces to left and right multiplication by the matrices𝐋𝐔−1\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}and𝐋𝐕−1\\mathbf\{L\}\_\{\\mathbf\{V\}\}^\{\-1\}such that

W𝐗,ε​\(𝐙\)=𝐋𝐔−1​𝐙andW𝐗,ε​\(𝐙\)=𝐙𝐋𝐕−1,W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}\\quad\\text\{ and \}\\quad W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{Z\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}^\{\-1\},\(19\)respectively, where𝐋𝐔:=𝐔𝐗​diag⁡\(λ1,…,λd1\)​𝐔𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{U\}\}:=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}and𝐋𝐕:=𝐕𝐗​diag⁡\(λ1,…,λd2\)​𝐕𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{V\}\}:=\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{2\}\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}are square matrices withλi:=max⁡\(σi,ε\)\\lambda\_\{i\}:=\\max\(\\sigma\_\{i\},\\varepsilon\)for eachi∈\[max⁡\(d1,d2\)\]i\\in\[\\max\(d\_\{1\},d\_\{2\}\)\]\. In fact, the left\-reweighted matrix𝐋𝐔−1​𝐙\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}was originally used by[Fornasier et al\. \(2011, equations \(2\.7\) and \(2\.10\)\)](https://arxiv.org/html/2608.23765#bib.bib54)without framing this as an action of a weight operator\. The contemporaneous work of[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48)used right\-reweighted matrices similar to𝐙𝐋𝐕−1\\mathbf\{Z\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}^\{\-1\}\. By transposition of the underlying matrices𝐗\\mathbf\{X\}and𝐙\\mathbf\{Z\}, which preserves the rank, the left\-sided and right\-sided weight operator notions recover each other, respectively\. The harmonic\-mean weight operatorW𝐗,εW\_\{\\mathbf\{X\},\\varepsilon\}using[eq\.6](https://arxiv.org/html/2608.23765#S3.E6), on the other hand,*cannot*be expressed as a simple left or right multiplication by a matrix and was proposed in[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)with the rationale of providing a reweighting that appropriately acts both on row and column spaces\. Acting on both row and column spaces in this balanced way also has desirable structural consequences, such as preserving symmetry in symmetric problems\.

We recall our observation[eq\.19](https://arxiv.org/html/2608.23765#S3.E19)that the left\-sided weight operator using[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)is only informed by the column space of the current iterate𝐗\\mathbf\{X\}, and that its action amounts to matrix multiplication of the variable𝐙\\mathbf\{Z\}from the left\. On the other hand, the right\-sided weight operator is only informed by the row space of the current iterate𝐗\\mathbf\{X\}\. This dichotomy between the row and column space information has undesirable consequences\. For example, when the left\-sided weights are applied to the transposed problem as opposed to the original problem[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)—i\.e\., when𝐗\\mathbf\{X\}is replaced by𝐗⊤\\mathbf\{X\}^\{\\top\}and the linear operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}is replaced by a compatible one𝒜tr:ℝd2×d1⟶ℝm\\mathcal\{A\}\_\{\\text\{tr\}\}:\\R^\{d\_\{2\}\\times d\_\{1\}\}\\longrightarrow\\R^\{m\}satisfying𝒜​\(𝐗\)ℓ=𝒜tr​\(𝐗⊤\)ℓ\\mathcal\{A\}\(\\mathbf\{X\}\)\_\{\\ell\}=\\mathcal\{A\}\_\{\\text\{tr\}\}\(\\mathbf\{X\}^\{\\top\}\)\_\{\\ell\}, the trajectory of the IRLS iterates\(𝐗\(k\)\)k≥0\(\\mathbf\{X\}^\{\(k\)\}\)\_\{k\\geq 0\}will be different from what it would be if the left\-sided weights were applied to the original problem\. The same is true for the right\-sided weights\.

The harmonic\-mean weight operator[eqs\.6](https://arxiv.org/html/2608.23765#S3.E6)and[7](https://arxiv.org/html/2608.23765#S3.E7)avoids this pitfall\. However, more generally, an entire family of weight operators that is not subject to this issue can be defined via the notion of*power means*\.

###### Definition 3\(Power means,[Bullen](https://arxiv.org/html/2608.23765#bib.bib42),[2003](https://arxiv.org/html/2608.23765#bib.bib42)\)\.

For−∞≤q≤∞\-\\infty\\leq q\\leq\\infty, the*qq\-power mean* ℳq​\(a,b\)\\mathcal\{M\}\_\{q\}\(a,b\)of two numbersa,b\>0a,b\>0is given by

ℳq​\(a,b\)=\{min⁡\(a,b\),if​q=−∞,\(aq\+bq2\)1q,if​q∈\(−∞,0\)∪\(0,∞\),a​b,if​q=0,max⁡\(a,b\),if​q=∞\.\\mathcal\{M\}\_\{q\}\(a,b\)=\\begin\{cases\}\\min\(a,b\),&\\text\{ if \}q=\-\\infty,\\\\ \\left\(\\frac\{a^\{q\}\+b^\{q\}\}\{2\}\\right\)^\{\\frac\{1\}\{q\}\},&\\text\{ if \}q\\in\(\-\\infty,0\)\\cup\(0,\\infty\),\\\\ \\sqrt\{ab\},&\\text\{ if \}q=0,\\\\ \\max\(a,b\),&\\text\{ if \}q=\\infty\.\\\\ \\end\{cases\}\(20\)

According to the previous definition, the arithmetic mean and the harmonic mean correspond to theqq\-power mean forq=1q=1andq=−1q=\-1, respectively\. By extension, power means of the two one\-sided weight variants induce an entire family of weight operators\. It is well\-known\([Bullen, 2003](https://arxiv.org/html/2608.23765#bib.bib42), Section III\.3, Theorem 1\)that fora,b≥0a,b\\geq 0and−∞≤q≤q′≤\+∞\-\\infty\\leq q\\leq q^\{\\prime\}\\leq\+\\infty,ℳq​\(a,b\)≤ℳq′​\(a,b\)\\mathcal\{M\}\_\{q\}\(a,b\)\\leq\\mathcal\{M\}\_\{q^\{\\prime\}\}\(a,b\)\. This implies a Loewner ordering for the family of power\-mean\-induced weight operators, which we formalize in[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)\.

###### Lemma 1\(Monotonicity of Power Mean Weight Operators\)\.

Fixε\>0\\varepsilon\>0and𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}with singular value decomposition𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\,\\diag\(\\bm\{\\sigma\}\)\\,\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}as in[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)\. Defineσ~i:=max⁡\(σi,ε\)−1\\widetilde\{\\sigma\}\_\{i\}:=\\max\(\\sigma\_\{i\},\\varepsilon\)^\{\-1\}fori∈\[d\]i\\in\[d\]andσ~i:=ε−1\\widetilde\{\\sigma\}\_\{i\}:=\\varepsilon^\{\-1\}fori∈\[D\]∖\[d\]i\\in\[D\]\\setminus\[d\]\. For eachq∈\[−∞,∞\]q\\in\[\-\\infty,\\infty\], let𝐇𝛔,ε\(q\)∈ℝd1×d2\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be given entrywise fori∈\[d1\]i\\in\[d\_\{1\}\]andj∈\[d2\]j\\in\[d\_\{2\}\]by

\(𝐇𝝈,ε\(q\)\)i​j:=ℳq​\(σ~i,σ~j\),\\bigl\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}:=\\mathcal\{M\}\_\{q\}\(\\widetilde\{\\sigma\}\_\{i\},\\widetilde\{\\sigma\}\_\{j\}\),\(21\)whereℳq​\(⋅,⋅\)\\mathcal\{M\}\_\{q\}\(\\cdot,\\cdot\)denotes theqq\-power mean from[eq\.20](https://arxiv.org/html/2608.23765#S3.E20)\. LetW𝐗,ε\(q\)W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q\)\}be the corresponding weight operator defined as in[eq\.7](https://arxiv.org/html/2608.23765#S3.E7), i\.e\., the operator mapping any𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}to

W𝐗,ε\(q\)​\(𝐙\):=𝐔𝐗​\(𝐇𝝈,ε\(q\)∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\)​𝐕𝐗⊤\.W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q\)\}\(\\mathbf\{Z\}\):=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\Bigl\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\Bigr\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\.\(22\)Then for any−∞≤q≤q′≤∞\-\\infty\\leq q\\leq q^\{\\prime\}\\leq\\infty, one has the Loewner orderingW𝐗,ε\(q\)⪯W𝐗,ε\(q′\)W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q\)\}\\preceq\\ W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q^\{\\prime\}\)\}and the associated quadratic modelsQε\(q\)Q\_\{\\varepsilon\}^\{\(q\)\}andQε\(q′\)Q\_\{\\varepsilon\}^\{\(q^\{\\prime\}\)\}\(cf\.[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)\) satisfy for all𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}that

Qε\(q\)​\(𝐙∣𝐗\)≤Qε\(q′\)​\(𝐙∣𝐗\)\.Q\_\{\\varepsilon\}^\{\(q\)\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\leq Q\_\{\\varepsilon\}^\{\(q^\{\\prime\}\)\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\.\(23\)

A proof of[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)is given in[SectionA\.2](https://arxiv.org/html/2608.23765#A1.SS2)\.

Recalling that the majorization property[eq\.16](https://arxiv.org/html/2608.23765#S3.E16)of the quadratic modelQε\(−1\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(\-1\)\}\(\\cdot\\mid\\mathbf\{X\}\)associated to the harmonic\-mean weight operator with respect to the smoothed nuclear norm objective was shown in[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1), we can infer from[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)that the majorization property holds for anyQε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)withq≥−1q\\geq\-1, which includes geometric and arithmetic mean weight operators\. On the other hand, one can ask whether the harmonic mean is at the boundary or whether otherqq\-means forq<−1q<\-1, possibly the most extreme min\-mean corresponding toq=−∞q=\-\\infty, also lead to majorization\. In[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), we establish that within the power\-mean weight operator family, the harmonic mean is indeed*optimal*: any power\-mean weight operator that is smaller in the Loewner ordering defines a quadratic model that*violates*majorization locally\.

###### Theorem 2\(Optimality of Harmonic\-Mean Weight Operator\)\.

Let𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}have SVD𝐗=𝐔​diag⁡\(𝛔\)​𝐕⊤\\mathbf\{X\}=\\mathbf\{U\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}^\{\\top\}such that there are two indicesi,j∈\[d\]i,j\\in\[d\]withi≠ji\\neq jwhose singular values satisfyσi≠σj\\sigma\_\{i\}\\neq\\sigma\_\{j\}andσi,σj\>ε\>0\\sigma\_\{i\},\\sigma\_\{j\}\>\\varepsilon\>0\. Fixq∈\[−∞,∞\]q\\in\[\-\\infty,\\infty\]and consider the weight operatorW𝐗,ε​\(⋅\):=W𝐗,ε\(q\)​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\):=W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q\)\}\(\\cdot\)of[eq\.22](https://arxiv.org/html/2608.23765#S3.E22)based on the weight core matrix𝐇𝛔,ε\(q\)\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}arising as entrywiseqq\-power mean of the left\- and right\-sided core matrices via[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)\. Then the associated quadratic modelQε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)\(cf\.[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)\) majorizes𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\), i\.e\.,Qε\(q\)​\(𝐙∣𝐗\)≥𝒥ε​\(𝐙\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)for each𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, if and only ifq≥−1q\\geq\-1\.

A full proof of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)is given in[SectionA\.2](https://arxiv.org/html/2608.23765#A1.SS2)\. We note that one direction of the necessary and sufficient conditionq≥−1q\\geq\-1is a direct consequence of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)and the Loewner ordering of[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)\. The other direction involves an explicit necessary condition that we obtain from an expression for the Hessian of spectral functions, applied to a perturbation of𝐗\\mathbf\{X\}that is confined to a two\-dimensional singular block\.

[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)states that the harmonic mean provides the tightest possible majorization within the class of power mean weight operators\. This tightness provides one ingredient towards explaining the superior performance ofMatrixIRLSwith harmonic\-mean weight operators, which we explore in[Section5](https://arxiv.org/html/2608.23765#S5), compared to the one\-sided or arithmetic mean IRLS variants—MatrixIRLSoptimizes the quadratic model with the pointwise smallest gap to the smoothed surrogate objective among all majorizing power mean variants, which is the mechanism we expect to drive its faster per\-iteration progress\. A provable consequence of this tightness is an improved, dimension\-free local linear convergence rate ofMatrixIRLS, which we show with[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)in[Section4\.3](https://arxiv.org/html/2608.23765#S4.SS3)\. Numerical experiments of[Sections5\.2](https://arxiv.org/html/2608.23765#S5.SS2)and[5\.3](https://arxiv.org/html/2608.23765#S5.SS3)indicate that the improved local linear rate cannot be observed for IRLS variants using larger power means such as the arithmetic mean variant, nor for the one\-sided IRLS variants, for which we provide a counterexample in[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)\.

[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)were the first to propose a harmonic\-mean weight operator and observed improved local convergence, but focusing on nonconvex Schatten\-ppquasi\-norm minimization for0<p<10<p<1\. However, we would like to point out that it follows from a generalization of the second\-order necessary argument outlined in[SectionA\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2), which is part of the proof of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), that the quadratic model associated to the harmonic\-mean weight operator is in fact*not*a valid global majorizer in the nonconvex case ofp<1p<1, which lies outside the scope of the present paper\.

## 4Linear Convergence Rates of IRLS for Nuclear Norm Minimization

Building on the majorization results of[Section3\.2](https://arxiv.org/html/2608.23765#S3.SS2), we now establish a detailed convergence analysis ofMatrixIRLSfor nuclear norm minimization \([Algorithm1](https://arxiv.org/html/2608.23765#alg1)\) and distinguish different linear convergence rates that can be shown for IRLS depending on the choice of the weight operator\.[Theorems3](https://arxiv.org/html/2608.23765#Thmtheorem3)and[4](https://arxiv.org/html/2608.23765#Thmtheorem4)establish the first global linear convergence rates for IRLS methods in this setting \([Section4\.2](https://arxiv.org/html/2608.23765#S4.SS2)\)\. For harmonic\-meanMatrixIRLS,[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)further gives a dimension\-free local linear rate, which, as we show via[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6), cannot be achieved for IRLS using one\-sided weight operators \([Section4\.3](https://arxiv.org/html/2608.23765#S4.SS3)\)\.

### 4\.1Preliminaries

Our results are based on a well\-studied regularity assumption on the measurement operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{m\}\. In particular, we make the assumption that the measurement operator𝒜\\mathcal\{A\}satisfies the NSP\([Recht et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib45);[Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Yi and Xu, 2020](https://arxiv.org/html/2608.23765#bib.bib46)\), which is a sufficient \(and in a very related form, also necessary\) condition for successful recovery of low\-rank matrices via nuclear norm minimization\.

###### Definition 4\(Null Space Property\)\.

A linear operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{m\}is said to satisfy the*NSP*of orderr∈ℕr\\in\\N,1≤r<d1\\leq r<dwith constant0<ηr<10<\\eta\_\{r\}<1if

∑i=1rσi​\(𝐍\)≤ηr​∑i=r\+1dσi​\(𝐍\)\\sum\_\{i=1\}^\{r\}\\sigma\_\{i\}\(\\mathbf\{N\}\)\\leq\\eta\_\{r\}\\sum\_\{i=r\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{N\}\)\(24\)for all𝐍∈ker⁡\(𝒜\)\\mathbf\{N\}\\in\\ker\(\\mathcal\{A\}\)\. Here,σi​\(𝐍\)\\sigma\_\{i\}\(\\mathbf\{N\}\)denotes theii\-th largest singular value of the matrix𝐍\\mathbf\{N\}\.

The NSP holds for several classes of linear measurement operators𝒜\\mathcal\{A\}\. For example, if the operator𝒜\\mathcal\{A\}satisfies the restricted isometry property\([Recht et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib65)\)of sufficient order, then the NSP holds\(see, e\.g\.,[Foucart and Rauhut, 2013](https://arxiv.org/html/2608.23765#bib.bib55), Exercise 6\.24\)\. In particular, the NSP of orderrrfor some constant0<ηr<10<\\eta\_\{r\}<1holds, e\.g\., with high probability, if the measurement matrices are random rank\-one matrices with Gaussian factors\([Cai and Zhang, 2015](https://arxiv.org/html/2608.23765#bib.bib83)\)and if the number of measurements satisfiesm=Ω⁡\(r⁡\(d1\+d2\)\)m=\\Omega\(r\(d\_\{1\}\+d\_\{2\}\)\), see also[Kueng et al\. \(2017\)](https://arxiv.org/html/2608.23765#bib.bib80);[Kabanava et al\. \(2016\)](https://arxiv.org/html/2608.23765#bib.bib81)\.111In fact,[Cai and Zhang \(2015\)](https://arxiv.org/html/2608.23765#bib.bib83)shows under the said assumption that a variant of[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)holds whereηr\\eta\_\{r\}in[eq\.24](https://arxiv.org/html/2608.23765#S4.E24)is11and the inequality is strict\. A standard modification yields[eq\.24](https://arxiv.org/html/2608.23765#S4.E24)for some0<ηr<10<\\eta\_\{r\}<1\.

Beyond guaranteeing that𝐗⋆\\mathbf\{X\}\_\{\\star\}is the unique nuclear\-norm minimizer compatible with𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\),[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)yields a linear error bound on the feasible set\. If𝐗⋆\\mathbf\{X\}\_\{\\star\}has rankrr, then every𝐗\\mathbf\{X\}with𝒜⁡\(𝐗\)=𝒜⁡\(𝐗⋆\)\\mathcal\{A\}\(\\mathbf\{X\}\)=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\)satisfies \(see[Lemma10](https://arxiv.org/html/2608.23765#Thmlemma10)for a more general version\)

‖𝐗−𝐗⋆‖∗≤1\+ηr1−ηr​\(‖𝐗‖∗−‖𝐗⋆‖∗\)\.\\,\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\.\(25\)In the language of optimization, this is a*sharpness*condition for the nuclear norm on the affine constraint\([Burke and Ferris, 1993](https://arxiv.org/html/2608.23765#bib.bib41);[Roulet et al\., 2020](https://arxiv.org/html/2608.23765#bib.bib20)\): the objective gap grows linearly with the nuclear\-norm distance to𝐗⋆\\mathbf\{X\}\_\{\\star\}\. Linear error bounds of this type are the standard landscape hypothesis for linear convergence of first\-order methods\([Goffin, 1977](https://arxiv.org/html/2608.23765#bib.bib39)\)\. The same phenomenon originates in sparse recovery, where the analogousℓ1\\ell\_\{1\}\-NSP characterizes exact recovery by basis pursuit\([Cohen et al\., 2009](https://arxiv.org/html/2608.23765#bib.bib34);[Foucart and Rauhut, 2013](https://arxiv.org/html/2608.23765#bib.bib55)\)and is the assumption behind the first global linear rates for IRLS forℓ1\\ell\_\{1\}\-minimization\([Kümmerle et al\., 2021](https://arxiv.org/html/2608.23765#bib.bib37)\)\.

For IRLS, however, sharpness alone does not produce a rate: In contrast to known sharpness\-based arguments,[Algorithm1](https://arxiv.org/html/2608.23765#alg1)takes no step size, and[eq\.25](https://arxiv.org/html/2608.23765#S4.E25)is oblivious to the choice of the weight operatorW\(k\)W^\{\(k\)\}in[eq\.2](https://arxiv.org/html/2608.23765#S1.E2)\. Controlling the nuclear norm gap via the majorization property together with appropriate, weight\-operator dependent upper bounds on the quadratic forms implied byW\(k\)W^\{\(k\)\}, which depend on their spectral structure, is key for obtaining linear convergence rates for IRLS algorithms like[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\.

### 4\.2Global Linear Convergence of IRLS for Nuclear Norm Minimization

We show that the iterates\(𝐗\(k\)\)k≥1\(\\mathbf\{X\}^\{\(k\)\}\)\_\{k\\geq 1\}of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)converge globally to a low\-rank ground truth matrix𝐗⋆\\mathbf\{X\}\_\{\\star\}with a linear rate whenever the measurement operator𝒜\\mathcal\{A\}satisfies the NSP, largely independently of the particular weight operator choice relevant for updatingW\(k\+1\):=W𝐗\(k\),εkW^\{\(k\+1\)\}:=W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\. We present in[Section4\.2\.1](https://arxiv.org/html/2608.23765#S4.SS2.SSS1)a result for the recovery of exactly low\-rank matrices𝐗⋆\\mathbf\{X\}\_\{\\star\}, before generalizing this in[Section4\.2\.2](https://arxiv.org/html/2608.23765#S4.SS2.SSS2)to approximately low\-rank ground truths\.

#### 4\.2\.1Rank\-rrGround Truth𝐗⋆\\mathbf\{X\}\_\{\\star\}

Specific to linear operators𝒜\\mathcal\{A\}satisfying the NSP of orderrrof[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)with constantηr\>0\\eta\_\{r\}\>0, we define the constants

Aηr=\(32\+ηr\)​\(1\+ηr\)1−ηrandCηr=\(34−2​ηr1\+ηr\)23\+2​ηr\.A\_\{\\eta\_\{r\}\}=\\frac\{\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\+\\eta\_\{r\}\\right\)\}\{1\-\\eta\_\{r\}\}\\quad\\text\{and\}\\quad C\_\{\\eta\_\{r\}\}=\\frac\{\\left\(\\frac\{3\}\{4\}\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{3\+2\\eta\_\{r\}\}\.\(26\)Unlike the local result of[Section4\.3](https://arxiv.org/html/2608.23765#S4.SS3), the global results of this section are not tied to one specific weighting scheme, but hold for an entire family of weight operators, which we fix first for later reference\. Define the the weight operator\-dependent constant

cq:=\{1,if harmonic, one\-sided or power mean weight operator with​q∈\[−∞,1\],22−1/q−1,if power mean weight operator with​q∈\(1,∞\),3,if power mean weight operator with​q=∞\.c\_\{q\}:=\\begin\{cases\}1,&\\text\{ if harmonic, one\-sided or power mean weight operator with \}q\\in\[\-\\infty,1\],\\\\ 2^\{2\-1/q\}\-1,&\\text\{ if power mean weight operator with \}q\\in\(1,\\infty\),\\\\ 3,&\\text\{ if power mean weight operator with \}q=\\infty\.\\end\{cases\}\(27\)
###### Definition 5\(Admissible Weight Operators\)\.

We call the weight operatorsW𝐗\(k\),εkW\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}of[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)used within[Algorithm1](https://arxiv.org/html/2608.23765#alg1)*admissible*with constantcqc\_\{q\}if, at each iterationkk, their weight operator core matrix𝐇𝛔,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}of[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)is \(i\) the*harmonic\-mean*core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6), \(ii\) a*one\-sided*core matrix[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)or[eq\.5](https://arxiv.org/html/2608.23765#S3.E5), or \(iii\) a*qq\-power mean*core matrix[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq∈\[−1,∞\]q\\in\[\-1,\\infty\], wherecqc\_\{q\}is the associated constant of[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)\.

Case \(i\) is the special caseq=−1q=\-1of case \(iii\), whereas the one\-sided operators of case \(ii\) are not power means\. In all cases,cq∈\[1,3\]c\_\{q\}\\in\[1,3\], andcq=1c\_\{q\}=1for the most common choices, cf\. the discussion after[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)\. Since[Algorithm1](https://arxiv.org/html/2608.23765#alg1)is stated with harmonic\-mean weights, a choice other than \(i\) is to be understood as the corresponding variant of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)in which only the weight operator update is replaced accordingly, while all other steps remain unchanged\.

###### Theorem 3\(Global Linear Convergence Rate, Low\-Rank Ground Truths\)\.

Let 𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}be a matrix of rankrr\. Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP[eq\.24](https://arxiv.org/html/2608.23765#S4.E24)of orderrrwith constantηr<3/5\\eta\_\{r\}<3/5\. Let\(𝐗\(k\)\)k≥0\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\_\{k\\geq 0\}and\(εk\)k≥0\\left\(\\varepsilon\_\{k\}\\right\)\_\{k\\geq 0\}be theMatrixIRLSiterates obtained from[Algorithm1](https://arxiv.org/html/2608.23765#alg1)with measurements𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\), rank estimater~=r\\widetilde\{r\}=rand initializing weight operatorW\(0\)W^\{\(0\)\}, and assume that the weight operatorsW𝐗\(k\),εkW\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}are admissible in the sense of[Definition5](https://arxiv.org/html/2608.23765#Thmdefinition5), with associated constantcqc\_\{q\}of[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)\.

Then it holds for all iterationsk∈ℕ0k\\in\\mathbb\{N\}\_\{0\}carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1)that

𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗≤\(1−Cηrcq​η1​d\)k​\(𝒥ε0​\(𝐗\(0\)\)−‖𝐗⋆‖∗\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{k\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{0\}\}\\left\(\\mathbf\{X\}^\{\(0\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\(28\)as well as

‖𝐗\(k\)−𝐗⋆‖∗≤Aηr​\(1−Cηrcq​η1​d\)k​‖𝐗\(0\)−𝐗⋆‖∗\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{k\}\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\(29\)whereAηrA\_\{\\eta\_\{r\}\}andCηrC\_\{\\eta\_\{r\}\}are as in[eq\.26](https://arxiv.org/html/2608.23765#S4.E26)and0<η1≤ηr0<\\eta\_\{1\}\\leq\\eta\_\{r\}denotes the NSP constant of𝒜\\mathcal\{A\}of order11\.

The proof of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)can be found in[SectionsA\.3\.2](https://arxiv.org/html/2608.23765#A1.SS3.SSS2)and[A\.3\.3](https://arxiv.org/html/2608.23765#A1.SS3.SSS3)\. Specifically, it is shown that the smoothed objective gap𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}convergesQQ\-linearly to zero, and that the iterate norm difference to𝐗⋆\\mathbf\{X\}\_\{\\star\}convergesRR\-linearly\. Note that this theorem implies that it holds that‖𝐗\(k\)−𝐗⋆‖∗≤δ\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\deltaafterOηr​\(1\+cq​η1​d​log\+⁡\(‖𝐗\(0\)−𝐗⋆‖∗δ\)\)O\_\{\\eta\_\{r\}\}\\left\(1\+c\_\{q\}\\eta\_\{1\}d\\log\_\{\+\}\\left\(\\frac\{\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\delta\}\\right\)\\right\)iterations; this corresponds to a number of iterations to reach a fixed accuracyδ\\deltathat depends linearly on the dimensiondd, across all considered weight operator choices\. A slight dependence on the weight operator choice can be inferred fromcqc\_\{q\}, which satisfiescq=1c\_\{q\}=1for the most common weight operator choices such as harmonic mean or one\-sided whereascq→3c\_\{q\}\\to 3as theqq\-parameter of a power mean weight operator increases withq→∞q\\to\\infty\. The NSP constantηr\\eta\_\{r\}\(as well asη1\\eta\_\{1\}\) can be generally considered as an \(undetermined\) dimension\-free constant, in which caseCηrC\_\{\\eta\_\{r\}\}is also dimension\-free: If, for example,ηr=1/10\\eta\_\{r\}=1/10, the constantCηrC\_\{\\eta\_\{r\}\}satisfiesCηr≈0\.101C\_\{\\eta\_\{r\}\}\\approx 0\.101\. The constantAηrA\_\{\\eta\_\{r\}\}is furthermore dimension\-free and ranges between1\.5≤Aηr<8\.41\.5\\leq A\_\{\\eta\_\{r\}\}<8\.4, depending on the value ofηr\\eta\_\{r\}\.

In[Section5](https://arxiv.org/html/2608.23765#S5), we present numerical experiments that corroborate that linear convergence rates of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)can be observed in practice, across all considered weight operator variants, and that a linear convergence rate with factor of order1−cd1\-\\frac\{c\}\{d\}could indeed describe the correct worst\-case behavior of the IRLS algorithm class for nuclear norm minimization; this is done by constructing an adversarial initialization \([Section5\.4](https://arxiv.org/html/2608.23765#S5.SS4)\) to define the initial weight operatorW\(0\)W^\{\(0\)\}of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\. On the other hand, we see in[Sections5\.2](https://arxiv.org/html/2608.23765#S5.SS2)and[5\.3](https://arxiv.org/html/2608.23765#S5.SS3)that for harmonic\-mean weight operators, the generic linear rate of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)is*dimension\-independent*in the sense that it does*not*depend on the dimensiondd\. We refer to[Section4\.3](https://arxiv.org/html/2608.23765#S4.SS3)for a local linear convergence result that better captures this generic behavior\.

#### 4\.2\.2Approximately Low\-Rank Ground Truth

We now generalize the result of[Section4\.2\.1](https://arxiv.org/html/2608.23765#S4.SS2.SSS1)to the setting in which the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}is only*approximately*low\-rank\. Define the𝒜\\mathcal\{A\}\-dependent constants

Bηr:=\(72\+ηr\)​\(1\+ηr\)1−ηrandC~ηr:=\(12−2​ηr1\+ηr\)23\+2​ηrB\_\{\\eta\_\{r\}\}:=\\frac\{\\left\(\\frac\{7\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\+\\eta\_\{r\}\\right\)\}\{1\-\\eta\_\{r\}\}\\quad\\text\{ and \}\\quad\\widetilde\{C\}\_\{\\eta\_\{r\}\}:=\\frac\{\\left\(\\frac\{1\}\{2\}\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{3\+2\\eta\_\{r\}\}\(30\)given its NSP constantηr\\eta\_\{r\}of orderrr\.

###### Theorem 4\(Linear Decay to Approximation Floor\)\.

Let𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be arbitrary\. Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP of orderrrwith constantηr<1/3\\eta\_\{r\}<1/3\. If\(𝐗\(k\)\)k≥0\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\_\{k\\geq 0\}and\(εk\)k≥0\\left\(\\varepsilon\_\{k\}\\right\)\_\{k\\geq 0\}are iterates and smoothing parameters ofMatrixIRLSwith input𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\), arbitrary initial weight operatorW\(0\)W^\{\(0\)\}, rank estimater~=r\\widetilde\{r\}=rand weight operatorsW𝐗\(k\),εkW\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}that are admissible in the sense of[Definition5](https://arxiv.org/html/2608.23765#Thmdefinition5), set

k^:=min⁡\{k∈ℕ0:βr​\(𝐗⋆\)∗\>19​‖𝐗\(k\)−𝐗⋆‖∗\},\\hat\{k\}:=\\min\\left\\\{k\\in\\mathbb\{N\}\_\{0\}:\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\>\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\\\},\(31\)with the conventionmin⁡∅=∞\\min\\varnothing=\\infty\(which holds wheneverβr​\(𝐗⋆\)∗=0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0\)\. Then the following two statements hold, in each case for the iterations carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\.

1. 1\.Forcqc\_\{q\}as in[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)andAηrA\_\{\\eta\_\{r\}\},BηrB\_\{\\eta\_\{r\}\}andC~ηr\\widetilde\{C\}\_\{\\eta\_\{r\}\}as in[eq\.26](https://arxiv.org/html/2608.23765#S4.E26)and[eq\.30](https://arxiv.org/html/2608.23765#S4.E30), it holds for allk∈ℕ0k\\in\\mathbb\{N\}\_\{0\}that 𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​\(𝒥ε0​\(𝐗\(0\)\)−‖𝐗⋆‖∗\),and\\displaystyle\\leq\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{0\}\}\(\\mathbf\{X\}^\{\(0\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\),\\text\{ and\}\(32\)‖𝐗\(k\)−𝐗⋆‖∗\\displaystyle\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤Aηr​\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​‖𝐗\(0\)−𝐗⋆‖∗\+Bηr​βr​\(𝐗⋆\)∗,\\displaystyle\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+B\_\{\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},\(33\)where0<η1≤ηr0<\\eta\_\{1\}\\leq\\eta\_\{r\}denotes the order\-one NSP constant of𝒜\\mathcal\{A\}\.
2. 2\.For allk≥k^k\\geq\\hat\{k\}withk^\\hat\{k\}as in[eq\.31](https://arxiv.org/html/2608.23765#S4.E31), it holds that ‖𝐗\(k\)−𝐗⋆‖∗≤41​βr​\(𝐗⋆\)∗\.\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq 41\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\.\(34\)

The proof of[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)is deferred to[SectionA\.3\.4](https://arxiv.org/html/2608.23765#A1.SS3.SSS4), where we also show that ifβr​\(𝐗⋆\)∗\>0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\>0, then

k^≤1\+cq​η1​dC~ηr​log\+⁡\(Aηr9−Bηr⋅‖𝐗\(0\)−𝐗⋆‖∗βr​\(𝐗⋆\)∗\),\\hat\{k\}\\leq 1\+\\frac\{c\_\{q\}\\eta\_\{1\}d\}\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\\log\_\{\+\}\\\!\\left\(\\frac\{A\_\{\\eta\_\{r\}\}\}\{9\-B\_\{\\eta\_\{r\}\}\}\\cdot\\frac\{\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\}\\right\),so inequality[eq\.34](https://arxiv.org/html/2608.23765#S4.E34)holds after at mostOηr​\(cq​η1​d​log\+⁡\(‖𝐗\(0\)−𝐗⋆‖∗/βr​\(𝐗⋆\)∗\)\)O\_\{\\eta\_\{r\}\}\\big\(c\_\{q\}\\eta\_\{1\}d\\log\_\{\+\}\\left\(\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}/\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\right\)\\big\)iterations, wherelog\+⁡\(x\)=max⁡\(0,log⁡\(x\)\)\\log\_\{\+\}\(x\)=\\max\(0,\\log\(x\)\)\. Ifβr​\(𝐗⋆\)∗=0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0, thenk^=∞\\hat\{k\}=\\inftyand[eqs\.32](https://arxiv.org/html/2608.23765#S4.E32)and[33](https://arxiv.org/html/2608.23765#S4.E33)hold withmin⁡\(k,k^\)=k\\min\(k,\\hat\{k\}\)=kfor allkk\.

### 4\.3Fast Local Linear Rate ofMatrixIRLSwith Harmonic\-Mean Weights

The global results of[Section4\.2](https://arxiv.org/html/2608.23765#S4.SS2)are largely indifferent to the underlying weighting scheme:[Theorems3](https://arxiv.org/html/2608.23765#Thmtheorem3)and[4](https://arxiv.org/html/2608.23765#Thmtheorem4)hold uniformly across the admissible weight operators of[Definition5](https://arxiv.org/html/2608.23765#Thmdefinition5), whose choice enters these statements only through the constantcq∈\[1,3\]c\_\{q\}\\in\[1,3\]of[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)\. In particular, no admissible choice is singled out by the global rates, and none of them escapes the factor1/d1/din the linear convergence factor\.

We now consider the scenario that an iterate ofMatrixIRLShas already entered a specific, local neighborhood of the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}, in which the picture is different, as the choice of the weight operator becomes decisive\. In this case, we show that, specifically for the algorithm variant[Algorithm1](https://arxiv.org/html/2608.23765#alg1)that uses harmonic\-mean weight operators, the iterates converge locally with a fast linear rate, which is independent of the ambient dimensiondd\. What drives this improvement is a sharp local upper bound on the weighted quadratic form of the quadratic model function, which is available for harmonic\-mean weight operators but*provably fails*for one\-sided ones\. We make this mechanism precise in[Corollary1](https://arxiv.org/html/2608.23765#Thmcorollary1)and[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)at the end of this section, and delineate in[Remark3](https://arxiv.org/html/2608.23765#Thmremark3)the weight operators for which a dimension\-free local rate remains available\.

###### Theorem 5\(Dimension\-Free Fast Linear Rate ofMatrixIRLS\)\.

Let𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}be a matrix of rankrr\. Assume that the measurement operator𝒜\\mathcal\{A\}satisfies the NSP of[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)of orderrrwith constantηr<3/5\\eta\_\{r\}<3/5and of order11with constantη1≤ηr\\eta\_\{1\}\\leq\\eta\_\{r\}\. Let\(𝐗\(k\)\)k≥0\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\_\{k\\geq 0\}and\(εk\)k≥0\\left\(\\varepsilon\_\{k\}\\right\)\_\{k\\geq 0\}be the iterates and smoothing parameters ofMatrixIRLSwith input𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\), arbitrary initial weight operatorW\(0\)W^\{\(0\)\}andr~=r\\widetilde\{r\}=r\. LetAηrA\_\{\\eta\_\{r\}\}be defined as in[eq\.26](https://arxiv.org/html/2608.23765#S4.E26)\. Assume thatMatrixIRLS’s weight operatorsW𝐗\(k\),εkW\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}are defined based on harmonic\-mean core matrices[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)and that there is a natural numberk~\\tilde\{k\}such that

‖𝐗\(k~\)−𝐗⋆‖∗≤σr​\(𝐗⋆\)9​max⁡\(η1​d,2\)\.\\left\\lVert\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{9\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\}\.\(35\)Then for all iterationsk≥k~k\\geq\\tilde\{k\}carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1)it holds that

𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗≤\(1−cηr\)k−k~​\(𝒥εk~​\(𝐗\(k~\)\)−‖𝐗⋆‖∗\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\-\\tilde\{k\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{\\tilde\{k\}\}\}\\left\(\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\(36\)as well as

‖𝐗\(k\)−𝐗⋆‖∗≤Aηr​\(1−cηr\)k−k~​‖𝐗\(k~\)−𝐗⋆‖∗\.\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\-\\tilde\{k\}\}\\left\\lVert\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(37\)Here, the constantcηrc\_\{\\eta\_\{r\}\}depends only on the NSP constantηr\\eta\_\{r\}and is given by

cηr=\(3−5​ηr\)2​\(1−ηr\)38​\(1\+ηr\)2​\(3\+2​ηr\)​\[8​\(1−ηr\)3\+\(3\+ηr\)2​\(1\+ηr\)​\(3\+2​ηr\)\]\.c\_\{\\eta\_\{r\}\}=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\(1\-\\eta\_\{r\}\)^\{3\}\}\{8\(1\+\\eta\_\{r\}\)^\{2\}\(3\+2\\eta\_\{r\}\)\\Bigl\[8\(1\-\\eta\_\{r\}\)^\{3\}\+\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\(3\+2\\eta\_\{r\}\)\\Bigr\]\}\.\(38\)

The proof of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)can be found in[SectionA\.4](https://arxiv.org/html/2608.23765#A1.SS4)\.

To parse the definition ofcηrc\_\{\\eta\_\{r\}\}in[eq\.38](https://arxiv.org/html/2608.23765#S4.E38), we consider again measurement operators with an orderrr\-NSP constant ofηr=1/10\\eta\_\{r\}=1/10\. In this case, we havecηr≈0\.0037c\_\{\\eta\_\{r\}\}\\approx 0\.0037\. Notably, this means that the decrease factor1−cηr1\-c\_\{\\eta\_\{r\}\}of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)*does not depend ondd*, unlike the decrease factor1−Cηrη1​d1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{\\eta\_\{1\}d\}of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)\. This means that for large dimensions ofd≫1d\\gg 1, the analysis of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)leads to a sharper bound than the global rates established above\. Remaining in theηr=1/10\\eta\_\{r\}=1/10example, assuming additionallyη1=ηr\\eta\_\{1\}=\\eta\_\{r\}, which is rather pessimistic,[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)leads to a faster rate than[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)ifd\>273d\>273\. We note that we did not attempt to optimize theηr\\eta\_\{r\}\-dependence ofcηrc\_\{\\eta\_\{r\}\}in our proof, which we leave for future work\.

By combining[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)with[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3), we obtain that‖𝐗\(k\)−𝐗⋆‖∗≤δ\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\deltaafter

Oηr​\(η1​d​log\+⁡\(max⁡\(η1​d,2\)​‖𝐗\(0\)−𝐗⋆‖∗σr​\(𝐗⋆\)\)\+log\+⁡\(σr​\(𝐗⋆\)max⁡\(η1​d,2\)​δ\)\)O\_\{\\eta\_\{r\}\}\\left\(\\eta\_\{1\}d\\log\_\{\+\}\\left\(\\frac\{\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\\right\)\+\\log\_\{\+\}\\left\(\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\\delta\}\\right\)\\right\)iterations, whereOηr​\(⋅\)O\_\{\\eta\_\{r\}\}\(\\cdot\)represents theηr\\eta\_\{r\}\-dependentOO\-notation andlog\+⁡\(x\)=max⁡\(0,log⁡\(x\)\)\\log\_\{\+\}\(x\)=\\max\(0,\\log\(x\)\)\.

We now make the mechanism behind[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)precise\. Its proof rests on a more precise estimate for the quadratic term⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}of the implied quadratic model functionQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)of[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)that becomes available in the case that𝐗\\mathbf\{X\}is close enough to the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}, and which is given by[Lemma17](https://arxiv.org/html/2608.23765#Thmlemma17)in[SectionA\.4](https://arxiv.org/html/2608.23765#A1.SS4)\. We state a simplified corollary of it below, which fixes the constantηr=1/10\\eta\_\{r\}=1/10for legibility\.

###### Corollary 1\(Sharp Local Upper Bound on Weighted Quadratic Form\)\.

Assume that the linear measurement operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\rightarrow\\mathbb\{R\}^\{m\}satisfies the NSP of orderrrwith constantηr=1/10\\eta\_\{r\}=1/10, that𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is of rankrrand that𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}satisfies𝐗−𝐗⋆∈ker⁡\(𝒜\)\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\in\\ker\(\\mathcal\{A\}\)\. Ifε=βr​\(𝐗\)∗d\>0\\varepsilon=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}\>0,‖𝐗−𝐗⋆‖∗≤σr​\(𝐗⋆\)max⁡\(η1​d,2\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\}\{\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\}, where0<η1≤ηr0<\\eta\_\{1\}\\leq\\eta\_\{r\}denotes again the order\-one NSP constant of𝒜\\mathcal\{A\}, andW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is the harmonic\-mean weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7), then

⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F≤16​‖𝐗−𝐗⋆‖∗\.\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\leq 16\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(39\)

[Corollary1](https://arxiv.org/html/2608.23765#Thmcorollary1)follows immediately from[Lemma17](https://arxiv.org/html/2608.23765#Thmlemma17)withq=−1q=\-1andϑ=1\\vartheta=1, usingD1/10=31/9D\_\{1/10\}=31/9\.

##### Impossibility ofdd\-Independent Fast Local Rate for One\-Sided Weight Operators\.

The dimension\-independent upper bound[eq\.39](https://arxiv.org/html/2608.23765#S4.E39)on⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}proportional to‖𝐗−𝐗⋆‖∗\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}is*specific to the harmonic\-mean weight operator*\(and toqq\-power mean weight operators withq∈\[−1,0\)q\\in\[\-1,0\), cf\.[Remark3](https://arxiv.org/html/2608.23765#Thmremark3)\): it*cannot hold*for one\-sided weight operators \(such as those with left\-sided or right\-sided core matrices[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5)\), even within smaller local neighborhoods of𝐗⋆\\mathbf\{X\}\_\{\\star\}than the one defined by[eq\.35](https://arxiv.org/html/2608.23765#S4.E35)\.[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)below establishes this through a counterexample for which add\-dependent*lower bound*on⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}holds if a left\-sided weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)with core matrix[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)is used; the details of this construction are provided in[SectionA\.4\.3](https://arxiv.org/html/2608.23765#A1.SS4.SSS3)\.

###### Theorem 6\(No Dimension\-Free Fast Local Rate for One\-Sided IRLS\)\.

For any r,d∈ℕr,d\\in\\mathbb\{N\}withd≥220​rd\\geq 220r, there exists a measurement operator𝒜:ℝd×d→ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\\times d\}\\rightarrow\\R^\{m\}satisfying the NSP of orderrrwith constantηr=1/10\\eta\_\{r\}=1/10and of order11withη1=1/\(11​r−1\)\\eta\_\{1\}=1/\(11r\-1\), a rank\-rrmatrix𝐗⋆∈ℝd×d\\mathbf\{X\}\_\{\\star\}\\in\\mathbb\{R\}^\{d\\times d\}and a matrix𝐗∈ℝd×d\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\\times d\}such that𝐗−𝐗⋆∈ker⁡\(𝒜\)\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\in\\ker\(\\mathcal\{A\}\)and‖𝐗−𝐗⋆‖∗≤σr​\(𝐗⋆\)max⁡\(η1​d,2\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\}\{\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\}, but also

⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F≥d220​r​‖𝐗−𝐗⋆‖∗,\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\geq\\frac\{d\}\{220r\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\},ifW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is the weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)with left\-sided core matrix[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)and the smoothing parameterε\\varepsilonsatisfiesε=βr​\(𝐗\)∗d\\varepsilon=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}\.

Furthermore, in this case, the subsequent IRLS iterate

𝐗\+=arg​min𝐙∈ℝd×d⁡⟨𝐙,W𝐗,ε​\(𝐙\)⟩F​subject to​𝒜​\(𝐙\)=𝒜⁡\(𝐗⋆\)\\mathbf\{X\}^\{\+\}=\\argmin\_\{\\mathbf\{Z\}\\in\\mathbb\{R\}^\{d\\times d\}\}\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\text\{ subject to \}\\mathcal\{A\}\(\\mathbf\{Z\}\)=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\)satisfies

‖𝐗\+−𝐗⋆‖∗≥\(1−220​rd\)​‖𝐗−𝐗⋆‖∗\.\\left\\lVert\\mathbf\{X\}^\{\+\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\geq\\left\(1\-\\frac\{220r\}\{d\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.

An analogous result to[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)can be shown for right\-sided weight operators\. The different local linear convergence rates of IRLS using harmonic\-mean weight operators on the one hand, and of IRLS using one\-sided, or large\-qqpower\-mean weight operators on the other hand, can also be observed in practice; for generic examples, the dimension\-free error decay of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)’s iterates can often be observed beyond the limited\-size local neighborhood defined by[eq\.35](https://arxiv.org/html/2608.23765#S4.E35)\. We elaborate on this fact below in[Sections5\.2](https://arxiv.org/html/2608.23765#S5.SS2)and[5\.3](https://arxiv.org/html/2608.23765#S5.SS3)\.

## 5Numerical Experiments

In this section, we explore the empirical behavior of IRLS for nuclear norm minimization to solve low\-rank matrix sensing problems\. We focus on the empirical speed of convergence of iterates of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)given different choices of the weight operator core matrix \(see[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)\), taking also into account the role of algorithmic initialization and the smoothing parameter update rule[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)\.

### 5\.1Setup

For all experiments, we consider low\-rank matrix recovery problems of*matrix sensing*type, where the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}of \([1](https://arxiv.org/html/2608.23765#S1.E1)\) consists of noiseless,*random Gaussian rank\-one measurements*\. In particular, theℓ\\ell\-th coordinate of𝒜⁡\(𝐗\)\\mathcal\{A\}\(\\mathbf\{X\}\)given the input matrix𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is𝒜​\(𝐗\)ℓ=⟨aℓ​bℓ⊤,𝐗⟩\\mathcal\{A\}\(\\mathbf\{X\}\)\_\{\\ell\}=\\langle a\_\{\\ell\}b\_\{\\ell\}^\{\\top\},\\mathbf\{X\}\\ranglefor eachℓ=1,…,m\\ell=1,\\ldots,m, whereaℓ,bℓa\_\{\\ell\},b\_\{\\ell\}are independent random vectors of lengthd1d\_\{1\}andd2d\_\{2\}, respectively, with independent, standard normal entries\. Such measurements correspond to a simplified, real\-valued variant of the measurement setting available in blind deconvolution problems\([Ahmed et al\., 2014](https://arxiv.org/html/2608.23765#bib.bib66);[Li et al\., 2019](https://arxiv.org/html/2608.23765#bib.bib71);[Ma et al\., 2020](https://arxiv.org/html/2608.23765#bib.bib62)\)\. We use such a setup as these random rank\-one measurements \(see[Section4\.1](https://arxiv.org/html/2608.23765#S4.SS1)\) are very likely to make𝒜\\mathcal\{A\}satisfy an NSP of orderrrand match the assumptions of the convergence theorems of[Section4](https://arxiv.org/html/2608.23765#S4), as long asmmis chosen to be at least proportional tor⁡\(d1\+d2\)r\(d\_\{1\}\+d\_\{2\}\)with a certain proportionality factorC\>1C\>1\. Another reason is the implied computational cost of rank\-one measurements\. WithO⁡\(m⁡\(d1\+d2\)\)O\(m\(d\_\{1\}\+d\_\{2\}\)\)entries, they have smaller memory requirements than measurement operators with, e\.g\., measurement matrices drawn from a dense Gaussian ensemble, and smaller evaluation costs on rank\-rrmatrices withO⁡\(m​r​\(d1\+d2\)\)O\(mr\(d\_\{1\}\+d\_\{2\}\)\)instead ofO⁡\(m​d1​d2\)O\(md\_\{1\}d\_\{2\}\)\.

### 5\.2Linear Convergence Rate Factors for IRLS with Weight Operator Variants

In the first experiment, we compare the performance of IRLS for nuclear norm minimization \(as defined in[Algorithm1](https://arxiv.org/html/2608.23765#alg1)\), using*different*weight operator variants on typical low\-rank matrix recovery problem instances\. In the following, we compare IRLS using the updatesW\(k\+1\):=W𝐗\(k\),εkW^\{\(k\+1\)\}:=W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}, whereW𝐗,εW\_\{\\mathbf\{X\},\\varepsilon\}is as in[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2), but with different choices for the weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}in[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)\. We consider

- •a harmonic\-mean weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}as in[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)of[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1),
- •a left\-sided weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}as in[eq\.4](https://arxiv.org/html/2608.23765#S3.E4),
- •a right\-sided weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}as in[eq\.5](https://arxiv.org/html/2608.23765#S3.E5), as well as
- •an arithmetic mean weight operator core matrix𝐇𝝈,ε=𝐇𝝈,ε\(1\)\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}=\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(1\)\}as in[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq=1q=1\.

For these choices,[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)provides an identical global linear convergence rate under suitable conditions ascqc\_\{q\}of[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)is equal to11in every case\. For different matrix dimensionsd∈\{30,60,90,110,140\}d\\in\\\{30,60,90,110,140\\\}withd1=d2=dd\_\{1\}=d\_\{2\}=d, we sample a random low\-rank matrix𝐗⋆=𝐔𝐕⊤∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}=\\mathbf\{U\}\\mathbf\{V\}^\{\\top\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}of rankr=2r=2with random factor matrices𝐔∈ℝd1×r\\mathbf\{U\}\\in\\R^\{d\_\{1\}\\times r\}and𝐕∈ℝd2×r\\mathbf\{V\}\\in\\R^\{d\_\{2\}\\times r\}with i\.i\.d\. standard Gaussian entries and run the respective variant of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)until convergence\. The IRLS variants take as input a realization of the rank\-one measurement operator𝒜\\mathcal\{A\}outlined in[Section5\.1](https://arxiv.org/html/2608.23765#S5.SS1), that is𝐲=𝒜⁡\(𝐗⋆\)∈ℝm\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\)\\in\\R^\{m\}with the number of measurementsm=4\.5​r​\(d1\+d2\)m=4\.5r\(d\_\{1\}\+d\_\{2\}\)\. Typical decay curves of the absolute smoothed nuclear norm gap𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}resulting from the respective variant of[Algorithm1](https://arxiv.org/html/2608.23765#alg1), are shown in[Figure3](https://arxiv.org/html/2608.23765#S5.F3)\. In all experiments, we use a*tangent space implementation*for solving the weighted least squares problem[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)and partial singular value decompositions for the weight operator update of[Algorithm1](https://arxiv.org/html/2608.23765#alg1), which generalizes the implementations of[Kümmerle and Mayrink Verdun \(2021\)](https://arxiv.org/html/2608.23765#bib.bib33)and[Ghosh et al\. \(2024\)](https://arxiv.org/html/2608.23765#bib.bib11)\. We refer to these papers and to research code associate to this paper for more details\.

Figure 3:Smoothed nuclear norm gap𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}of IRLS iterates𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}for different weight operator variants \(harmonic, left\-sided, right\-sided, arithmetic\) across matrix dimensionsd∈\{30,60,90,110,140\}d\\in\\\{30,60,90,110,140\\\}, rank\-22ground truth𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, random Gaussian rank\-one measurements with linear scaling\.We observe that while all variants exhibit a linear convergence rate, their multiplicative decrease factors differ substantially, with the harmonic\-mean weight operator variant of IRLS \(in blue\) converging fastest for all considered dimensions, followed by the arithmetic mean variant, while the left\-sided and right\-sided variants are the slowest\. For example, ford=30d=30, the harmonic\-mean variant of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)reaches an error threshold of10−510^\{\-5\}after 16 iterations, whereas the arithmetic, left\-sided, and right\-sided variants require 49, 68, and 70 iterations, respectively\. For each of the larger dimensionsd∈\{60,90,110,140\}d\\in\\\{60,90,110,140\\\}, this gap widens: the harmonic\-mean variant needs at most 40 iterations to reach the threshold, whereas none of the other variants reaches it within 300 iterations, which is the maximal iteration count used in the experiment\.

Figure 4:Successive decrease\-factor trajectories for the standard initialization𝐗\(0\)\\mathbf\{X\}^\{\(0\)\}in the random Gaussian rank\-one measurement setup, shown ford∈\{30,60,140\}d\\in\\\{30,60,140\\\}\. The color pattern and variant order match[Figure3](https://arxiv.org/html/2608.23765#S5.F3): harmonic \(blue\), left\-sided \(green\), right\-sided \(red\), arithmetic \(orange\)\. Ford=60d=60andd=140d=140, only iteration indices00through150150are displayed\.To make the contraction explicit, we define for each iterationk≥0k\\geq 0the pairwise decrease factors

μ⁡\(k\)=𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mu\(k\)=\\frac\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\(40\)and visualize the corresponding decrease\-factor trajectories across iterationskkin[Figure4](https://arxiv.org/html/2608.23765#S5.F4)for thed∈\{30,60,140\}d\\in\\\{30,60,140\\\}experiments\. In this figure, values farther below11indicate a stronger per\-iteration decrease of the smoothed nuclear norm gap and thus faster linear convergence\.[Figure4](https://arxiv.org/html/2608.23765#S5.F4)illustrates that at initialization, the decrease factor is similar for all IRLS variants, but deteriorates quickly for the left\-sided, right\-sided and arithmetic mean variants\. On the other hand, for[Algorithm1](https://arxiv.org/html/2608.23765#alg1)with the harmonic\-mean weights, we observe stability or a slight improvement quickly after initialization\. This is consistent with the global linear rate analysis of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3), which characterizes the global behavior of any IRLS variant and which applies already for the first iteration, and the fact that an improved local linear rate can be shown \(cf\.[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)\) for the harmonic\-mean IRLS variant once the iterates get closer to the ground truth\. The deterioration of the decrease factor at later iterations for the left\-sided, right\-sided and arithmetic mean variants asddgrows suggests that a dimension\-independent, fast local convergence rate such as shown in[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)does not hold for these variants\.[Figure3](https://arxiv.org/html/2608.23765#S5.F3)and[Figure4](https://arxiv.org/html/2608.23765#S5.F4)suggest that a slight increase in the number of iterations is needed for harmonic\-mean IRLS to reach the error threshold of10−510^\{\-5\}\. Similarly, the limitingμ⁡\(k\)\\mu\(k\)for largekkincreases with increasing dimensiondd, namely from around0\.360\.36ford=30d=30, through0\.550\.55ford=60d=60, to0\.630\.63ford=140d=140\. However, this behavior is entirely compatible with[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)\. As discussed after[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5), its factor1−cηr1\-c\_\{\\eta\_\{r\}\}is an upper bound with a non\-optimized constant, which equals≈0\.996\\approx 0\.996forηr=1/10\\eta\_\{r\}=1/10and thus lies above all factors observed here\. It is moreover dimension\-free for a*fixed*NSP constant, whereas our experiments keep the oversampling factorm/\(r⁡\(d1\+d2\)\)m/\(r\(d\_\{1\}\+d\_\{2\}\)\)fixed, so thatηr\\eta\_\{r\}andη1\\eta\_\{1\}may still vary withdd\.

### 5\.3Recovery of Rectangular Low\-Rank Matrices

In the recent literature on IRLS for low\-rank matrix recovery, one\-sided variants of IRLS are still often considered\([Kraemer, 2025](https://arxiv.org/html/2608.23765#bib.bib13);[Radhakrishnan et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib86)\), partially motivated by the rectangular structure of many problem instances, where, for example,d1d\_\{1\}can be considered as the number of data samples andd2d\_\{2\}as the number of features in which case[Radhakrishnan et al\. \(2025\)](https://arxiv.org/html/2608.23765#bib.bib86)recommends right\-sided reweighting\. This could also be justified specifically for the case ofd1≫d2d\_\{1\}\\gg d\_\{2\}, as in this case the right\-sided variant of IRLS requires a matrix decomposition of a smaller matrix,222[Radhakrishnan et al\. \(2025\)](https://arxiv.org/html/2608.23765#bib.bib86)discusses SVD\-free variants of one\-sided IRLS under the name*SVD\-free lin\-RFM*\(recursive feature machines\)\. However, this implementation variant is not available for the case of the nuclear norm surrogate \(as this corresponds toα=1/4\\alpha=1/4in the framework of[Radhakrishnan et al\. \(2025\)](https://arxiv.org/html/2608.23765#bib.bib86), whereas the SVD\-free variant is only available for integer multiples ofα=1/2\\alpha=1/2\)\.whereas this perspective would motivate left\-sided reweighting ford1≪d2d\_\{1\}\\ll d\_\{2\}\.

Figure 5:Smoothed nuclear norm gap trajectories𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}for the tall rectangular recovery experiment with standard initialization and dimensionsd∈\{20,40,80\}d\\in\\\{20,40,80\\\}\. In each panel, a rectangular ground truth𝐗⋆∈ℝ4​d×d\\mathbf\{X\}\_\{\\star\}\\in\\R^\{4d\\times d\}of rankr=2r=2is recovered fromm=4\.5​r​\(d1\+d2\)m=4\.5r\(d\_\{1\}\+d\_\{2\}\)random Gaussian rank\-one measurements\.In any case, the question arises whether one\-sided IRLS variants perform sufficiently well for highly rectangular problem instances withd1≫d2d\_\{1\}\\gg d\_\{2\}, or whether the optimal harmonic\-mean variant is still advantageous in this case\. To this end, we consider the setup of[Section5\.2](https://arxiv.org/html/2608.23765#S5.SS2)to recover a rankr=2r=2ground truth𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}fromm=4\.5​r​\(d1\+d2\)m=4\.5r\(d\_\{1\}\+d\_\{2\}\)rank\-one measurements, but with the tall rectangular dimensionsd1=4​dd\_\{1\}=4dandd2=dd\_\{2\}=dford∈\{20,40,80\}d\\in\\\{20,40,80\\\}, so that the aspect ratiod1/d2d\_\{1\}/d\_\{2\}is the only parameter in which the two setups differ\.

Figure 6:Successive decrease\-factor trajectoriesμ⁡\(k\)\\mu\(k\)of[eq\.40](https://arxiv.org/html/2608.23765#S5.E40)for the tall rectangular recovery experiment with standard initialization, shown ford∈\{20,40,80\}d\\in\\\{20,40,80\\\}and rankr=2r=2ground truth𝐗⋆∈ℝ4​d×d\\mathbf\{X\}\_\{\\star\}\\in\\R^\{4d\\times d\}\. Each trajectory ends once the respective variant terminates or the maximal iteration count of500500is reached\.We observe in[Figure5](https://arxiv.org/html/2608.23765#S5.F5)that, ford=20d=20, the right\-sided IRLS variant using[eq\.5](https://arxiv.org/html/2608.23765#S3.E5)in[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)clearly converges faster than the left\-sided variant using[eq\.4](https://arxiv.org/html/2608.23765#S3.E4), passing the error threshold of10−510^\{\-5\}after 40 instead of 81 iterations, whereas the arithmetic mean variant lies in between with 51 iterations and harmonic\-mean IRLS is fastest with 19 iterations\. This ordering persists for the larger dimensions, but the gaps*widen*substantially asddgrows: harmonic\-mean IRLS passes the threshold after 28 and 38 iterations ford=40d=40andd=80d=80, while the right\-sided variant, the best of the alternatives, requires 113 and 324 iterations, and the left\-sided variant does not reach the threshold ford=80d=80within the maximal iteration count of 500 used in this experiment\.

The decrease factor trajectoriesμ⁡\(k\)\\mu\(k\)of[eq\.40](https://arxiv.org/html/2608.23765#S5.E40)depicted in[Figure6](https://arxiv.org/html/2608.23765#S5.F6)visualize this deterioration in a different manner\. In each panel, the four variants start from an almost identicalμ⁡\(0\)\\mu\(0\), in line with the global rate of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)that applies to all of them alike, but settle at markedly different levels: the limiting decrease factor of harmonic\-mean IRLS increases only mildly from about0\.420\.42ford=20d=20to about0\.620\.62ford=80d=80, cf\.[Remark3](https://arxiv.org/html/2608.23765#Thmremark3), whereas those of the other three variants reach values between0\.960\.96and0\.980\.98ford=80d=80, which again suggests that a fast local linear rate such as the one of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)does not hold for these variants\.

Overall, this experiment shows that for rectangular low\-rank matrix recovery, among the one\-sided variants, using nontrivial weights on the*smaller*matrix dimension pays off relatively speaking, but the optimal harmonic\-mean variant still is significantly superior, and increasingly so for instances of higher dimensions\.

### 5\.4Adversarial Initialization

[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)guarantees a global linear decrease factor of the form1−Cηrcq​η1​d1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\. A natural question is whether the ambient\-dimension factorddin this bound is essentially sharp, or merely an artifact of the proof\. While the guarantee applies for every choice of the initial positive definite weight operatorW\(0\)W^\{\(0\)\}, the default choiceW\(0\)=IdW^\{\(0\)\}=\\Idin[Algorithm1](https://arxiv.org/html/2608.23765#alg1)typically yields a first iterate that is already somewhat aligned with the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}\.333It is easy to see that the solution𝐗\(0\)\\mathbf\{X\}^\{\(0\)\}of \([12](https://arxiv.org/html/2608.23765#S3.E12)\) is simply𝐗\(0\)=𝒜†​\(𝐲\)\\mathbf\{X\}^\{\(0\)\}=\\mathcal\{A\}^\{\\dagger\}\(\\mathbf\{y\}\)in this case, where𝒜†\\mathcal\{A\}^\{\\dagger\}denotes the pseudo\-inverse of the measurement operator𝒜\\mathcal\{A\}\.To probe the worst\-case behavior, we therefore track the initial decrease factorμ⁡\(0\)\\mu\(0\)of[eq\.40](https://arxiv.org/html/2608.23765#S5.E40)for an initial weight operator constructed from an auxiliary reference matrix that is deliberately chosen to be as poorly aligned with𝐗⋆\\mathbf\{X\}\_\{\\star\}as possible\.

In particular, to create such an*adversarial initialization*, we first compute the standard nuclear norm minimizer𝐗nuc\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}of \([1](https://arxiv.org/html/2608.23765#S1.E1)\)\. Let𝐗nuc\(r\)=𝐔nuc\(r\)​diag⁡\(σnuc\(r\)\)​𝐕nuc\(r\)⊤\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}^\{\(r\)\}=\\mathbf\{U\}\_\{\\mathrm\{nuc\}\}^\{\(r\)\}\\diag\(\\sigma\_\{\\mathrm\{nuc\}\}^\{\(r\)\}\)\\mathbf\{V\}\_\{\\mathrm\{nuc\}\}^\{\(r\)^\{\\top\}\}denote its best rank\-rrapproximation, and letT=\{𝐗=𝐔nuc\(r\)​𝐌\+𝐍𝐕nuc\(r\)⊤,𝐌∈ℝr×d2,𝐍∈ℝd1×r\}⊂ℝd1×d2T=\\left\\\{\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathrm\{nuc\}\}^\{\(r\)\}\\mathbf\{M\}\+\\mathbf\{N\}\\mathbf\{V\}\_\{\\mathrm\{nuc\}\}^\{\(r\)^\{\\top\}\},\\mathbf\{M\}\\in\\R^\{r\\times d\_\{2\}\},\\mathbf\{N\}\\in\\R^\{d\_\{1\}\\times r\}\\right\\\}\\subset\\R^\{d\_\{1\}\\times d\_\{2\}\}be the tangent space of the fixed\-rank manifold at𝐗nuc\(r\)\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}^\{\(r\)\}\([Vandereycken, 2013](https://arxiv.org/html/2608.23765#bib.bib12)\)\. WritingPT⟂​\(⋅\)P\_\{T^\{\\perp\}\}\(\\cdot\)for the orthogonal projection onto the complementary spaceT⟂T^\{\\perp\}, we then define

𝐗\(−1\)=arg​min𝐗∈ℝd1×d2⁡‖𝐗‖∗subject to𝒜⁡\(PT⟂​\(𝐗\)\)=𝐲\.\\mathbf\{X\}^\{\(\-1\)\}\\;=\\;\\argmin\_\{\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\}\\;\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\quad\\text\{subject to\}\\quad\\mathcal\{A\}\\bigl\(P\_\{T^\{\\perp\}\}\(\\mathbf\{X\}\)\\bigr\)=\\mathbf\{y\}\.Relative to the geometry suggested by𝐗nuc\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}, this forces the measurements to be explained through theT⟂T^\{\\perp\}component and thereby yields an auxiliary reference matrix that is poorly aligned with𝐗⋆\\mathbf\{X\}\_\{\\star\}\. Since the two\-sided orthogonal projectionPT⟂​\(⋅\)P\_\{T^\{\\perp\}\}\(\\cdot\)is nuclear\-norm nonexpansive, a minimizer can be chosen inT⟂T^\{\\perp\}; in particular,𝒜⁡\(𝐗\(−1\)\)=𝐲\\mathcal\{A\}\(\\mathbf\{X\}^\{\(\-1\)\}\)=\\mathbf\{y\}\. We initialize the experimental recurrence byε−1:=βr​\(𝐗\(−1\)\)∗/d\\varepsilon\_\{\-1\}:=\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\-1\)\}\)\_\{\*\}/dandW\(0\):=W𝐗\(−1\),ε−1W^\{\(0\)\}:=W\_\{\\mathbf\{X\}^\{\(\-1\)\},\\varepsilon\_\{\-1\}\}\. The weighted least\-squares step[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)then produces𝐗\(0\)\\mathbf\{X\}^\{\(0\)\}\.

Figure 7:Reciprocal first\-algorithmic contraction gaps1/\(1−μ⁡\(0\)\)1/\(1\-\\mu\(0\)\)\(for initial\), measuring the transition from𝐗\(0\)\\mathbf\{X\}^\{\(0\)\}to𝐗\(1\)\\mathbf\{X\}^\{\(1\)\}, for the extended recurrence initialized by\(𝐗\(−1\),ε−1\)\(\\mathbf\{X\}^\{\(\-1\)\},\\varepsilon\_\{\-1\}\)and the resultingW\(0\)W^\{\(0\)\}, across dimensionsd∈\{25,30,40,50,60,70,80,90,100,110,120,130,140\}d\\in\\\{25,30,40,50,60,70,80,90,100,110,120,130,140\\\}in the square random Gaussian rank\-one measurement setup with rank\-22ground truth and linear measurement scaling; five terminal iteration average of1/\(1−μ⁡\(k\)\)1/\(1\-\\mu\(k\)\)for the harmonic\-mean variant\. Median values across1010seeds with25%25\\%and75%75\\%quantiles\.With this extended adversarial initialization, we revisit the experimental setup of[Section5\.2](https://arxiv.org/html/2608.23765#S5.SS2)and recover rankr=2r=2ground truth matrices𝐗⋆∈ℝd×d\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\\times d\}of different sizesd∈\{25,30,40,50,60,70,80,d\\in\\\{25,30,40,50,60,70,80,90,100,110,120,130,140\}90,100,110,120,130,140\\\}fromm=9​r​dm=9rdrank\-one measurements\. As in[Section5\.2](https://arxiv.org/html/2608.23765#S5.SS2), we consider different weight operator variants\. Revisiting the contraction factorμ⁡\(0\)\\mu\(0\)of first non\-initialization iteration as defined in[eq\.40](https://arxiv.org/html/2608.23765#S5.E40), we plot the dimension\-dependent behavior of1/\(1−μ⁡\(0\)\)1/\(1\-\\mu\(0\)\)in[Figure7](https://arxiv.org/html/2608.23765#S5.F7)in box plots across1010random seeds\.

We observe that across all IRLS variants,1/\(1−μ⁡\(0\)\)1/\(1\-\\mu\(0\)\)grows approximately linearly with the dimensiondd, especially over the larger dimensions, with the reference lined/7−3d/7\-3amounting to a suitable fit for the setup considered\. We note that the first algorithmic decrease for this adversarial setup is markedly slower than for the standard least\-squares initialization illustrated in[Figure4](https://arxiv.org/html/2608.23765#S5.F4)of[Section5\.2](https://arxiv.org/html/2608.23765#S5.SS2), particularly asddgrows\.

For harmonic mean IRLS specifically, however, that this deterioration is a*transient*effect, as the value ofμ⁡\(k\)\\mu\(k\)quickly falls to a lower, terminal value\. We indicate in[Figure7](https://arxiv.org/html/2608.23765#S5.F7)as*Harmonic, terminal*the distribution of the values of15​∑k​among last five iterations1/\(1−μ⁡\(k\)\)\\frac\{1\}\{5\}\\sum\_\{k\\text\{ among last five iterations\}\}1/\(1\-\\mu\(k\)\)for the harmonic\-mean variant\. We observe that this value grows only marginally with the dimensiondd, staying well below3\.03\.0even in the larger dimension range ofd∈\{100,110,120,130,140\}d\\in\\\{100,110,120,130,140\\\}\. On the other hand, for the other variants, the in\-trajectory and terminal values of1/\(1−μ⁡\(k\)\)1/\(1\-\\mu\(k\)\)grow linearly withddsignificantly*faster*than1/\(1−μ⁡\(k\)\)1/\(1\-\\mu\(k\)\), similar to[Figure6](https://arxiv.org/html/2608.23765#S5.F6), with the terminal five\-iteration averages growing above5050for the one\-sided variants and to almost4040for the arithmetic mean variant\. For harmonic mean IRLS, the observed transition toward the same terminal behavior as for the standard initialization is consistent with the distinction between the global and local regimes in[Section4](https://arxiv.org/html/2608.23765#S4): the local factor1−cηr1\-c\_\{\\eta\_\{r\}\}of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)depends neither on the dimensionddnor on the initialization once a neighborhood as in[eq\.35](https://arxiv.org/html/2608.23765#S4.E35)has been reached\.

### 5\.5Role of the Smoothing Parameter Update Rules

As discussed in[Section3\.3](https://arxiv.org/html/2608.23765#S3.SS3), the smoothing\-parameter update rule is a second important design choice for IRLS\. We revisit the experimental setup of[Section5\.2](https://arxiv.org/html/2608.23765#S5.SS2)and compare theℓ1\\ell\_\{1\}\-tail update \([13](https://arxiv.org/html/2608.23765#S3.E13)\), theℓ2\\ell\_\{2\}\-tail update \([18](https://arxiv.org/html/2608.23765#S3.E18)\), and theℓ∞\\ell\_\{\\infty\}\-tail update \([17](https://arxiv.org/html/2608.23765#S3.E17)\)\.

In the left panel of[Figure8](https://arxiv.org/html/2608.23765#S5.F8), we show a typical trajectory for the oversampling factor ofC=3C=3\. We observe that theℓ∞\\ell\_\{\\infty\}\-tail rule does not yield convergence\. However, the advantage via faster convergence of the harmonic\-mean variant observed in[Sections5\.2](https://arxiv.org/html/2608.23765#S5.SS2),[5\.3](https://arxiv.org/html/2608.23765#S5.SS3)and[5\.4](https://arxiv.org/html/2608.23765#S5.SS4)relative to the one\-sided variants persists across all smoothing\-parameter update rules\. Increasing the oversampling factor toC=4C=4improves the per\-iteration gap decrease for all variants, as illustrated in the right panel of[Figure8](https://arxiv.org/html/2608.23765#S5.F8), and enables ground truth convergence for theℓ∞\\ell\_\{\\infty\}\-tail rule\-based variants as well\.

Figure 8:Selected trajectory plots of IRLS variants with different smoothing parameter update rules at dimensiond=60d=60with rank\-22ground truth and Gaussian rank\-one measurements\. Left: Objective gap\|𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\|\\left\|\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\|vs\. iterationkkforC=3\.0C=3\.0\. Right: Analogous experiment with oversampling factor ofC=4\.0C=4\.0\.A better understanding of the sampling\-data dependence of the IRLS variants with different smoothing parameter update rules is provided in[Figure9](https://arxiv.org/html/2608.23765#S5.F9)\. In this experiment, we run1212problem instances for each combination of dimensiond∈\{30,60,100\}d\\in\\\{30,60,100\\\}and a selection of different oversampling factorsC∈\[2\.0,5\.0\]C\\in\[2\.0,5\.0\]for the different smoothing parameter update rules\. The distribution of the relative Frobenius errors‖𝐗\(k~\)−𝐗⋆‖F/‖𝐗⋆‖F\\\|\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\\|\_\{F\}/\\\|\\mathbf\{X\}\_\{\\star\}\\\|\_\{F\}is visualized using box plots, wherek~\\tilde\{k\}is the minimum of500500and of the first iteration index for which two subsequent algorithm iterates differ by a relative change in Frobenius norm of less than10−1010^\{\-10\}\. Naturally, the setup is such that the problem becomes easier to solve asCCincreases\. In addition, with the black star, we visualize the smallest value ofCCfor which the nuclear norm minimizer𝐗nuc\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}of[eq\.1](https://arxiv.org/html/2608.23765#S1.E1)recovers the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}\(up to a relative error of at most10−410^\{\-4\}\) using the splitting conic solverSCS\([O’Donoghue et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib7)\)viaCVXPY\([Diamond and Boyd, 2016](https://arxiv.org/html/2608.23765#bib.bib8)\)\.

Figure 9:Phase transition experiment for nuclear norm MatrixIRLS with harmonic\-mean weight operator, showing distribution of relative Frobenius errors across1212problem instances for different oversampling factorsCCand different smoothing parameter update rules\.We observe that the box plots for theℓ1\\ell\_\{1\}\-tail andℓ2\\ell\_\{2\}\-tail variants of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)track closely the performance of the nuclear norm minimizer𝐗nuc\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}in the setting of[Figure9](https://arxiv.org/html/2608.23765#S5.F9), recovering𝐗⋆\\mathbf\{X\}\_\{\\star\}whenever the nuclear norm minimizer𝐗nuc\\mathbf\{X\}\_\{\\mathrm\{nuc\}\}does, with a slight edge of theℓ1\\ell\_\{1\}\-tail variant over theℓ2\\ell\_\{2\}\-tail as can be seen, e\.g\., for thed=100d=100andC=3\.0C=3\.0case\. Theℓ∞\\ell\_\{\\infty\}\-tail variant, on the other hand, trails in a low\-sampling regime, but transitions also to consistent recovery for aroundC=2\.9C=2\.9ford=30d=30,C=3\.4C=3\.4ford=60d=60andC=3\.7C=3\.7ford=100d=100, respectively\.

Although theℓ∞\\ell\_\{\\infty\}\-tail rule is less robust at low sampling levels, it can substantially reduce the cost of an IRLS iteration\. For iterationkk, define the rank envelope and objective gap by

renv\(k\):=\#⁡\{i:σi​\(𝐗\(k\)\)\>εk\},gk:=\|𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\|,r\_\{\\mathrm\{env\}\}^\{\(k\)\}:=\\\#\\\{i:\\sigma\_\{i\}\(\\mathbf\{X\}^\{\(k\)\}\)\>\\varepsilon\_\{k\}\\\},\\qquad g\_\{k\}:=\\left\|\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\\|\\mathbf\{X\}\_\{\\star\}\\\|\_\{\*\}\\right\|,respectively\. As discussed in[Section3\.3](https://arxiv.org/html/2608.23765#S3.SS3), both the memory and time requirements of the tangent\-space implementation depend onrenv\(k\)r\_\{\\mathrm\{env\}\}^\{\(k\)\}\. For a fixed iterate, elementary tail\-norm inequalities make the candidateℓ∞\\ell\_\{\\infty\}\-smoothing parameter often larger than itsℓ1\\ell\_\{1\}andℓ2\\ell\_\{2\}counterparts; in the representative runs below, this produces substantially smaller rank envelopes\.[Table2](https://arxiv.org/html/2608.23765#S5.T2)quantifies the resulting tradeoff ford=100d=100, rank\-22instances withC∈\{3\.0,4\.0\}C\\in\\\{3\.0,4\.0\\\}and the harmonic\-mean weight operator\.

Forτ=10−5\\tau=10^\{\-5\}, letkτ:=min⁡\{k:gk≤τ\}k\_\{\\tau\}:=\\min\\\{k:g\_\{k\}\\leq\\tau\\\}\. Because𝐗\(0\)\\mathbf\{X\}^\{\(0\)\}is the least\-squares initialization,kτk\_\{\\tau\}is also the number of completed outer IRLS solves needed to reach the threshold\. The IRLS runtimes in[Table2](https://arxiv.org/html/2608.23765#S5.T2)are the median cumulative solver times over five timing repetitions after one warmup\. TheSCSrows report the numerical solve time returned byCVXPY’s solver statistics for the same instance of \([1](https://arxiv.org/html/2608.23765#S1.E1)\), using a solver tolerance of10−910^\{\-9\}and at most50,00050\{,\}000iterations; CVXPY canonicalization and SCS setup time are excluded\. All timings were obtained in CPU double precision on a MacBook Pro with an Apple M4 Pro processor \(14 cores, 24 GB RAM\)\.

Table 2:Envelope ranks and computational effort for the representatived=100d=100, rank\-22, seed\-4242smoothing\-parameter experiment with the harmonic mean weight operator\. For IRLS, runtime is the median cumulative solver time to reach the objective\-gap thresholdτ=10−5\\tau=10^\{\-5\}over 5 timing repetitions after 1 warmup\. ForSCS, runtime is the solver\-reported numerical solve time; CVXPY canonicalization and SCS setup are excluded\. A dash denotes a quantity that is not applicable or an objective\-gap threshold not reached before termination\.Theℓ∞\\ell\_\{\\infty\}\-tail rule maintainsrenv\(k\)=r~=2r\_\{\\mathrm\{env\}\}^\{\(k\)\}=\\widetilde\{r\}=2throughout both representative runs, whereas theℓ1\\ell\_\{1\}\- andℓ2\\ell\_\{2\}\-tail rules use substantially larger envelopes\. ForC=4\.0C=4\.0, the lower per\-iteration cost of theℓ∞\\ell\_\{\\infty\}\-tail rule more than offsets its larger iteration count: it reaches the prescribed gap after125125iterations in2\.292\.29seconds, compared with4343iterations and4\.884\.88seconds for theℓ1\\ell\_\{1\}\-tail rule and4747iterations and3\.533\.53seconds for theℓ2\\ell\_\{2\}\-tail rule\. The corresponding SCS numerical solve time is29\.3229\.32seconds; forC=3\.0C=3\.0, it is25\.3725\.37seconds\. At the lower sampling levelC=3\.0C=3\.0, however, only theℓ1\\ell\_\{1\}\- andℓ2\\ell\_\{2\}\-tail rules reach the IRLS objective\-gap threshold; theℓ∞\\ell\_\{\\infty\}\-tail run terminates atk~=191\\tilde\{k\}=191withgk~=5\.99×101g\_\{\\tilde\{k\}\}=5\.99\\times 10^\{1\}\. Thus, the computational advantage of a smaller rank envelope must be balanced against robustness at lower sampling levels\. In fact, theℓ1\\ell\_\{1\}\-tail rule remains the conservative, theoretically supported default\. Comparing theMatrixIRLSruntimes with the SCS solve times of25\.3725\.37and29\.3229\.32seconds, respectively, we observe that IRLS can achieve order\-of\-magnitude speedups over generic SDP solvers, especially in setups farther above the sample complexity phase transition point\.

## 6Conclusion

In this work, we provide comprehensive answers to fundamental questions regarding both the design and analysis of iteratively reweighted least squares methods for the central spectral optimization problem of nuclear norm minimization, which had remained unanswered since the initial works of[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54)and[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48)despite the recent interest in the methodology in the literature: we frame the algorithm design around the*weight operator*notion and show that a particular choice of weight operator, the harmonic mean weight operator, constitutes an arguably*optimal*choice, which we substantiate with the rigorous majorization and tightness results of[Section3](https://arxiv.org/html/2608.23765#S3)\.

Building on these results, we provide the first convergence rate analysis of IRLS methods for the problem that is furthermore fine\-grained as it, \(a\) shows that IRLS methods using a variety of weight operators, including traditional ones as well as the harmonic mean one \(see[Definition5](https://arxiv.org/html/2608.23765#Thmdefinition5)\), exhibit*global linear convergence*from any initialization, and \(b\) shows that the convergence rate can be improved to a*dimension\-free linear rate*once the iterates reach a neighborhood of the low\-rank solution, if the correct weight operator notion is used\.

The numerical experiments of[Section5](https://arxiv.org/html/2608.23765#S5)corroborate this separation: across rectangular and square instances, IRLS with harmonic mean reweighting exhibits a substantially faster empirically observed linear rate than the one\-sided or arithmetic mean variants\. We further presented adversarial initialization experiments indicating that the dimension dependence appearing in the global linear rate of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)across all IRLS variants is not merely an artifact of the proof, but also, that it tends to be a transient phenomenon associated with the initial phase of the algorithmic trajectory\. Finally, our results highlight the role of the smoothing parameter continuation strategy as a second important algorithmic design choice\. The nuclear\-norm\-tail update used in our theory couples the smoothing parameter directly to the quantity needed in the global convergence argument and, empirically, tracks the nuclear\-norm recovery threshold reliably\. More computationally economical updates are not presently covered by the convergence theory, so that the generalization of linear rate results to IRLS withℓ2\\ell\_\{2\}\- orℓ∞\\ell\_\{\\infty\}\-tail updates remains open for future research\.

Several other interesting questions remain open: the presented convergence analysis crucially depends on the fact that the linear measurement operator𝒜\\mathcal\{A\}satisfies a suitable null space property\. However, it is well\-known that for some important application scenarios such as low\-rank matrix completion, this assumption does not hold\. It would be interesting to develop a theory which also covers this scenario\. Another observation is that although the local contraction factor of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)is dimension\-independent, the certified neighborhood in which the local behavior applies still depends on the dimension\. The experiments, however, suggest that the favorable fast linear rate regime forMatrixIRLScan arise substantially earlier than this worst\-case basin predicts\. Understanding whether the methodology and/or proof technique can be improved to establish a similar result for a dimension\-free basin size in a worst\-case scenario remains an open problem\. We also note that our quadratic model majorization and tightness analysis currently only applies to the smoothed nuclear norm objective that is conducive to[eq\.1](https://arxiv.org/html/2608.23765#S1.E1), but not to the optimization of other spectral objectives such as non\-convex Schatten\-ppquasi\-norms with0<p<10<p<1or smoothed log\-determinants\. It remains unknown which weight operator notion defines a tight or the tightest majorizing quadratic model for these spectral objectives, with the geometric mean constituting a popular, but so far theoretically unjustified choice in the latter context\([Kümmerle and Mayrink Verdun, 2021](https://arxiv.org/html/2608.23765#bib.bib33);[Kraemer, 2025](https://arxiv.org/html/2608.23765#bib.bib13)\)\. Finally, it would be interesting to explore how the insights of this work can improve either the design or analysis of related methodologies in machine learning, such as recursive feature machines \(RFM\)\([Radhakrishnan et al\., 2024](https://arxiv.org/html/2608.23765#bib.bib85)\), iteratively reweighted kernel machines\([Zhu et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib87)\), orMuon\-like spectral optimizers in deep learning\. The fact that the specialization of RFM to low\-rank optimization\([Radhakrishnan et al\., 2025](https://arxiv.org/html/2608.23765#bib.bib86)\)leads to suboptimal IRLS variants indicates that there can be room for improvement via the use of tight majorizing quadratic models\.

## Acknowledgments

The authors thank Felix Krahmer for insightful discussions around the paper subject\. C\.K\. and T\.M\. acknowledge the support of this research by the Mathematisches Forschungsinstitut Oberwolfach through the*Oberwolfach Research Fellows*program\. C\.K\. was supported in part by the grant NSF\-2549926\.

## Appendix AProofs of Main Results

In this section, we provide all proofs establishing the main theoretical results of the paper, starting with the global majorization statement[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)of[Section3\.2](https://arxiv.org/html/2608.23765#S3.SS2)in[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)\. We continue with the proof of the optimality result[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)in[SectionA\.2](https://arxiv.org/html/2608.23765#A1.SS2), before detailing the global linear convergence results[Theorems3](https://arxiv.org/html/2608.23765#Thmtheorem3)and[4](https://arxiv.org/html/2608.23765#Thmtheorem4)in[SectionA\.3\.3](https://arxiv.org/html/2608.23765#A1.SS3.SSS3)and[SectionA\.3\.4](https://arxiv.org/html/2608.23765#A1.SS3.SSS4), respectively\. Finally, establishing the fast local linear convergence result of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)is done in[SectionA\.4](https://arxiv.org/html/2608.23765#A1.SS4)\. The latter section also contains the proof of the impossibility result[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)for dimension\-independent fast local rate for one\-sided weight operators\.

### A\.1Proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)\(Harmonic\-Mean Quadratic Model Majorization\)

The major challenge in the proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)in the case of harmonic\-mean weights is to derive a suitable lower bound for the weighted inner product⟨W𝐗,ε​\(𝐙\),𝐙⟩F\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\. The argument below isolates this lower bound and then proves the majorization property directly\. Afterwards, in[SectionB\.3\.3](https://arxiv.org/html/2608.23765#A2.SS3.SSS3), we explain how the same proof scheme specializes to the simpler one\-sided weights, where the required lower bound follows immediately from the Cauchy–Schwarz inequality\.

However, in the case of harmonic\-mean weights proving a lower bound is significantly more challenging due to the more complicated structure of the weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)\. The following lemma provides such a bound\.

###### Lemma 2\(Lower Bound for Weighted Inner Product of Harmonic Mean\)\.

Letε\>0\\varepsilon\>0and let𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}with SVD given by𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}, wherediag⁡\(𝛔\)∈ℝd1×d2\\diag\(\\bm\{\\sigma\}\)\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is the rectangular diagonal matrix with the extended vector of nonincreasing singular values𝛔∈ℝD\\bm\{\\sigma\}\\in\\R^\{D\}of𝐗\\mathbf\{X\}on the diagonal\. Consider the weight operatorW𝐗,ε:ℝd1×d2→ℝd1×d2W\_\{\\mathbf\{X\},\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}of[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)with harmonic\-mean core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)\. Define the matrices𝐋𝐔=𝐔𝐗​diag⁡\(λ1,…,λd1\)​𝐔𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{U\}\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}and𝐋𝐕=𝐕𝐗​diag⁡\(λ1,…,λd2\)​𝐕𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{V\}\}=\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{2\}\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}, whereλi:=max⁡\(σi,ε\)\\lambda\_\{i\}:=\\max\\left\(\\sigma\_\{i\},\\varepsilon\\right\)for1≤i≤max⁡\(d1,d2\)1\\leq i\\leq\\max\\left\(d\_\{1\},d\_\{2\}\\right\)with the convention ofσi=0\\sigma\_\{i\}=0fori\>di\>d\.

Let now𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be arbitrary with SVD given by𝐙=∑k=1dσk​\(𝐙\)​𝐮k​𝐯k⊤\\mathbf\{Z\}=\\sum\_\{k=1\}^\{d\}\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}, whereσk​\(𝐙\)\\sigma\_\{k\}\(\\mathbf\{Z\}\)denotes thekk\-th singular value of𝐙\\mathbf\{Z\}\. Then, it holds that

⟨W𝐗,ε​\(𝐙\),𝐙⟩F≥∑k=1dσk2​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F/2\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F/2\.\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\\geq\\sum\_\{k=1\}^\{d\}\\frac\{\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2\}\.

As already mentioned, the proof of Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2)is significantly more involved than the corresponding lower bound for one\-sided weights\. We provide the proof of[Lemma2](https://arxiv.org/html/2608.23765#Thmlemma2)in[SectionA\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1)below\. The major challenge in the proof of this lemma is to deal with the fact that the inner product⟨W𝐗,ε​\(𝐙\),𝐙⟩F\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}no longer has a simple structure as in the case of one\-sided weights, where it could be expressed as∑k=1dσk2​\(𝐙\)​⟨𝐋𝐔−1,𝐮k​𝐮k⊤⟩F\\sum\_\{k=1\}^\{d\}\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\.

With Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2)at hand, we can now prove[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)in the case of harmonic\-mean weight operators\.

Proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)We first compute the terms in the simplified representation \([11](https://arxiv.org/html/2608.23765#S3.E11)\) ofQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)\. Since harmonic\-mean weights satisfy\(𝐇𝝈,ε\)i​i=1/max⁡\(σi​\(𝐗\),ε\)\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=1/\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)on the diagonal, we have

⟨W𝐗,ε​\(𝐗\),𝐗⟩F=\\displaystyle\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}=⟨𝐇𝝈,ε∘diag⁡\(𝝈\),diag⁡\(𝝈\)⟩F\\displaystyle\\langle\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\diag\(\\bm\{\\sigma\}\),\\diag\(\\bm\{\\sigma\}\)\\rangle\_\{F\}=\\displaystyle=∑k∈\[d\]:σk​\(𝐗\)≥εσk\(𝐗\)\+∑k∈\[d\]:σk​\(𝐗\)<εσk2​\(𝐗\)ε\.\\displaystyle\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{X\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)<\\varepsilon\}\\frac\{\\sigma\_\{k\}^\{2\}\(\\mathbf\{X\}\)\}\{\\varepsilon\}\.Moreover, by definition of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\},

𝒥ε\(𝐗\)=∑k∈\[d\]:σk​\(𝐗\)≥εσk\(𝐗\)\+∑k∈\[d\]:σk​\(𝐗\)<ε\(σk2​\(𝐗\)2​ε\+ε2\)\.\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)=\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{X\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)<\\varepsilon\}\\left\(\\frac\{\\sigma\_\{k\}^\{2\}\(\\mathbf\{X\}\)\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\}\\right\)\.Inserting these two identities into \([11](https://arxiv.org/html/2608.23765#S3.E11)\), we obtain that

Qε​\(𝐙∣𝐗\)=\\displaystyle Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)=12∑k∈\[d\]:σk​\(𝐗\)≥εσk\(𝐗\)\+12∑k∈\[d\]:σk​\(𝐗\)<εε\+12⟨W𝐗,ε\(𝐙\),𝐙⟩F\.\\displaystyle\\frac\{1\}\{2\}\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{X\}\)\+\\frac\{1\}\{2\}\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)<\\varepsilon\}\\varepsilon\+\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\.\(41\)Now, note that by definition of𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}, it holds that

∑k=1d⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F​≤\(a\)\\displaystyle\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\overset\{\(a\)\}\{\\leq\}max𝐳1,…,𝐳d​orthonormal​∑k=1d⟨𝐋𝐔,𝐳k​𝐳k⊤⟩F​=\(b\)​∑k=1dλk\\displaystyle\\underset\{\\mathbf\{z\}\_\{1\},\\ldots,\\mathbf\{z\}\_\{d\}\\text\{ orthonormal\}\}\{\\max\}\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{z\}\_\{k\}\\mathbf\{z\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\overset\{\(b\)\}\{=\}\\sum\_\{k=1\}^\{d\}\\lambda\_\{k\}=\(c\)\\displaystyle\\overset\{\(c\)\}\{=\}∑k∈\[d\]:σk​\(𝐗\)≥εσk\(𝐗\)\+∑k∈\[d\]:σk​\(𝐗\)<εε,\\displaystyle\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{X\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)<\\varepsilon\}\\varepsilon,where in inequality \(a\), we have used that\(𝐮k\)k=1d\(\\mathbf\{u\}\_\{k\}\)\_\{k=1\}^\{d\}is an orthonormal set inℝd1\\R^\{d\_\{1\}\}\. In equality \(b\), we have used the Wielandt minimax principle, see\([Bhatia, 1997](https://arxiv.org/html/2608.23765#bib.bib60), Theorem III\.3\.5\), and thatλ1≥λ2≥…≥λd1≥0\\lambda\_\{1\}\\geq\\lambda\_\{2\}\\geq\\ldots\\geq\\lambda\_\{d\_\{1\}\}\\geq 0are the eigenvalues of𝐋𝐔=𝐔𝐗​diag⁡\(λ1,…,λd1\)​𝐔𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{U\}\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\. In equation \(c\), we used thatλk=max⁡\(σk​\(𝐗\),ε\)\\lambda\_\{k\}=\\max\(\\sigma\_\{k\}\(\\mathbf\{X\}\),\\varepsilon\)fork=1,2,…,d1k=1,2,\\ldots,d\_\{1\}\. Analogously, we obtain that

∑k=1d⟨𝐋𝐕,𝐯k𝐯k⊤⟩F≤∑k∈\[d\]:σk​\(𝐗\)≥εσk\(𝐗\)\+∑k∈\[d\]:σk​\(𝐗\)<εε\.\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\leq\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{X\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{X\}\)<\\varepsilon\}\\varepsilon\.Inserting these two inequalities into equation \([41](https://arxiv.org/html/2608.23765#A1.E41)\) forQε​\(𝐙∣𝐗\)Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\), we obtain that

Qε​\(𝐙\|𝐗\)≥14​∑k=1d⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\+14​∑k=1d⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F\+12​⟨W𝐗,ε​\(𝐙\),𝐙⟩F\.Q\_\{\\varepsilon\}\\left\(\\mathbf\{Z\}\|\\mathbf\{X\}\\right\)\\geq\\frac\{1\}\{4\}\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\frac\{1\}\{4\}\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\.By applying Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2), we obtain that

Qε​\(𝐙∣𝐗\)≥\\displaystyle Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq12∑k=1d\[⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F2\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F2⏟=:αk\+σk2​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F/2\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F/2\]\\displaystyle\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{d\}\\Big\[\\underset\{\{=:\\alpha\_\{k\}\}\}\{\\underbrace\{\\frac\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\{2\}\+\\frac\{\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\{2\}\}\}\+\\frac\{\\sigma^\{2\}\_\{k\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2\}\\Big\]=\\displaystyle=12​∑k=1d1αk​\(αk2\+σk2​\(𝐙\)\)\.\\displaystyle\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{d\}\\frac\{1\}\{\\alpha\_\{k\}\}\\left\(\\alpha\_\{k\}^\{2\}\+\\sigma^\{2\}\_\{k\}\(\\mathbf\{Z\}\)\\right\)\.\(42\)Now note that𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are positive semidefinite matrices with eigenvalues lower bounded byε\\varepsilon\. Thus, we have that⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F≥ε\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\geq\\varepsilonand⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F≥ε\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\geq\\varepsilonfor allk=1,2,…,dk=1,2,\\ldots,d\. In particular, we have thatαk≥ε\\alpha\_\{k\}\\geq\\varepsilonfor allk=1,2,…,dk=1,2,\\ldots,d\. We now estimate the scalar summands in \([42](https://arxiv.org/html/2608.23765#A1.E42)\)\. Ifσk​\(𝐙\)≥ε\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\geq\\varepsilon, thenαk2\+σk2​\(𝐙\)≥2​αk​σk​\(𝐙\)\\alpha\_\{k\}^\{2\}\+\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\geq 2\\alpha\_\{k\}\\sigma\_\{k\}\(\\mathbf\{Z\}\), and hence

12​αk​\(αk2\+σk2​\(𝐙\)\)≥σk​\(𝐙\)\.\\frac\{1\}\{2\\alpha\_\{k\}\}\\left\(\\alpha\_\{k\}^\{2\}\+\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\right\)\\geq\\sigma\_\{k\}\(\\mathbf\{Z\}\)\.Ifσk​\(𝐙\)<ε\\sigma\_\{k\}\(\\mathbf\{Z\}\)<\\varepsilon, thenαk≥ε\>σk​\(𝐙\)\\alpha\_\{k\}\\geq\\varepsilon\>\\sigma\_\{k\}\(\\mathbf\{Z\}\)\. For fixedσk​\(𝐙\)\\sigma\_\{k\}\(\\mathbf\{Z\}\), the functionx↦\(x2\+σk2​\(𝐙\)\)/\(2​x\)x\\mapsto\(x^\{2\}\+\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\)/\(2x\)is monotonically increasing forx≥σk​\(𝐙\)x\\geq\\sigma\_\{k\}\(\\mathbf\{Z\}\)\. Therefore,

12​αk​\(αk2\+σk2​\(𝐙\)\)≥12​ε​\(ε2\+σk2​\(𝐙\)\)=σk2​\(𝐙\)2​ε\+ε2\.\\frac\{1\}\{2\\alpha\_\{k\}\}\\left\(\\alpha\_\{k\}^\{2\}\+\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\right\)\\geq\\frac\{1\}\{2\\varepsilon\}\\left\(\\varepsilon^\{2\}\+\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\right\)=\\frac\{\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\}\.Summing these two cases gives

12∑k=1d1αk\(αk2\+σk2\(𝐙\)\)≥∑k∈\[d\]:σk​\(𝐙\)≥εσk\(𝐙\)\+∑k∈\[d\]:σk​\(𝐙\)<ε\(σk2​\(𝐙\)2​ε\+ε2\)\.\\frac\{1\}\{2\}\\sum\_\{k=1\}^\{d\}\\frac\{1\}\{\\alpha\_\{k\}\}\\left\(\\alpha\_\{k\}^\{2\}\+\\sigma^\{2\}\_\{k\}\(\\mathbf\{Z\}\)\\right\)\\geq\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{Z\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{Z\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{Z\}\)<\\varepsilon\}\\left\(\\frac\{\\sigma^\{2\}\_\{k\}\(\\mathbf\{Z\}\)\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\}\\right\)\.Inserting this case distinction into Equation \([42](https://arxiv.org/html/2608.23765#A1.E42)\), we obtain that

Qε\(𝐙∣𝐗\)≥∑k∈\[d\]:σk​\(𝐙\)≥εσk\(𝐙\)\+∑k∈\[d\]:σk​\(𝐙\)<ε\(σk2​\(𝐙\)2​ε\+ε2\)=∑k=1djε\(σk\(𝐙\)\)=𝒥ε\(𝐙\)\.Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{Z\}\)\\geq\\varepsilon\}\\sigma\_\{k\}\(\\mathbf\{Z\}\)\+\\sum\_\{k\\in\[d\]:\\ \\sigma\_\{k\}\(\\mathbf\{Z\}\)<\\varepsilon\}\\left\(\\frac\{\\sigma^\{2\}\_\{k\}\(\\mathbf\{Z\}\)\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\}\\right\)=\\sum\_\{k=1\}^\{d\}j\_\{\\varepsilon\}\(\\sigma\_\{k\}\(\\mathbf\{Z\}\)\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\.This completes the proof\.

#### A\.1\.1Proof of Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2)via Iterative Pinching

In this section, we provide a complete, constructive proof of[Lemma2](https://arxiv.org/html/2608.23765#Thmlemma2)\. We first establish a Sylvester equation characterization of the weight operator \([Lemma3](https://arxiv.org/html/2608.23765#Thmlemma3)\) and an abstract majorization lemma \([Lemma4](https://arxiv.org/html/2608.23765#Thmlemma4)\), that we subsequently specialize to the case of the relevant Sylvester equation \([Lemma5](https://arxiv.org/html/2608.23765#Thmlemma5)\), before introducing the iterative pinching argument via[Lemma6](https://arxiv.org/html/2608.23765#Thmlemma6)and[Lemma7](https://arxiv.org/html/2608.23765#Thmlemma7)\. The proof of[Lemma2](https://arxiv.org/html/2608.23765#Thmlemma2)is completed at the end of this subsection\.

##### Sylvester Equation and Abstract Majorization Lemma\.

The first observation we make is that the harmonic\-mean weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)satisfies the Sylvester equation[eq\.43](https://arxiv.org/html/2608.23765#A1.E43)below\.

###### Lemma 3\(Sylvester Equation Characterization\)\.

Let𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}andε\>0\\varepsilon\>0\. Let the SVD of𝐗\\mathbf\{X\}be given by𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}, where𝛔=\(σ1,…,σd\)∈ℝd\\bm\{\\sigma\}=\(\\sigma\_\{1\},\\ldots,\\sigma\_\{d\}\)\\in\\R^\{d\}are the singular values of𝐗\\mathbf\{X\}ordered in nonincreasing order\. Fori∈\[max⁡\(d1,d2\)\]i\\in\[\\max\\left\(d\_\{1\},d\_\{2\}\\right\)\], setλi:=max⁡\(σi,ε\)\\lambda\_\{i\}:=\\max\\left\(\\sigma\_\{i\},\\varepsilon\\right\)with the convention ofσi=0\\sigma\_\{i\}=0fori\>di\>d\. Define the vectors𝛌1:=\(λ1,…,λd1\)∈ℝd1\\bm\{\\lambda\}\_\{1\}:=\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\in\\R^\{d\_\{1\}\}and𝛌2:=\(λ1,…,λd2\)∈ℝd2\\bm\{\\lambda\}\_\{2\}:=\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{2\}\}\)\\in\\R^\{d\_\{2\}\}\. Then, for any𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, its image via the harmonic\-mean weight operatorW𝐗,ε​\(𝐙\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)with core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)is the unique solution of the Sylvester equation

𝐋𝐔​W𝐗,ε​\(𝐙\)\+W𝐗,ε​\(𝐙\)​𝐋𝐕=2​𝐙,\\mathbf\{L\}\_\{\\mathbf\{U\}\}W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\+W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}=2\\mathbf\{Z\},\(43\)where𝐋𝐔=𝐔𝐗​diag⁡\(𝛌1\)​𝐔𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{U\}\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\lambda\}\_\{1\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}and𝐋𝐕=𝐕𝐗​diag⁡\(𝛌2\)​𝐕𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{V\}\}=\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\lambda\}\_\{2\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\.

We expect that Lemma[3](https://arxiv.org/html/2608.23765#Thmlemma3)is known in the literature\([Bhatia, 1997](https://arxiv.org/html/2608.23765#bib.bib60), see, e\.g\.,\)\. However, for the sake of completeness, we provide a proof here\.

Proof of[Lemma3](https://arxiv.org/html/2608.23765#Thmlemma3)First, we note that both𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are positive definite\. Then it follows from[Bhatia \(1997, Theorem VII\.2\.3\)](https://arxiv.org/html/2608.23765#bib.bib60)that the Sylvester equation𝐋𝐔​𝐖\+𝐖𝐋𝐕=2​𝐙\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{W\}\+\\mathbf\{W\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}=2\\mathbf\{Z\}has a unique solution𝐖∈ℝd1×d2\\mathbf\{W\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.

In order to show that𝐖=W𝐗,ε​\(𝐙\)\\mathbf\{W\}=W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)solves the Sylvester equation, we compute that

𝐔𝐗⊤​𝐋𝐔​W𝐗,ε​\(𝐙\)​𝐕𝐗​=\(a\)\\displaystyle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\overset\{\(a\)\}\{=\}𝐔𝐗⊤​𝐋𝐔​\(𝐔𝐗​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\]\)​𝐕𝐗⊤​𝐕𝐗\\displaystyle\\mathbf\{U\}^\{\\top\}\_\{\\mathbf\{X\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\left\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\]\\right\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}=\(b\)\\displaystyle\\overset\{\(b\)\}\{=\}𝐔𝐗⊤​\(𝐔𝐗​diag⁡\(𝝀1\)​𝐔𝐗⊤\)​\(𝐔𝐗​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\]​𝐕𝐗⊤\)​𝐕𝐗\\displaystyle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\left\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\lambda\}\_\{1\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\right\)\\left\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\right\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}=\\displaystyle=diag⁡\(𝝀1\)​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\]\.\\displaystyle\\diag\(\\bm\{\\lambda\}\_\{1\}\)\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\]\.In step \(a\), we have used the definition of the weight operatorW𝐗,ε​\(𝐙\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\), see equation \([7](https://arxiv.org/html/2608.23765#S3.E7)\), and in step \(b\), we have used the definition of𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}\. Note that sinceλi=max⁡\(σi,ε\)\\lambda\_\{i\}=\\max\(\\sigma\_\{i\},\\varepsilon\), the\(i,j\)\(i,j\)\-th entry of the last expression is then given by

\(𝐔𝐗⊤​𝐋𝐔​W𝐗,ε​\(𝐙\)​𝐕𝐗\)i,j\\displaystyle\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}=2​max⁡\(σi,ε\)max⁡\(σi,ε\)\+max⁡\(σj,ε\)⋅\(𝐔𝐗⊤​𝐙𝐕𝐗\)i,j\.\\displaystyle=\\frac\{2\\max\(\\sigma\_\{i\},\\varepsilon\)\}\{\\max\(\\sigma\_\{i\},\\varepsilon\)\+\\max\(\\sigma\_\{j\},\\varepsilon\)\}\\cdot\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}\.Analogously, we compute that

\(𝐔𝐗⊤​W𝐗,ε​\(𝐙\)​𝐋𝐕​𝐕𝐗\)i,j=\\displaystyle\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}=2​max⁡\(σj,ε\)max⁡\(σi,ε\)\+max⁡\(σj,ε\)⋅\(𝐔𝐗⊤​𝐙𝐕𝐗\)i,j\.\\displaystyle\\frac\{2\\max\(\\sigma\_\{j\},\\varepsilon\)\}\{\\max\(\\sigma\_\{i\},\\varepsilon\)\+\\max\(\\sigma\_\{j\},\\varepsilon\)\}\\cdot\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}\.Summing up both terms, we obtain that

𝐔𝐗⊤​\(𝐋𝐔​W𝐗,ε​\(𝐙\)\+W𝐗,ε​\(𝐙\)​𝐋𝐕\)​𝐕𝐗=2​𝐔𝐗⊤​𝐙𝐕𝐗\.\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\left\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\+W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\right\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}=2\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\.Multiplying from the left with𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}and from the right with𝐕𝐗⊤\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}, we obtain equation \([43](https://arxiv.org/html/2608.23765#A1.E43)\)\. This completes the proof\. We now present a constructive proof of[Lemma2](https://arxiv.org/html/2608.23765#Thmlemma2)\. With the Sylvester characterization of the weight operator of[Lemma3](https://arxiv.org/html/2608.23765#Thmlemma3)at hand, it proceeds by an iterative pinching argument\. It provides structural insights into explicit intermediate solutions and the Loewner\-monotonicity structure underlying the power\-mean family\. We outline the overall proof strategy for[Lemma2](https://arxiv.org/html/2608.23765#Thmlemma2)\.

To this end, we start with the following general majorization lemma\.

###### Lemma 4\(Abstract Majorization Lemma\)\.

Letℬ,ℬ~:ℝd1×d2→ℝd1×d2\\mathcal\{B\},\\tilde\{\\mathcal\{B\}\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}be linear operators\. Let𝐏𝐔∈ℝd1×d1,𝐏𝐕∈ℝd2×d2\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\},\\mathbf\{P\}\_\{\\mathbf\{V\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}be orthogonal projection matrices\. Denote by𝐏𝐔,⊥:=𝐈−𝐏𝐔\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}:=\\mathbf\{I\}\-\\mathbf\{P\}\_\{\\mathbf\{U\}\}and𝐏𝐕,⊥:=𝐈−𝐏𝐕\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}:=\\mathbf\{I\}\-\\mathbf\{P\}\_\{\\mathbf\{V\}\}the projections onto the orthogonal complement\. Define the operation𝒞⁡\(𝐀\):=𝐏𝐔​𝐀𝐏𝐕\+𝐏𝐔,⊥​𝐀𝐏𝐕,⊥\\mathcal\{C\}\(\\mathbf\{A\}\):=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}for any matrix𝐀∈ℝd1×d2\\mathbf\{A\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. Let now𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be fixed\. Assume that the following properties hold:

1. 1\.Assume that𝐙=𝒞⁡\(𝐙\)\\mathbf\{Z\}=\\mathcal\{C\}\(\\mathbf\{Z\}\)\.
2. 2\.ℬ\\mathcal\{B\}is self\-adjoint, i\.e\.,⟨ℬ⁡\(𝐖1\),𝐖2⟩F=⟨𝐖1,ℬ⁡\(𝐖2\)⟩F\\langle\\mathcal\{B\}\(\\mathbf\{W\}\_\{1\}\),\\mathbf\{W\}\_\{2\}\\rangle\_\{F\}=\\langle\\mathbf\{W\}\_\{1\},\\mathcal\{B\}\(\\mathbf\{W\}\_\{2\}\)\\rangle\_\{F\}for all𝐖1,𝐖2∈ℝd1×d2\\mathbf\{W\}\_\{1\},\\mathbf\{W\}\_\{2\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. In addition,ℬ\\mathcal\{B\}is positive definite, i\.e\.,⟨ℬ⁡\(𝐖1\),𝐖1⟩F\>0\\langle\\mathcal\{B\}\(\\mathbf\{W\}\_\{1\}\),\\mathbf\{W\}\_\{1\}\\rangle\_\{F\}\>0for all𝐖1∈ℝd1×d2\\\{𝟎\}\\mathbf\{W\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\\backslash\\\{\\mathbf\{0\}\\\}\.
3. 3\.ℬ~\\tilde\{\\mathcal\{B\}\}is self\-adjoint and positive definite\.
4. 4\.It holds that 𝒞⁡\(ℬ~​\(𝒞⁡\(𝐖^\)\)\)=𝒞⁡\(ℬ⁡\(𝒞⁡\(𝐖^\)\)\)\\mathcal\{C\}\(\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)=\\mathcal\{C\}\(\\mathcal\{B\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)for any matrix𝐖^∈ℝd1×d2\\hat\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.
5. 5\.It holds that 𝐏𝐔​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕,⊥\+𝐏𝐔,⊥​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}\(44\)for any matrix𝐖^∈ℝd1×d2\\hat\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.

Define𝐖:=ℬ−1​\(𝐙\)\\mathbf\{W\}:=\\mathcal\{B\}^\{\-1\}\(\\mathbf\{Z\}\)and𝐖~:=ℬ~−1​\(𝐙\)\\widetilde\{\\mathbf\{W\}\}:=\\tilde\{\\mathcal\{B\}\}^\{\-1\}\(\\mathbf\{Z\}\)\. Then it holds that

⟨𝐙,𝐖⟩F≥⟨𝐙,𝐖~⟩F\.\\langle\\mathbf\{Z\},\\mathbf\{W\}\\rangle\_\{F\}\\geq\\langle\\mathbf\{Z\},\\widetilde\{\\mathbf\{W\}\}\\rangle\_\{F\}\.

ProofLet𝒯:\(ℝd1×d2,⟨⋅,⋅⟩F\)→\(ℝd1​d2,⟨⋅,⋅⟩ℓ2\)\\mathcal\{T\}:\(\\R^\{d\_\{1\}\\times d\_\{2\}\},\\langle\\cdot,\\cdot\\rangle\_\{F\}\)\\rightarrow\(\\mathbb\{R\}^\{d\_\{1\}d\_\{2\}\},\\langle\\cdot,\\cdot\\rangle\_\{\\ell^\{2\}\}\)be an isometric vectorization operator such that

𝒯⁡\(𝐀\)=\(𝐚diag𝐚off\)\\mathcal\{T\}\(\\mathbf\{A\}\)=\\begin\{pmatrix\}\\mathbf\{a\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{a\}\_\{\\text\{off\}\}\\end\{pmatrix\}and such that𝐚diag=𝟎\\mathbf\{a\}\_\{\\text\{diag\}\}=\\mathbf\{0\}if and only if𝒞⁡\(𝐀\)=𝟎\\mathcal\{C\}\(\\mathbf\{A\}\)=\\mathbf\{0\}and𝐚off=𝟎\\mathbf\{a\}\_\{\\text\{off\}\}=\\mathbf\{0\}if and only if\(ℐ−𝒞\)​\(𝐀\)=𝟎\\left\(\\mathcal\{I\}\-\\mathcal\{C\}\\right\)\(\\mathbf\{A\}\)=\\mathbf\{0\}for any matrix𝐀∈ℝd1×d2\\mathbf\{A\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. \(In other words, the operator𝒯\\mathcal\{T\}splits the vectorization of a matrix𝐀\\mathbf\{A\}into the “diagonal” part𝐏𝐔​𝐀𝐏𝐕\+𝐏𝐔,⊥​𝐀𝐏𝐕,⊥\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}and its “off\-diagonal” part𝐏𝐔​𝐀𝐏𝐕,⊥\+𝐏𝐔,⊥​𝐀𝐏𝐕\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}\.\) Then we can denote the vectorization of the matrices𝐙,𝐖,𝐖~∈ℝd1×d2\\mathbf\{Z\},\\mathbf\{W\},\\widetilde\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}as

𝒯⁡\(𝐙\)=\(𝐳diag𝐳off\),𝒯⁡\(𝐖\)=\(𝐰diag𝐰off\),𝒯⁡\(𝐖~\)=\(𝐰~diag𝐰~off\)\.\\mathcal\{T\}\(\\mathbf\{Z\}\)=\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{z\}\_\{\\text\{off\}\}\\end\{pmatrix\},\\quad\\mathcal\{T\}\(\\mathbf\{W\}\)=\\begin\{pmatrix\}\\mathbf\{w\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{w\}\_\{\\text\{off\}\}\\end\{pmatrix\},\\quad\\mathcal\{T\}\(\\widetilde\{\\mathbf\{W\}\}\)=\\begin\{pmatrix\}\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\\\ \\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}\\end\{pmatrix\}\.Using the operator𝒯\\mathcal\{T\}, the equation𝐙=ℬ⁡\(𝐖\)\\mathbf\{Z\}=\\mathcal\{B\}\(\\mathbf\{W\}\)can then be rewritten as

\(𝐳diag𝐳off\)=\(𝒯∘ℬ∘𝒯−1\)​\(𝐰diag𝐰off\)=\(𝐀1,1𝐀1,2𝐀1,2⊤𝐀2,2\)​\(𝐰diag𝐰off\)\.\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{z\}\_\{\\text\{off\}\}\\end\{pmatrix\}=\(\\mathcal\{T\}\\circ\\mathcal\{B\}\\circ\\mathcal\{T\}^\{\-1\}\)\\begin\{pmatrix\}\\mathbf\{w\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{w\}\_\{\\text\{off\}\}\\end\{pmatrix\}=\\begin\{pmatrix\}\\mathbf\{A\}\_\{\\text\{1,1\}\}&\\mathbf\{A\}\_\{\\text\{1,2\}\}\\\\ \\mathbf\{A\}\_\{\\text\{1,2\}\}^\{\\top\}&\\mathbf\{A\}\_\{\\text\{2,2\}\}\\end\{pmatrix\}\\begin\{pmatrix\}\\mathbf\{w\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{w\}\_\{\\text\{off\}\}\\end\{pmatrix\}\.\(45\)In the last step, we have used that the operator𝒯∘ℬ∘𝒯−1\\mathcal\{T\}\\circ\\mathcal\{B\}\\circ\\mathcal\{T\}^\{\-1\}can be represented as a symmetric block matrix sinceℬ\\mathcal\{B\}is linear and self\-adjoint\. In a similar way, the equation𝐖~=ℬ~−1​\(𝐙\)\\widetilde\{\\mathbf\{W\}\}=\\tilde\{\\mathcal\{B\}\}^\{\-1\}\(\\mathbf\{Z\}\)can be rewritten as

\(𝐳diag𝐳off\)=\(𝒯∘ℬ~∘𝒯−1\)​\(𝐰~diag𝐰~off\)=\(𝐀~1,1𝐀~1,2𝐀~1,2⊤𝐀~2,2\)​\(𝐰~diag𝐰~off\)\.\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{z\}\_\{\\text\{off\}\}\\end\{pmatrix\}=\(\\mathcal\{T\}\\circ\\tilde\{\\mathcal\{B\}\}\\circ\\mathcal\{T\}^\{\-1\}\)\\begin\{pmatrix\}\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\\\ \\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}\\end\{pmatrix\}=\\begin\{pmatrix\}\\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,1\}\}&\\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,2\}\}\\\\ \\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,2\}\}^\{\\top\}&\\tilde\{\\mathbf\{A\}\}\_\{\\text\{2,2\}\}\\end\{pmatrix\}\\begin\{pmatrix\}\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\\\ \\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}\\end\{pmatrix\}\.\(46\)Again, in the last equation, we have used that the operator𝒯∘ℬ~∘𝒯−1\\mathcal\{T\}\\circ\\tilde\{\\mathcal\{B\}\}\\circ\\mathcal\{T\}^\{\-1\}can be represented as a symmetric block matrix sinceℬ~\\tilde\{\\mathcal\{B\}\}is linear and self\-adjoint\. Now we note that since𝐙=𝒞⁡\(𝐙\)\\mathbf\{Z\}=\\mathcal\{C\}\(\\mathbf\{Z\}\)we have that𝐏𝐔,⊥​𝐙𝐏𝐕=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}and𝐏𝐔​𝐙𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}\. Thus, by the definition of the operator𝒯\\mathcal\{T\}, it holds that𝐳off=𝟎\\mathbf\{z\}\_\{\\text\{off\}\}=\\mathbf\{0\}\.

Now recall the assumption that, for any matrix𝐖^∈ℝd1×d2\\hat\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, we have that𝒞⁡\(ℬ~​\(𝒞⁡\(𝐖^\)\)\)=𝒞⁡\(ℬ⁡\(𝒞⁡\(𝐖^\)\)\)\\mathcal\{C\}\(\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)=\\mathcal\{C\}\(\\mathcal\{B\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)\. Now note that, since𝒯\\mathcal\{T\}is an isometric isomorphism, this implies that

𝒯∘\(𝒞∘ℬ~∘𝒞\)∘𝒯−1=𝒯∘\(𝒞∘ℬ∘𝒞\)∘𝒯−1\.\\mathcal\{T\}\\circ\(\\mathcal\{C\}\\circ\\tilde\{\\mathcal\{B\}\}\\circ\\mathcal\{C\}\)\\circ\\mathcal\{T\}^\{\-1\}=\\mathcal\{T\}\\circ\(\\mathcal\{C\}\\circ\\mathcal\{B\}\\circ\\mathcal\{C\}\)\\circ\\mathcal\{T\}^\{\-1\}\.We compute that

𝒯∘\(𝒞∘ℬ~∘𝒞\)∘𝒯−1=\\displaystyle\\mathcal\{T\}\\circ\(\\mathcal\{C\}\\circ\\tilde\{\\mathcal\{B\}\}\\circ\\mathcal\{C\}\)\\circ\\mathcal\{T\}^\{\-1\}=\(𝒯∘𝒞∘𝒯−1\)∘\(𝒯∘ℬ~∘𝒯−1\)∘\(𝒯∘𝒞∘𝒯−1\)\\displaystyle\\left\(\\mathcal\{T\}\\circ\\mathcal\{C\}\\circ\\mathcal\{T\}^\{\-1\}\\right\)\\circ\\left\(\\mathcal\{T\}\\circ\\tilde\{\\mathcal\{B\}\}\\circ\\mathcal\{T\}^\{\-1\}\\right\)\\circ\\left\(\\mathcal\{T\}\\circ\\mathcal\{C\}\\circ\\mathcal\{T\}^\{\-1\}\\right\)=\\displaystyle=\(𝐈𝟎𝟎𝟎\)​\(𝐀~1,1𝐀~1,2𝐀~1,2⊤𝐀~2,2\)​\(𝐈𝟎𝟎𝟎\)=\(𝐀~1,1𝟎𝟎𝟎\),\\displaystyle\\begin\{pmatrix\}\\mathbf\{I\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}\\end\{pmatrix\}\\begin\{pmatrix\}\\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,1\}\}&\\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,2\}\}\\\\ \\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,2\}\}^\{\\top\}&\\tilde\{\\mathbf\{A\}\}\_\{\\text\{2,2\}\}\\end\{pmatrix\}\\begin\{pmatrix\}\\mathbf\{I\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}\\end\{pmatrix\}=\\begin\{pmatrix\}\\tilde\{\\mathbf\{A\}\}\_\{\\text\{1,1\}\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}\\end\{pmatrix\},where we have identified the operators with their matrix representations\. In a similar way, we obtain that

𝒯∘\(𝒞∘ℬ∘𝒞\)∘𝒯−1=\(𝐀1,1𝟎𝟎𝟎\)\.\\mathcal\{T\}\\circ\(\\mathcal\{C\}\\circ\\mathcal\{B\}\\circ\\mathcal\{C\}\)\\circ\\mathcal\{T\}^\{\-1\}=\\begin\{pmatrix\}\\mathbf\{A\}\_\{\\text\{1,1\}\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}\\end\{pmatrix\}\.Thus, the assumption𝒞⁡\(ℬ~​\(𝒞⁡\(𝐖^\)\)\)=𝒞⁡\(ℬ⁡\(𝒞⁡\(𝐖^\)\)\)\\mathcal\{C\}\(\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)=\\mathcal\{C\}\(\\mathcal\{B\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)is equivalent to𝐀~1,1=𝐀1,1\\tilde\{\\mathbf\{A\}\}\_\{1,1\}=\\mathbf\{A\}\_\{1,1\}in the vectorized representation\. Next, assumption \([44](https://arxiv.org/html/2608.23765#A1.E44)\) is equivalent to𝐀~2,1=𝐀~1,2⊤=𝟎\\tilde\{\\mathbf\{A\}\}\_\{2,1\}=\\tilde\{\\mathbf\{A\}\}\_\{1,2\}^\{\\top\}=\\mathbf\{0\}in the vectorized representation\. Note that𝐀~1,2=𝟎\\tilde\{\\mathbf\{A\}\}\_\{1,2\}=\\mathbf\{0\}also implies that𝐰~off=𝟎\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}=\\mathbf\{0\}due to equation \([46](https://arxiv.org/html/2608.23765#A1.E46)\) and due to the fact that𝐳off=𝟎\\mathbf\{z\}\_\{\\text\{off\}\}=\\mathbf\{0\}, together with the invertibility of𝐀~2,2\\tilde\{\\mathbf\{A\}\}\_\{2,2\}\(which is a principal block of the positive definite matrix representingℬ~\\tilde\{\\mathcal\{B\}\}\)\.

Inserting these identities into \([45](https://arxiv.org/html/2608.23765#A1.E45)\) and \([46](https://arxiv.org/html/2608.23765#A1.E46)\), we obtain that

\(𝐳diag𝟎\)=\(𝐀1,1𝐀1,2𝐀1,2⊤𝐀2,2\)​\(𝐰diag𝐰off\),\(𝐳diag𝟎\)=\(𝐀1,1𝟎𝟎𝐀~2,2\)​\(𝐰~diag𝐰~off\)\.\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{0\}\\end\{pmatrix\}=\\begin\{pmatrix\}\\mathbf\{A\}\_\{\\text\{1,1\}\}&\\mathbf\{A\}\_\{\\text\{1,2\}\}\\\\ \\mathbf\{A\}\_\{\\text\{1,2\}\}^\{\\top\}&\\mathbf\{A\}\_\{\\text\{2,2\}\}\\end\{pmatrix\}\\begin\{pmatrix\}\\mathbf\{w\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{w\}\_\{\\text\{off\}\}\\end\{pmatrix\},\\quad\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{0\}\\end\{pmatrix\}=\\begin\{pmatrix\}\\mathbf\{A\}\_\{\\text\{1,1\}\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\tilde\{\\mathbf\{A\}\}\_\{\\text\{2,2\}\}\\end\{pmatrix\}\\begin\{pmatrix\}\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\\\ \\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}\\end\{pmatrix\}\.From the first equation, we obtain that

𝐳diag=𝐀1,1​𝐰diag\+𝐀1,2​𝐰off,\\displaystyle\\mathbf\{z\}\_\{\\text\{diag\}\}=\\mathbf\{A\}\_\{\\text\{1,1\}\}\\mathbf\{w\}\_\{\\text\{diag\}\}\+\\mathbf\{A\}\_\{\\text\{1,2\}\}\\mathbf\{w\}\_\{\\text\{off\}\},𝟎=𝐀1,2⊤​𝐰diag\+𝐀2,2​𝐰off\.\\displaystyle\\mathbf\{0\}=\\mathbf\{A\}\_\{\\text\{1,2\}\}^\{\\top\}\\mathbf\{w\}\_\{\\text\{diag\}\}\+\\mathbf\{A\}\_\{\\text\{2,2\}\}\\mathbf\{w\}\_\{\\text\{off\}\}\.Sinceℬ\\mathcal\{B\}is positive definite, the block matrix in equation \([45](https://arxiv.org/html/2608.23765#A1.E45)\) is positive definite as well\. Then the Schur complement𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}is positive definite as well and thus invertible\. Therefore, we obtain by a direct calculation that

𝐰diag=\(𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\)−1​𝐳diag\.\\mathbf\{w\}\_\{\\text\{diag\}\}=\\left\(\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\right\)^\{\-1\}\\mathbf\{z\}\_\{\\text\{diag\}\}\.Moreover, from equation \([46](https://arxiv.org/html/2608.23765#A1.E46)\) we obtain due to𝐳off=𝟎\\mathbf\{z\}\_\{\\text\{off\}\}=\\mathbf\{0\}and𝐀~1,2=𝟎\\tilde\{\\mathbf\{A\}\}\_\{1,2\}=\\mathbf\{0\}that

𝐰~diag=𝐀1,1−1​𝐳diag,\\displaystyle\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}=\\mathbf\{A\}\_\{1,1\}^\{\-1\}\\mathbf\{z\}\_\{\\text\{diag\}\},since𝐀1,1\\mathbf\{A\}\_\{1,1\}is invertible as a principal submatrix of a positive definite matrix\. Now we note that

⟨𝐙,𝐖−𝐖~⟩F​=\(a\)\\displaystyle\\langle\\mathbf\{Z\},\\mathbf\{W\}\-\\widetilde\{\\mathbf\{W\}\}\\rangle\_\{F\}\\overset\{\(a\)\}\{=\}⟨𝒯⁡\(𝐙\),𝒯⁡\(𝐖\)−𝒯⁡\(𝐖~\)⟩ℓ2\\displaystyle\\langle\{\\mathcal\{T\}\(\\mathbf\{Z\}\),\\mathcal\{T\}\(\\mathbf\{W\}\)\-\\mathcal\{T\}\(\\widetilde\{\\mathbf\{W\}\}\)\}\\rangle\_\{\\ell\_\{2\}\}=\(b\)\\displaystyle\\overset\{\(b\)\}\{=\}⟨\(𝐳diag𝟎\),\(𝐰diag−𝐰~diag𝐰off−𝐰~off\)⟩ℓ2\\displaystyle\\langle\\begin\{pmatrix\}\\mathbf\{z\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{0\}\\end\{pmatrix\},\\begin\{pmatrix\}\\mathbf\{w\}\_\{\\text\{diag\}\}\-\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\\\ \\mathbf\{w\}\_\{\\text\{off\}\}\-\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{off\}\}\\end\{pmatrix\}\\rangle\_\{\\ell\_\{2\}\}=\\displaystyle=⟨𝐳diag,𝐰diag−𝐰~diag⟩ℓ2\\displaystyle\\langle\\mathbf\{z\}\_\{\\text\{diag\}\},\\mathbf\{w\}\_\{\\text\{diag\}\}\-\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}\\rangle\_\{\\ell\_\{2\}\}=\(c\)\\displaystyle\\overset\{\(c\)\}\{=\}⟨𝐳diag,\[\(𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\)−1−𝐀1,1−1\]​𝐳diag⟩ℓ2\.\\displaystyle\\langle\\mathbf\{z\}\_\{\\text\{diag\}\},\\left\[\\left\(\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\right\)^\{\-1\}\-\\mathbf\{A\}\_\{1,1\}^\{\-1\}\\right\]\\mathbf\{z\}\_\{\\text\{diag\}\}\\rangle\_\{\\ell\_\{2\}\}\.\(47\)Here, in step \(a\) we have used that𝒯\\mathcal\{T\}is an isometric operator, and in step \(b\) we have used the vectorized representations of𝐙,𝐖\\mathbf\{Z\},\\mathbf\{W\}, and𝐖~\\widetilde\{\\mathbf\{W\}\}\. Step \(c\) follows from the formulas for𝐰diag\\mathbf\{w\}\_\{\\text\{diag\}\}and𝐰~diag\\widetilde\{\\mathbf\{w\}\}\_\{\\text\{diag\}\}derived above\.

Now, sinceℬ\\mathcal\{B\}is positive definite, it follows that the block matrix𝐀2,2\\mathbf\{A\}\_\{2,2\}is positive definite as well\. Thus,𝐀1,2​𝐀2,2−1​𝐀1,2⊤\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}is positive semidefinite\. It follows that

𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤⪯𝐀1,1,\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\preceq\\mathbf\{A\}\_\{1,1\},where⪯\\preceqdenotes the Loewner partial order on symmetric matrices, i\.e\.,𝐀⪯𝐁\\mathbf\{A\}\\preceq\\mathbf\{B\}if and only if𝐁−𝐀\\mathbf\{B\}\-\\mathbf\{A\}is positive semidefinite\. By the operator monotonicity of the inverse function on the cone of positive definite matrices we obtain that

\(𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\)−1⪰𝐀1,1−1\.\\left\(\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\right\)^\{\-1\}\\succeq\\mathbf\{A\}\_\{1,1\}^\{\-1\}\.It follows that

\(𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\)−1−𝐀1,1−1⪰𝟎,\\left\(\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\right\)^\{\-1\}\-\\mathbf\{A\}\_\{1,1\}^\{\-1\}\\succeq\\mathbf\{0\},which finally implies that

⟨𝐳diag,\[\(𝐀1,1−𝐀1,2​𝐀2,2−1​𝐀1,2⊤\)−1−𝐀1,1−1\]​𝐳diag⟩ℓ2≥0\.\\langle\\mathbf\{z\}\_\{\\text\{diag\}\},\\left\[\\left\(\\mathbf\{A\}\_\{1,1\}\-\\mathbf\{A\}\_\{1,2\}\\mathbf\{A\}\_\{2,2\}^\{\-1\}\\mathbf\{A\}\_\{1,2\}^\{\\top\}\\right\)^\{\-1\}\-\\mathbf\{A\}\_\{1,1\}^\{\-1\}\\right\]\\mathbf\{z\}\_\{\\text\{diag\}\}\\rangle\_\{\\ell\_\{2\}\}\\geq 0\.Combined with \([47](https://arxiv.org/html/2608.23765#A1.E47)\) this implies the claim\.

As a next step, we specialize Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)to the case of Sylvester equations\.

###### Lemma 5\(Abstract Majorization Lemma for Sylvester Equations\)\.

Let𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be arbitrary but fixed\. Let𝐋𝐔∈ℝd1×d1\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐋𝐕∈ℝd2×d2\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}be positive definite matrices\. Let𝐏𝐔∈ℝd1×d1,𝐏𝐕∈ℝd2×d2\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\},\\mathbf\{P\}\_\{\\mathbf\{V\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}be orthogonal projection matrices\. Denote by𝐏𝐔,⊥:=𝐈−𝐏𝐔\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}:=\\mathbf\{I\}\-\\mathbf\{P\}\_\{\\mathbf\{U\}\}and𝐏𝐕,⊥:=𝐈−𝐏𝐕\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}:=\\mathbf\{I\}\-\\mathbf\{P\}\_\{\\mathbf\{V\}\}the projections onto the orthogonal complement\. Assume that𝐏𝐔​𝐙𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}and𝐏𝐔,⊥​𝐙𝐏𝐕=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}\. Define the operator𝒞:ℝd1×d2→ℝd1×d2\\mathcal\{C\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}by𝒞⁡\(𝐀\):=𝐏𝐔​𝐀𝐏𝐕\+𝐏𝐔,⊥​𝐀𝐏𝐕,⊥\\mathcal\{C\}\(\\mathbf\{A\}\):=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{A\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}for any matrix𝐀∈ℝd1×d2\\mathbf\{A\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. Then the following statements hold true:

1. 1\.The equation 𝐋𝐔​𝐖\+𝐖𝐋𝐕=2​𝐙\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{W\}\+\\mathbf\{W\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}=2\\mathbf\{Z\}\(48\)has a unique solution𝐖∈ℝd1×d2\\mathbf\{W\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.
2. 2\.Set𝒞𝐔:ℝd1×d1→ℝd1×d1\\mathcal\{C\}\_\{\\mathbf\{U\}\}:\\R^\{d\_\{1\}\\times d\_\{1\}\}\\to\\R^\{d\_\{1\}\\times d\_\{1\}\}with𝒞𝐔​\(𝐀1\):=𝐏𝐔​𝐀1​𝐏𝐔\+𝐏𝐔,⊥​𝐀1​𝐏𝐔,⊥\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{A\}\_\{1\}\):=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}and𝒞𝐕:ℝd2×d2→ℝd2×d2\\mathcal\{C\}\_\{\\mathbf\{V\}\}:\\R^\{d\_\{2\}\\times d\_\{2\}\}\\to\\R^\{d\_\{2\}\\times d\_\{2\}\}with𝒞𝐕​\(𝐀2\):=𝐏𝐕​𝐀2​𝐏𝐕\+𝐏𝐕,⊥​𝐀2​𝐏𝐕,⊥\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{A\}\_\{2\}\):=\\mathbf\{P\}\_\{\\mathbf\{V\}\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}for any matrices𝐀1∈ℝd1×d1\\mathbf\{A\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐀2∈ℝd2×d2\\mathbf\{A\}\_\{2\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}, respectively\. Then, the equation 𝒞𝐔​\(𝐋𝐔\)​𝐖~\+𝐖~​𝒞𝐕​\(𝐋𝐕\)=2​𝐙\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\widetilde\{\\mathbf\{W\}\}\+\\widetilde\{\\mathbf\{W\}\}\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)=2\\mathbf\{Z\}\(49\)has a unique solution𝐖~∈ℝd1×d2\\widetilde\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.
3. 3\.It holds that ⟨𝐙,𝐖−𝐖~⟩F≥0\.\\langle\\mathbf\{Z\},\\mathbf\{W\}\-\\widetilde\{\\mathbf\{W\}\}\\rangle\_\{F\}\\geq 0\.

ProofWith the same argument as in the proof of Lemma[3](https://arxiv.org/html/2608.23765#Thmlemma3), it follows from the theory of Sylvester equations that Equation \([48](https://arxiv.org/html/2608.23765#A1.E48)\) has a unique solution𝐖∈ℝd1×d2\\mathbf\{W\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}since𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are positive definite,\(see, e\.g\.,[Bhatia, 1997](https://arxiv.org/html/2608.23765#bib.bib60), Theorem VII\.2\.3\)\. This proves the first statement\.

Now, note that𝒞𝐔​\(𝐋𝐔\)\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)and𝒞𝐕​\(𝐋𝐕\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)are positive definite as well since the pinching operations𝒞𝐔\\mathcal\{C\}\_\{\\mathbf\{U\}\}and𝒞𝐕\\mathcal\{C\}\_\{\\mathbf\{V\}\}preserve positive definiteness as one can easily verify\. This fact implies that equation \([49](https://arxiv.org/html/2608.23765#A1.E49)\) has a unique solution𝐖~∈ℝd1×d2\\widetilde\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}as well\. This proves the second statement\.

It remains to prove the third statement\. Define the operatorℬ:ℝd1×d2→ℝd1×d2\\mathcal\{B\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}by

ℬ⁡\(𝐀\):=12​\(𝐋𝐔​𝐀\+𝐀𝐋𝐕\)\\mathcal\{B\}\(\\mathbf\{A\}\):=\\frac\{1\}\{2\}\\left\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{A\}\+\\mathbf\{A\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\right\)and define the operatorℬ~:ℝd1×d2→ℝd1×d2\\tilde\{\\mathcal\{B\}\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R^\{d\_\{1\}\\times d\_\{2\}\}by

ℬ~​\(𝐀\):=12​\(𝒞𝐔​\(𝐋𝐔\)​𝐀\+𝐀​𝒞𝐕​\(𝐋𝐕\)\)\.\\tilde\{\\mathcal\{B\}\}\(\\mathbf\{A\}\):=\\frac\{1\}\{2\}\\left\(\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathbf\{A\}\+\\mathbf\{A\}\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\right\)\.Our goal is to apply the previous Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)\. For this, we need to verify that the assumptions of Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)are satisfied\. The first assumption is satisfied since we have assumed that𝐏𝐔​𝐙𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}and𝐏𝐔,⊥​𝐙𝐏𝐕=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}, which is equivalent to𝐙=𝒞⁡\(𝐙\)\\mathbf\{Z\}=\\mathcal\{C\}\(\\mathbf\{Z\}\)\. Next, note that for all matrices𝐙1,𝐙2∈ℝd1×d2\\mathbf\{Z\}\_\{1\},\\mathbf\{Z\}\_\{2\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}it holds that

⟨ℬ⁡\(𝐙1\),𝐙2⟩F=\\displaystyle\\langle\\mathcal\{B\}\(\\mathbf\{Z\}\_\{1\}\),\\mathbf\{Z\}\_\{2\}\\rangle\_\{F\}=12​⟨𝐋𝐔​𝐙1,𝐙2⟩F\+12​⟨𝐙1​𝐋𝐕,𝐙2⟩F\\displaystyle\\frac\{1\}\{2\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{1\},\\mathbf\{Z\}\_\{2\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\}\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{Z\}\_\{2\}\\rangle\_\{F\}=\\displaystyle=12​⟨𝐙1,𝐋𝐔​𝐙2⟩F\+12​⟨𝐙1,𝐙2​𝐋𝐕⟩F\\displaystyle\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\},\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{2\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\},\\mathbf\{Z\}\_\{2\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\rangle\_\{F\}=\\displaystyle=⟨𝐙1,ℬ⁡\(𝐙2\)⟩F\.\\displaystyle\\langle\\mathbf\{Z\}\_\{1\},\\mathcal\{B\}\(\\mathbf\{Z\}\_\{2\}\)\\rangle\_\{F\}\.Moreover, we note that

⟨ℬ⁡\(𝐙1\),𝐙1⟩F=\\displaystyle\\langle\\mathcal\{B\}\(\\mathbf\{Z\}\_\{1\}\),\\mathbf\{Z\}\_\{1\}\\rangle\_\{F\}=12​⟨𝐋𝐔​𝐙1,𝐙1⟩F\+12​⟨𝐙1​𝐋𝐕,𝐙1⟩F\\displaystyle\\frac\{1\}\{2\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{1\},\\mathbf\{Z\}\_\{1\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\}\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{Z\}\_\{1\}\\rangle\_\{F\}=\\displaystyle=12​⟨𝐙1,𝐋𝐔​𝐙1⟩F\+12​⟨𝐙1,𝐙1​𝐋𝐕⟩F\\displaystyle\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\},\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{1\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\_\{1\},\\mathbf\{Z\}\_\{1\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\rangle\_\{F\}=\\displaystyle=12​tr⁡\(𝐙1⊤​𝐋𝐔​𝐙1\)\+12​tr⁡\(𝐙1​𝐋𝐕​𝐙1⊤\)\\displaystyle\\frac\{1\}\{2\}\\trace\(\\mathbf\{Z\}\_\{1\}^\{\\top\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{1\}\)\+\\frac\{1\}\{2\}\\trace\(\\mathbf\{Z\}\_\{1\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\mathbf\{Z\}\_\{1\}^\{\\top\}\)for any𝐙1∈ℝd1×d2\\\{𝟎\}\\mathbf\{Z\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\\backslash\\\{\\mathbf\{0\}\\\}\. Since𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are positive definite matrices, we obtain that both𝐙1⊤​𝐋𝐔​𝐙1\\mathbf\{Z\}\_\{1\}^\{\\top\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\_\{1\}and𝐙1​𝐋𝐕​𝐙1⊤\\mathbf\{Z\}\_\{1\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\mathbf\{Z\}\_\{1\}^\{\\top\}are positive semidefinite matrices which are nonzero\. Thus, both terms in the last equation are positive which implies that

⟨ℬ⁡\(𝐙1\),𝐙1⟩F\>0\\langle\\mathcal\{B\}\(\\mathbf\{Z\}\_\{1\}\),\\mathbf\{Z\}\_\{1\}\\rangle\_\{F\}\>0for all𝐙1∈ℝd1×d2\\\{𝟎\}\\mathbf\{Z\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\\backslash\\\{\\mathbf\{0\}\\\}\. Thus, we have shown that the operatorℬ\\mathcal\{B\}is self\-adjoint and positive definite\. In an analogous way, one can show that the operatorℬ~\\tilde\{\\mathcal\{B\}\}is self\-adjoint and positive definite as well since we have that𝒞𝐔​\(𝐋𝐔\)\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)and𝒞𝐕​\(𝐋𝐕\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)are positive definite\. This shows that the second and third assumptions of Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)are satisfied\.

Now note that

𝐏𝐔​𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)​𝐏𝐕​=\(a\)​𝐏𝐔​𝐋𝐔​𝐏𝐔​𝒞​\(𝐖^\)​𝐏𝐕​=\(b\)​𝐏𝐔​𝐋𝐔​𝒞​\(𝐖^\)​𝐏𝐕,\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}\\overset\{\(a\)\}\{=\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}\\overset\{\(b\)\}\{=\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\},where in step \(a\) we have used that𝐏𝐔​𝒞𝐔​\(𝐋𝐔\)=𝐏𝐔​𝐋𝐔​𝐏𝐔\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}and in step \(b\) we have used that𝒞⁡\(𝐖^\)​𝐏𝐕=𝐏𝐔​𝒞​\(𝐖^\)​𝐏𝐕\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}\. In a similar way, one can verify the following three identities:

𝐏𝐔​𝒞​\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)​𝐏𝐕=\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}=𝐏𝐔​𝒞​\(𝐖^\)​𝐋𝐕​𝐏𝐕,\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\mathbf\{P\}\_\{\\mathbf\{V\}\},𝐏𝐔,⊥​𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)​𝐏𝐕,⊥=\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=𝐏𝐔,⊥​𝐋𝐔​𝒞​\(𝐖^\)​𝐏𝐕,⊥,\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\},𝐏𝐔,⊥​𝒞​\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)​𝐏𝐕,⊥=\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=𝐏𝐔,⊥​𝒞​\(𝐖^\)​𝐋𝐕​𝐏𝐕,⊥\.\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\.Using these equations we can show that for all matrices𝐖^∈ℝd1×d2\\hat\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, it holds that

𝒞​\(ℬ~​\(𝒞​\(𝐖^\)\)\)\\displaystyle\\mathcal\{C\}\(\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)=\\displaystyle=12​𝒞​\(𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)\)\\displaystyle\\frac\{1\}\{2\}\\mathcal\{C\}\\left\(\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\right\)=\\displaystyle=12​𝐏𝐔​\(𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)\)​𝐏𝐕\\displaystyle\\frac\{1\}\{2\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\left\(\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\right\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+12​𝐏𝐔,⊥​\(𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)\)​𝐏𝐕,⊥\\displaystyle\+\\frac\{1\}\{2\}\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\left\(\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\right\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\displaystyle=12​𝐏𝐔​\(𝐋𝐔​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝐋𝐕\)​𝐏𝐕\+12​𝐏𝐔,⊥​\(𝐋𝐔​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝐋𝐕\)​𝐏𝐕,⊥\\displaystyle\\frac\{1\}\{2\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\left\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\right\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}\+\\frac\{1\}\{2\}\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\left\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\right\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\displaystyle=12​𝒞​\(𝐋𝐔​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝐋𝐕\)=𝒞⁡\(ℬ⁡\(𝒞⁡\(𝐖^\)\)\)\.\\displaystyle\\frac\{1\}\{2\}\\mathcal\{C\}\\left\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\right\)=\\mathcal\{C\}\(\\mathcal\{B\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\)\.
This shows that the fourth assumption of Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)is satisfied\.

Finally, we need to verify the fifth assumption of Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)\. We note that, for any matrix𝐖^∈ℝd1×d2\\hat\{\\mathbf\{W\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, it holds that

𝐏𝐔​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕,⊥=12​𝐏𝐔​\(𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)\+𝒞⁡\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)\)​𝐏𝐕,⊥\.\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\frac\{1\}\{2\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\left\(\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\+\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\right\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\.Now note that𝐏𝐔​𝒞𝐔​\(𝐋𝐔\)=𝐏𝐔​𝐋𝐔​𝐏𝐔\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)=\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\}\}and𝒞⁡\(𝐖^\)​𝐏𝐕,⊥=𝐏𝐔,⊥​𝐖^​𝐏𝐕,⊥\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\hat\{\\mathbf\{W\}\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\. Consequently, we obtain that𝐏𝐔​𝒞𝐔​\(𝐋𝐔\)​𝒞​\(𝐖^\)​𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\_\{\\mathbf\{U\}\}\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}\. Similarly, we have𝐏𝐔​𝒞​\(𝐖^\)​𝒞𝐕​\(𝐋𝐕\)​𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\\mathcal\{C\}\_\{\\mathbf\{V\}\}\(\\mathbf\{L\}\_\{\\mathbf\{V\}\}\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}\. Thus, we obtain that𝐏𝐔​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}\. In a similar way, one can also show that𝐏𝐔,⊥​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕=𝟎\.\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}\.It follows that

𝐏𝐔​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕,⊥\+𝐏𝐔,⊥​ℬ~​\(𝒞⁡\(𝐖^\)\)​𝐏𝐕=𝟎\.\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\tilde\{\\mathcal\{B\}\}\(\\mathcal\{C\}\(\\hat\{\\mathbf\{W\}\}\)\)\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{0\}\.Thus, the fifth assumption of Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)is satisfied as well and we are in a position to apply Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)\.

Since, by definition,𝐖\\mathbf\{W\}and𝐖~\\widetilde\{\\mathbf\{W\}\}satisfyℬ⁡\(𝐖\)=𝐙\\mathcal\{B\}\(\\mathbf\{W\}\)=\\mathbf\{Z\}andℬ~​\(𝐖~\)=𝐙\\tilde\{\\mathcal\{B\}\}\(\\widetilde\{\\mathbf\{W\}\}\)=\\mathbf\{Z\}, it follows from Lemma[4](https://arxiv.org/html/2608.23765#Thmlemma4)that

⟨𝐙,𝐖−𝐖~⟩F=⟨𝐙,ℬ−1​\(𝐙\)−ℬ~−1​\(𝐙\)⟩F≥0\.\\displaystyle\\langle\\mathbf\{Z\},\\mathbf\{W\}\-\\widetilde\{\\mathbf\{W\}\}\\rangle\_\{F\}=\\langle\\mathbf\{Z\},\\mathcal\{B\}^\{\-1\}\(\\mathbf\{Z\}\)\-\\tilde\{\\mathcal\{B\}\}^\{\-1\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\geq 0\.This completes the proof\.

##### Iterative Pinching Argument\.

Let𝐋𝐔∈ℝd1×d1,𝐋𝐕∈ℝd2×d2\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\},\\mathbf\{L\}\_\{\\mathbf\{V\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}be positive definite matrices\. Let𝐮1,…,𝐮d1∈ℝd1\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\in\\R^\{d\_\{1\}\}be an orthonormal basis ofℝd1\\R^\{d\_\{1\}\}and let𝐯1,…,𝐯d2∈ℝd2\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\in\\R^\{d\_\{2\}\}be an orthonormal basis ofℝd2\\R^\{d\_\{2\}\}\. The goal is now to apply the previous lemma iteratively to obtain a majorization result for the case that both𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively\. For this, we define the orthogonal projection matrices

𝐏𝐔,i:j:=∑k=ij𝐮k𝐮k⊤for1≤i≤j≤d1,\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},i:j\}:=\\sum\_\{k=i\}^\{j\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\quad\\text\{for \}1\\leq i\\leq j\\leq d\_\{1\},𝐏𝐕,i:j:=∑k=ij𝐯k𝐯k⊤for1≤i≤j≤d2\.\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{V\},i:j\}:=\\sum\_\{k=i\}^\{j\}\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\quad\\text\{for \}1\\leq i\\leq j\\leq d\_\{2\}\.Next, we define the operators

𝒞𝐔,i​\(𝐀1\)\\displaystyle\\mathcal\{C\}\_\{\\mathbf\{U\},i\}\(\\mathbf\{A\}\_\{1\}\):=𝐏𝐔,1:i𝐀1𝐏𝐔,1:i\+𝐏𝐔,i\+1:d1𝐀1𝐏𝐔,i\+1:d1for1≤i<d1\\displaystyle:=\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\quad\\text\{for \}1\\leq i<d\_\{1\}and

𝒞𝐕,i​\(𝐀2\)\\displaystyle\\mathcal\{C\}\_\{\\mathbf\{V\},i\}\(\\mathbf\{A\}\_\{2\}\):=𝐏𝐕,1:i𝐀2𝐏𝐕,1:i\+𝐏𝐕,i\+1:d2𝐀2𝐏𝐕,i\+1:d2for1≤i<d2,\\displaystyle:=\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\}\+\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}\\quad\\text\{for \}1\\leq i<d\_\{2\},for any𝐀1∈ℝd1×d1\\mathbf\{A\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐀2∈ℝd2×d2\\mathbf\{A\}\_\{2\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}, respectively\. Then, we define recursively the matrices𝐋𝐔,i:=𝒞𝐔,i​\(𝐋𝐔,i−1\)\\mathbf\{L\}\_\{\\mathbf\{U\},i\}:=\\mathcal\{C\}\_\{\\mathbf\{U\},i\}\(\\mathbf\{L\}\_\{\\mathbf\{U\},i\-1\}\)fori=1,2,…,d1−1i=1,2,\\ldots,d\_\{1\}\-1and𝐋𝐕,i:=𝒞𝐕,i​\(𝐋𝐕,i−1\)\\mathbf\{L\}\_\{\\mathbf\{V\},i\}:=\\mathcal\{C\}\_\{\\mathbf\{V\},i\}\(\\mathbf\{L\}\_\{\\mathbf\{V\},i\-1\}\)fori=1,2,…,d2−1i=1,2,\\ldots,d\_\{2\}\-1, where we set𝐋𝐔,0:=𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\},0\}:=\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕,0:=𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\},0\}:=\\mathbf\{L\}\_\{\\mathbf\{V\}\}\.

The following lemma shows that afterd1−1d\_\{1\}\-1andd2−1d\_\{2\}\-1iterations, the matrices𝐋𝐔,d1−1\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}and𝐋𝐕,d2−1\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively\.

###### Lemma 6\.

It holds that

𝐋𝐔,d1−1=∑k=1d1⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F​𝐮k​𝐮k⊤,𝐋𝐕,d2−1=∑k=1d2⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F​𝐯k​𝐯k⊤\.\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}=\\sum\_\{k=1\}^\{d\_\{1\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\},\\quad\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}=\\sum\_\{k=1\}^\{d\_\{2\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\.In other words, the matrices𝐋𝐔,d1−1\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}and𝐋𝐕,d2−1\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively\.

ProofWe show by induction that fori=0,1,2,…,d1−1i=0,1,2,\\ldots,d\_\{1\}\-1, it holds that

𝐋𝐔,i=∑k=1i⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+𝐏𝐔,i\+1:d1𝐋𝐔𝐏𝐔,i\+1:d1\.\\mathbf\{L\}\_\{\\mathbf\{U\},i\}=\\sum\_\{k=1\}^\{i\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\.The base casei=0i=0follows directly from the definition of𝐋𝐔,0\\mathbf\{L\}\_\{\\mathbf\{U\},0\}\. Now assume that the statement holds for somei∈\{0,1,2,…,d1−2\}i\\in\\\{0,1,2,\\ldots,d\_\{1\}\-2\\\}\. We compute that

𝐋𝐔,i\+1=\\displaystyle\\mathbf\{L\}\_\{\\mathbf\{U\},i\+1\}=𝒞𝐔,i\+1​\(𝐋𝐔,i\)\\displaystyle\\mathcal\{C\}\_\{\\mathbf\{U\},i\+1\}\(\\mathbf\{L\}\_\{\\mathbf\{U\},i\}\)=\\displaystyle=𝐏𝐔,1:i\+1𝐋𝐔,i𝐏𝐔,1:i\+1\+𝐏𝐔,i\+2:d1𝐋𝐔,i𝐏𝐔,i\+2:d1\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\mathbf\{L\}\_\{\\mathbf\{U\},i\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\},i\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}=\\displaystyle=𝐏𝐔,1:i\+1\(∑k=1i⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+𝐏𝐔,i\+1:d1𝐋𝐔𝐏𝐔,i\+1:d1\)𝐏𝐔,1:i\+1\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\left\(\\sum\_\{k=1\}^\{i\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\right\)\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\+𝐏𝐔,i\+2:d1\(∑k=1i⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+𝐏𝐔,i\+1:d1𝐋𝐔𝐏𝐔,i\+1:d1\)𝐏𝐔,i\+2:d1\\displaystyle\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\left\(\\sum\_\{k=1\}^\{i\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\right\)\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}=\(a\)\\displaystyle\\overset\{\(a\)\}\{=\}𝐏𝐔,1:i\+1\(∑k=1i⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\)𝐏𝐔,1:i\+1\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\left\(\\sum\_\{k=1\}^\{i\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\right\)\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\+𝐏𝐔,1:i\+1𝐏𝐔,i\+1:d1𝐋𝐔𝐏𝐔,i\+1:d1𝐏𝐔,1:i\+1\\displaystyle\+\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\+𝐏𝐔,i\+2:d1𝐏𝐔,i\+1:d1𝐋𝐔𝐏𝐔,i\+1:d1𝐏𝐔,i\+2:d1\\displaystyle\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}=\(b\)\\displaystyle\\overset\{\(b\)\}\{=\}∑k=1i⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+⟨𝐋𝐔,𝐮i\+1𝐮i\+1⊤⟩F𝐮i\+1𝐮i\+1⊤\+𝐏𝐔,i\+2:d1𝐋𝐔𝐏𝐔,i\+2:d1\\displaystyle\\sum\_\{k=1\}^\{i\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{i\+1\}\\mathbf\{u\}\_\{i\+1\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{i\+1\}\\mathbf\{u\}\_\{i\+1\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}=\\displaystyle=∑k=1i\+1⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+𝐏𝐔,i\+2:d1𝐋𝐔𝐏𝐔,i\+2:d1\.\\displaystyle\\sum\_\{k=1\}^\{i\+1\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\.In equation \(a\), we used that𝐏𝐔,i\+2:d1⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤𝐏𝐔,i\+2:d1=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}=\\mathbf\{0\}for allk=1,…,ik=1,\\ldots,i\. In equation \(b\), we then used that𝐏𝐔,1:i\+1𝐮k𝐮k⊤𝐏𝐔,1:i\+1=𝐮k𝐮k⊤\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}=\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}fork=1,2,…,ik=1,2,\\ldots,i, that𝐏𝐔,1:i\+1𝐏𝐔,i\+1:d1=𝐮i\+1𝐮i\+1⊤\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\+1\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}=\\mathbf\{u\}\_\{i\+1\}\\mathbf\{u\}\_\{i\+1\}^\{\\top\}, and that𝐏𝐔,i\+2:d1𝐏𝐔,i\+1:d1=𝐏𝐔,i\+2:d1\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}=\\mathbf\{P\}\_\{\\mathbf\{U\},i\+2:d\_\{1\}\}\. This shows the induction step and we have shown that the displayed equation holds for alli=0,1,…,d1−1i=0,1,\\ldots,d\_\{1\}\-1\.

Thus, we obtain fori=d1−1i=d\_\{1\}\-1that

𝐋𝐔,d1−1=∑k=1d1−1⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\+𝐏𝐔,d1:d1𝐋𝐔𝐏𝐔,d1:d1=∑k=1d1⟨𝐋𝐔,𝐮k𝐮k⊤⟩F𝐮k𝐮k⊤\.\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}=\\sum\_\{k=1\}^\{d\_\{1\}\-1\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},d\_\{1\}:d\_\{1\}\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{P\}\_\{\\mathbf\{U\},d\_\{1\}:d\_\{1\}\}=\\sum\_\{k=1\}^\{d\_\{1\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\.This proves the claim for𝐋𝐔,d1−1\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}\. The proof for𝐋𝐕,d2−1\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}is analogous\. The next lemma shows that after arriving at a Sylvester equation where both𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\}\}are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively, we can give an explicit expression for the solution of the Sylvester equation\.

###### Lemma 7\(Explicit Harmonic\-Mean Sylvester Equation Solution\)\.

Let𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be an arbitrary matrix with SVD given by𝐙=∑i=1dσi​\(𝐙\)​𝐮i​𝐯i⊤\\mathbf\{Z\}=\\sum\_\{i=1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{Z\}\)\\mathbf\{u\}\_\{i\}\\mathbf\{v\}\_\{i\}^\{\\top\}, whereσ1​\(𝐙\)≥σ2​\(𝐙\)≥…≥σd​\(𝐙\)≥0\\sigma\_\{1\}\(\\mathbf\{Z\}\)\\geq\\sigma\_\{2\}\(\\mathbf\{Z\}\)\\geq\\ldots\\geq\\sigma\_\{d\}\(\\mathbf\{Z\}\)\\geq 0are the singular values of𝐙\\mathbf\{Z\}\. Here,\{𝐮1,…,𝐮d1\}⊂ℝd1\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}\\subset\\R^\{d\_\{1\}\}and\{𝐯1,…,𝐯d2\}⊂ℝd2\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}\\subset\\R^\{d\_\{2\}\}are orthonormal bases ofℝd1\\R^\{d\_\{1\}\}andℝd2\\R^\{d\_\{2\}\}, consisting of left and right singular vectors of𝐙\\mathbf\{Z\}, respectively\. Let𝐋~𝐔∈ℝd1×d1,𝐋~𝐕∈ℝd2×d2\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\},\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}be positive definite matrices which are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively\. Then, the unique solution𝐖∈ℝd1×d2\\mathbf\{W\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}of the Sylvester equation

𝐋~𝐔​𝐖\+𝐖​𝐋~𝐕=2​𝐙\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\}\\mathbf\{W\}\+\\mathbf\{W\}\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\}=2\\mathbf\{Z\}is given by

𝐖=∑k=1d2​σk​\(𝐙\)⟨𝐋~𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋~𝐕,𝐯k​𝐯k⊤⟩F​𝐮k​𝐯k⊤\.\\mathbf\{W\}=\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}\(\\mathbf\{Z\}\)\}\{\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\.

ProofUniqueness follows again from[Bhatia \(1997, Theorem VII\.2\.3\)](https://arxiv.org/html/2608.23765#bib.bib60)since𝐋~𝐔\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\}and𝐋~𝐕\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\}are positive definite\. To show that the given expression for𝐖\\mathbf\{W\}is indeed the solution of the Sylvester equation, we compute

𝐋~𝐔​𝐖\+𝐖​𝐋~𝐕\\displaystyle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\}\\mathbf\{W\}\+\\mathbf\{W\}\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\}=\\displaystyle=\(∑k=1d1⟨𝐋~𝐔,𝐮k​𝐮k⊤⟩F​𝐮k​𝐮k⊤\)​\(∑j=1d2​σj​\(𝐙\)⟨𝐋~𝐔,𝐮j​𝐮j⊤⟩F\+⟨𝐋~𝐕,𝐯j​𝐯j⊤⟩F​𝐮j​𝐯j⊤\)\\displaystyle\\left\(\\sum\_\{k=1\}^\{d\_\{1\}\}\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\right\)\\left\(\\sum\_\{j=1\}^\{d\}\\frac\{2\\sigma\_\{j\}\(\\mathbf\{Z\}\)\}\{\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{j\}\\mathbf\{u\}\_\{j\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{j\}\\mathbf\{v\}\_\{j\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{j\}\\mathbf\{v\}\_\{j\}^\{\\top\}\\right\)\+\(∑j=1d2​σj​\(𝐙\)⟨𝐋~𝐔,𝐮j​𝐮j⊤⟩F\+⟨𝐋~𝐕,𝐯j​𝐯j⊤⟩F​𝐮j​𝐯j⊤\)​\(∑k=1d2⟨𝐋~𝐕,𝐯k​𝐯k⊤⟩F​𝐯k​𝐯k⊤\)\\displaystyle\+\\left\(\\sum\_\{j=1\}^\{d\}\\frac\{2\\sigma\_\{j\}\(\\mathbf\{Z\}\)\}\{\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{j\}\\mathbf\{u\}\_\{j\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{j\}\\mathbf\{v\}\_\{j\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{j\}\\mathbf\{v\}\_\{j\}^\{\\top\}\\right\)\\left\(\\sum\_\{k=1\}^\{d\_\{2\}\}\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\right\)=\\displaystyle=∑k=1d2​σk​\(𝐙\)​⟨𝐋~𝐔,𝐮k​𝐮k⊤⟩F⟨𝐋~𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋~𝐕,𝐯k​𝐯k⊤⟩F​𝐮k​𝐯k⊤\+∑k=1d2​σk​\(𝐙\)​⟨𝐋~𝐕,𝐯k​𝐯k⊤⟩F⟨𝐋~𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋~𝐕,𝐯k​𝐯k⊤⟩F​𝐮k​𝐯k⊤\\displaystyle\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\{\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\+\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\{\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}=\\displaystyle=∑k=1d2​σk​\(𝐙\)​𝐮k​𝐯k⊤=2​𝐙\.\\displaystyle\\sum\_\{k=1\}^\{d\}2\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}=2\\mathbf\{Z\}\.In the first equation we have used that the matrices𝐋~𝐔\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{U\}\}and𝐋~𝐕\\widetilde\{\\mathbf\{L\}\}\_\{\\mathbf\{V\}\}are diagonal in the bases\{𝐮1,…,𝐮d1\}\\\{\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}\\\}and\{𝐯1,…,𝐯d2\}\\\{\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}\\\}, respectively\. We have shown the desired equation, which completes the proof\. Now we have all the ingredients in place to prove Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2), the lower bound on the weighted inner product⟨𝐙,W𝐗,ε​\(𝐙\)⟩F\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\.

Proof of Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2)In the following, we assume that𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}is arbitrary but fixed\. Then we have seen in Lemma[3](https://arxiv.org/html/2608.23765#Thmlemma3)that the weight matrix𝐖0=W𝐗,ε​\(𝐙\)\\mathbf\{W\}\_\{0\}=W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)as defined in Lemma[3](https://arxiv.org/html/2608.23765#Thmlemma3)is the unique solution of the Sylvester equation

𝐋𝐔​𝐖0\+𝐖0​𝐋𝐕=2​𝐙\.\\mathbf\{L\}\_\{\\mathbf\{U\}\}\\mathbf\{W\}\_\{0\}\+\\mathbf\{W\}\_\{0\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}=2\\mathbf\{Z\}\.
Now we denote the SVD of𝐙\\mathbf\{Z\}by𝐙=𝐔𝐙​diag⁡\(𝝁\)​𝐕𝐙⊤\\mathbf\{Z\}=\\mathbf\{U\}\_\{\\mathbf\{Z\}\}\\diag\(\\bm\{\\mu\}\)\\mathbf\{V\}^\{\\top\}\_\{\\mathbf\{Z\}\}, where𝝁=\(σ1​\(𝐙\),…,σd​\(𝐙\)\)∈ℝd\\bm\{\\mu\}=\(\\sigma\_\{1\}\(\\mathbf\{Z\}\),\\ldots,\\sigma\_\{d\}\(\\mathbf\{Z\}\)\)\\in\\R^\{d\}contains the singular values of𝐙\\mathbf\{Z\}\. We denote by𝐮1,…,𝐮d1\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}the columns of𝐔𝐙\\mathbf\{U\}\_\{\\mathbf\{Z\}\}and by𝐯1,…,𝐯d2\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}the columns of𝐕𝐙\\mathbf\{V\}\_\{\\mathbf\{Z\}\}\. We define the orthogonal projection matrices by

𝐏𝐔,i:j:=∑k=ij𝐮k𝐮k⊤for1≤i≤j≤d1,\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{U\},i:j\}:=\\sum\_\{k=i\}^\{j\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\quad\\text\{for \}1\\leq i\\leq j\\leq d\_\{1\},𝐏𝐕,i:j:=∑k=ij𝐯k𝐯k⊤for1≤i≤j≤d2\.\\displaystyle\\mathbf\{P\}\_\{\\mathbf\{V\},i:j\}:=\\sum\_\{k=i\}^\{j\}\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\quad\\text\{for \}1\\leq i\\leq j\\leq d\_\{2\}\.Now recall thatd=min⁡\(d1,d2\)d=\\min\(d\_\{1\},d\_\{2\}\)andD=max⁡\(d1,d2\)D=\\max\(d\_\{1\},d\_\{2\}\)\. Next, we define the operators

𝒞𝐔,i​\(𝐀1\)\\displaystyle\\mathcal\{C\}\_\{\\mathbf\{U\},i\}\(\\mathbf\{A\}\_\{1\}\):=\{𝐏𝐔,1:i𝐀1𝐏𝐔,1:i\+𝐏𝐔,i\+1:d1𝐀1𝐏𝐔,i\+1:d1for​1≤i<d1,𝐀1for​d1≤i<D\\displaystyle:=\\begin\{cases\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\+\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{A\}\_\{1\}\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\quad&\\text\{for \}1\\leq i<d\_\{1\},\\\\ \\mathbf\{A\}\_\{1\}\\quad&\\text\{for \}d\_\{1\}\\leq i<D\\end\{cases\}𝒞𝐕,i​\(𝐀2\)\\displaystyle\\mathcal\{C\}\_\{\\mathbf\{V\},i\}\(\\mathbf\{A\}\_\{2\}\):=\{𝐏𝐕,1:i𝐀2𝐏𝐕,1:i\+𝐏𝐕,i\+1:d2𝐀2𝐏𝐕,i\+1:d2for​1≤i<d2,𝐀2for​d2≤i<D\\displaystyle:=\\begin\{cases\}\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\}\+\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}\\mathbf\{A\}\_\{2\}\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}\\quad&\\text\{for \}1\\leq i<d\_\{2\},\\\\ \\mathbf\{A\}\_\{2\}\\quad&\\text\{for \}d\_\{2\}\\leq i<D\\end\{cases\}for matrices𝐀1∈ℝd1×d1\\mathbf\{A\}\_\{1\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐀2∈ℝd2×d2\\mathbf\{A\}\_\{2\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}\. Set𝐋𝐔,0:=𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\},0\}:=\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕,0:=𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\},0\}:=\\mathbf\{L\}\_\{\\mathbf\{V\}\}\. We define recursively

𝐋𝐔,i:=𝒞𝐔,i​\(𝐋𝐔,i−1\)\\displaystyle\\mathbf\{L\}\_\{\\mathbf\{U\},i\}:=\\mathcal\{C\}\_\{\\mathbf\{U\},i\}\(\\mathbf\{L\}\_\{\\mathbf\{U\},i\-1\}\)\\quadfor​i=1,2,…,D−1,\\displaystyle\\text\{ for \}i=1,2,\\ldots,D\-1,𝐋𝐕,i:=𝒞𝐕,i​\(𝐋𝐕,i−1\)\\displaystyle\\mathbf\{L\}\_\{\\mathbf\{V\},i\}:=\\mathcal\{C\}\_\{\\mathbf\{V\},i\}\(\\mathbf\{L\}\_\{\\mathbf\{V\},i\-1\}\)\\quadfor​i=1,2,…,D−1\.\\displaystyle\\text\{ for \}i=1,2,\\ldots,D\-1\.Note that since𝐋𝐔,0=𝐋𝐔\\mathbf\{L\}\_\{\\mathbf\{U\},0\}=\\mathbf\{L\}\_\{\\mathbf\{U\}\}and𝐋𝐕,0=𝐋𝐕\\mathbf\{L\}\_\{\\mathbf\{V\},0\}=\\mathbf\{L\}\_\{\\mathbf\{V\}\}are positive definite, it follows by construction that𝐋𝐔,i\\mathbf\{L\}\_\{\\mathbf\{U\},i\}and𝐋𝐕,i\\mathbf\{L\}\_\{\\mathbf\{V\},i\}are positive definite for alli=0,…,D−1i=0,\\ldots,D\-1as well since the pinching operators𝒞𝐔,i​\(⋅\)\\mathcal\{C\}\_\{\\mathbf\{U\},i\}\(\\cdot\)and𝒞𝐕,i​\(⋅\)\\mathcal\{C\}\_\{\\mathbf\{V\},i\}\(\\cdot\)preserve positive definiteness\.

By construction of the projection matrices𝐏𝐔,1:i,𝐏𝐔,i\+1:d1,𝐏𝐕,1:i,𝐏𝐕,i\+1:d2\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\},\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\},\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\},\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}it holds that

𝐏𝐔,1:i𝐙𝐏𝐕,i\+1:d2=𝟎,𝐏𝐔,i\+1:d1𝐙𝐏𝐕,1:i=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},i\+1:d\_\{2\}\}=\\mathbf\{0\},\\quad\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:d\_\{1\}\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},1:i\}=\\mathbf\{0\}for alli=1,2,…,d−1i=1,2,\\ldots,d\-1\. Thus, we can apply the previous Lemma[5](https://arxiv.org/html/2608.23765#Thmlemma5)for1≤i≤d−11\\leq i\\leq d\-1\. Now consider the scenariod≤i<Dd\\leq i<D\. IfD=d1D=d\_\{1\}, we aim to apply Lemma[5](https://arxiv.org/html/2608.23765#Thmlemma5)with𝐏𝐕=Id\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\Idand𝐏𝐔=𝐏𝐔,1:i\\mathbf\{P\}\_\{\\mathbf\{U\}\}=\\mathbf\{P\}\_\{\\mathbf\{U\},1:i\}\. It follows that𝐏𝐔​𝐙𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\}\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}since𝐏𝐕,⊥=𝟎\\mathbf\{P\}\_\{\\mathbf\{V\},\\bot\}=\\mathbf\{0\}\. Moreover, we have𝐏𝐔,⊥𝐙𝐏𝐕=𝐏𝐔,i\+1:D𝐙=𝟎\\mathbf\{P\}\_\{\\mathbf\{U\},\\bot\}\\mathbf\{Z\}\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\mathbf\{P\}\_\{\\mathbf\{U\},i\+1:D\}\\mathbf\{Z\}=\\mathbf\{0\}, where we have used𝐏𝐕=Id\\mathbf\{P\}\_\{\\mathbf\{V\}\}=\\Idin the first step and the SVD of𝐙\\mathbf\{Z\}in the second step\. This shows that we can apply[Lemma5](https://arxiv.org/html/2608.23765#Thmlemma5)ifD=d1D=d\_\{1\}\. IfD=d2D=d\_\{2\}, we can argue analogously\. Thus, we can also apply[Lemma5](https://arxiv.org/html/2608.23765#Thmlemma5)in the cased≤i<Dd\\leq i<Dboth in the caseD=d1D=d\_\{1\}andD=d2D=d\_\{2\}\.

This implies that fori=1,2,…,D−1i=1,2,\\ldots,D\-1the Sylvester equation

𝐋𝐔,i​𝐖i\+𝐖i​𝐋𝐕,i=2​𝐙\\mathbf\{L\}\_\{\\mathbf\{U\},i\}\\mathbf\{W\}\_\{i\}\+\\mathbf\{W\}\_\{i\}\\mathbf\{L\}\_\{\\mathbf\{V\},i\}=2\\mathbf\{Z\}\(50\)has a unique solution𝐖i∈ℝd1×d2\\mathbf\{W\}\_\{i\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}and that

⟨𝐖i−1,𝐙⟩F≥⟨𝐖i,𝐙⟩F\.\\langle\\mathbf\{W\}\_\{i\-1\},\\mathbf\{Z\}\\rangle\_\{F\}\\geq\\langle\\mathbf\{W\}\_\{i\},\\mathbf\{Z\}\\rangle\_\{F\}\.It follows that

⟨𝐖0,𝐙⟩≥⟨𝐖1,𝐙⟩≥⟨𝐖2,𝐙⟩≥…≥⟨𝐖D−1,𝐙⟩\.\\langle\\mathbf\{W\}\_\{0\},\\mathbf\{Z\}\\rangle\\geq\\langle\\mathbf\{W\}\_\{1\},\\mathbf\{Z\}\\rangle\\geq\\langle\\mathbf\{W\}\_\{2\},\\mathbf\{Z\}\\rangle\\geq\\ldots\\geq\\langle\\mathbf\{W\}\_\{D\-1\},\\mathbf\{Z\}\\rangle\.\(51\)Note that by construction we have that𝐋𝐔,D−1=𝐋𝐔,d1−1\\mathbf\{L\}\_\{\\mathbf\{U\},D\-1\}=\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}and𝐋𝐕,D−1=𝐋𝐕,d2−1\\mathbf\{L\}\_\{\\mathbf\{V\},D\-1\}=\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}\. By Lemma[6](https://arxiv.org/html/2608.23765#Thmlemma6), we have that

𝐋𝐔,d1−1=∑k=1d1⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F​𝐮k​𝐮k⊤,𝐋𝐕,d2−1=∑k=1d2⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F​𝐯k​𝐯k⊤\.\\mathbf\{L\}\_\{\\mathbf\{U\},d\_\{1\}\-1\}=\\sum\_\{k=1\}^\{d\_\{1\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\},\\quad\\mathbf\{L\}\_\{\\mathbf\{V\},d\_\{2\}\-1\}=\\sum\_\{k=1\}^\{d\_\{2\}\}\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\.Thus, fori=D−1i=D\-1the solution𝐖D−1\\mathbf\{W\}\_\{D\-1\}of the Sylvester equation \([50](https://arxiv.org/html/2608.23765#A1.E50)\) can be computed explicitly and we obtain that

𝐖D−1=∑k=1d2​σk​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F​𝐮k​𝐯k⊤\.\\mathbf\{W\}\_\{D\-1\}=\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\.It follows that

⟨𝐖D−1,𝐙⟩F=∑k=1d2​σk2​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F\.\\langle\\mathbf\{W\}\_\{D\-1\},\\mathbf\{Z\}\\rangle\_\{F\}=\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\.Combining this equation with the inequality chain \([51](https://arxiv.org/html/2608.23765#A1.E51)\) and with𝐖0=W𝐗,ε​\(𝐙\)\\mathbf\{W\}\_\{0\}=W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\), we have shown that

⟨W𝐗,ε​\(𝐙\),𝐙⟩F≥∑k=1d2​σk2​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F\.\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\\geq\\sum\_\{k=1\}^\{d\}\\frac\{2\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}\.This completes the proof\.

### A\.2Proof of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)\(Optimality of Harmonic Mean Quadratic Model\)

We now prove the optimality result[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)of the quadratic model induced by the harmonic\-mean weight operator, which has been presented in[Section3\.4](https://arxiv.org/html/2608.23765#S3.SS4)\.

#### A\.2\.1Proof of[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)\(Monotonicity of Power Mean Weight Operators\)

We begin with proving the Loewner ordering and domination properties of[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1), which is a tool to understand optimality of the quadratic models defined by power mean\-induced weight operators\.

Proof of[Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)We want to show that

⟨𝐙,W𝐗,ε\(q\)​\(𝐙\)⟩F≤⟨𝐙,W𝐗,ε\(q′\)​\(𝐙\)⟩Ffor all​𝐙∈ℝd1×d2,\\displaystyle\\quad\\langle\\mathbf\{Z\},W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\leq\\langle\\mathbf\{Z\},W^\{\(q^\{\\prime\}\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\ \\ \\text\{for all \}\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\},whereW𝐗,ε\(q\)W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}andW𝐗,ε\(q′\)W^\{\(q^\{\\prime\}\)\}\_\{\\mathbf\{X\},\\varepsilon\}are the power mean\-induced weight operators[eq\.22](https://arxiv.org/html/2608.23765#S3.E22)associated with power means of orderqqandq′q^\{\\prime\}withq≤q′q\\leq q^\{\\prime\}\. Fix𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}and set𝐙~:=𝐔𝐗⊤​𝐙𝐕𝐗∈ℝd1×d2\\widetilde\{\\mathbf\{Z\}\}:=\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, with𝐔𝐗∈ℝd1×d1\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and𝐕𝐗∈ℝd2×d2\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}being the left and right singular vector matrices of the singular value decomposition of𝐗\\mathbf\{X\}\. By orthogonality of𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}and𝐕𝐗\\mathbf\{V\}\_\{\\mathbf\{X\}\}and the definition ofW𝐗,ε\(q\)W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}, it holds that

⟨𝐙,W𝐗,ε\(q\)​\(𝐙\)⟩F=\\displaystyle\\langle\\mathbf\{Z\},W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}=⟨𝐙,𝐔𝐗​\[𝐇𝝈,ε\(q\)∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)\]​𝐕𝐗⊤⟩F=⟨𝐔𝐗⊤​𝐙𝐕𝐗,𝐇𝝈,ε\(q\)∘\(𝐔𝐗⊤​𝐙𝐕𝐗\)⟩F\\displaystyle\\langle\\mathbf\{Z\},\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\bigl\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\bigr\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\rangle\_\{F\}=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\},\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\bigl\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\bigr\)\\rangle\_\{F\}=\\displaystyle=⟨𝐙~,𝐇𝝈,ε\(q\)∘𝐙~⟩F=∑i=1d1∑j=1d2\(𝐇𝝈,ε\(q\)\)i​j​𝐙~i​j2\.\\displaystyle\\bigl\\langle\\widetilde\{\\mathbf\{Z\}\},\\ \\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\widetilde\{\\mathbf\{Z\}\}\\bigr\\rangle\_\{F\}=\\sum\_\{i=1\}^\{d\_\{1\}\}\\sum\_\{j=1\}^\{d\_\{2\}\}\\bigl\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\\,\\widetilde\{\\mathbf\{Z\}\}\_\{ij\}^\{\\,2\}\.Analogously, we have

⟨𝐙,W𝐗,ε\(q′\)​\(𝐙\)⟩F=∑i=1d1∑j=1d2\(𝐇𝝈,ε\(q′\)\)i​j​𝐙~i​j2\.\\langle\\mathbf\{Z\},W^\{\(q^\{\\prime\}\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}=\\sum\_\{i=1\}^\{d\_\{1\}\}\\sum\_\{j=1\}^\{d\_\{2\}\}\\bigl\(\\mathbf\{H\}^\{\(q^\{\\prime\}\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\\,\\widetilde\{\\mathbf\{Z\}\}\_\{ij\}^\{\\,2\}\.Asq≤q′q\\leq q^\{\\prime\}, it follows from the monotonicity of power means\([Bullen, 2003](https://arxiv.org/html/2608.23765#bib.bib42), Section III\.3, Theorem 1\)that for alli∈\[d1\]i\\in\[d\_\{1\}\]andj∈\[d2\]j\\in\[d\_\{2\}\],ℳq​\(σ~i,σ~j\)≤ℳq′​\(σ~i,σ~j\)\\mathcal\{M\}\_\{q\}\(\\widetilde\{\\sigma\}\_\{i\},\\widetilde\{\\sigma\}\_\{j\}\)\\leq\\mathcal\{M\}\_\{q^\{\\prime\}\}\(\\widetilde\{\\sigma\}\_\{i\},\\widetilde\{\\sigma\}\_\{j\}\)and, hence,\(𝐇𝝈,ε\(q\)\)i​j≤\(𝐇𝝈,ε\(q′\)\)i​j\\bigl\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\\leq\\bigl\(\\mathbf\{H\}^\{\(q^\{\\prime\}\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\. Thus, we obtain the desired Loewner ordering since

⟨𝐙,W𝐗,ε\(q\)​\(𝐙\)⟩F=∑i=1d1∑j=1d2\(𝐇𝝈,ε\(q\)\)i​j​𝐙~i​j2≤∑i=1d1∑j=1d2\(𝐇𝝈,ε\(q′\)\)i​j​𝐙~i​j2=⟨𝐙,W𝐗,ε\(q′\)​\(𝐙\)⟩F\.\\displaystyle\\langle\\mathbf\{Z\},W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}=\\sum\_\{i=1\}^\{d\_\{1\}\}\\sum\_\{j=1\}^\{d\_\{2\}\}\\bigl\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\\,\\widetilde\{\\mathbf\{Z\}\}\_\{ij\}^\{\\,2\}\\leq\\sum\_\{i=1\}^\{d\_\{1\}\}\\sum\_\{j=1\}^\{d\_\{2\}\}\\bigl\(\\mathbf\{H\}^\{\(q^\{\\prime\}\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\bigr\)\_\{ij\}\\,\\widetilde\{\\mathbf\{Z\}\}\_\{ij\}^\{\\,2\}=\\langle\\mathbf\{Z\},W^\{\(q^\{\\prime\}\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\.Finally, the domination[eq\.23](https://arxiv.org/html/2608.23765#S3.E23)of the quadratic modelQε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)byQε\(q′\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q^\{\\prime\}\)\}\(\\cdot\\mid\\mathbf\{X\}\)follows from the Loewner ordering and the formula[eq\.11](https://arxiv.org/html/2608.23765#S3.E11)as⟨𝐗,W𝐗,ε\(q\)​\(𝐗\)⟩F\\langle\\mathbf\{X\},W^\{\(q\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}and⟨𝐗,W𝐗,ε\(q′\)​\(𝐗\)⟩F\\langle\\mathbf\{X\},W^\{\(q^\{\\prime\}\)\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}both coincide sincediag⁡\(𝐇𝝈,ε\(q\)\)=diag⁡\(𝐇𝝈,ε\(q′\)\)\\diag\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)=\\diag\(\\mathbf\{H\}^\{\(q^\{\\prime\}\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\.

#### A\.2\.2Proofs of[Lemmas8](https://arxiv.org/html/2608.23765#Thmlemma8)and[9](https://arxiv.org/html/2608.23765#Thmlemma9)\(Second Order Necessary Condition for Majorization\)

Given the established results, the main burden for proving[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)is to show that majorization is violated whenq<−1q<\-1\. As we see below, this can be shown via a second order analysis of the smoothed objective𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}\. To this end, we first define the*symmetrization operator*S:ℝd×d→ℝd×dS:\\R^\{d\\times d\}\\to\\R^\{d\\times d\}and the*antisymmetrization operator*T:ℝd×d→ℝd×dT:\\R^\{d\\times d\}\\to\\R^\{d\\times d\}by

S⁡\(𝐙\):=12​\(𝐙\+𝐙⊤\)andT⁡\(𝐙\):=12​\(𝐙−𝐙⊤\)S\(\\mathbf\{Z\}\):=\\frac\{1\}\{2\}\(\\mathbf\{Z\}\+\\mathbf\{Z\}^\{\\top\}\)\\qquad\\text\{and\}\\qquad T\(\\mathbf\{Z\}\):=\\frac\{1\}\{2\}\(\\mathbf\{Z\}\-\\mathbf\{Z\}^\{\\top\}\)\(52\)for any𝐙∈ℝd×d\\mathbf\{Z\}\\in\\R^\{d\\times d\}, which enables us to state an explicit formula for the Hessian of the spectral functionFFin[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)\.

###### Lemma 8\(Hessian of Spectral Functions\)\.

Letf:ℝ≥0→ℝf:\\R\_\{\\geq 0\}\\to\\Rbe a differentiable function withLL\-Lipschitz first derivativef′f^\{\\prime\}such thatf′f^\{\\prime\}is right differentiable at00andf′​\(0\)=0f^\{\\prime\}\(0\)=0\. Then the spectral functionF:ℝd1×d2→ℝF:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\R,F⁡\(𝐗\)=∑i=1df⁡\(σi​\(𝐗\)\)F\(\\mathbf\{X\}\)=\\sum\_\{i=1\}^\{d\}f\(\\sigma\_\{i\}\(\\mathbf\{X\}\)\)is differentiable withLL\-Lipschitz gradients∇F​\(𝐗\)∈ℝd1×d2\\nabla F\(\\mathbf\{X\}\)\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}and furthermore almost everywhere twice differentiable\. Moreover,FFis twice differentiable at𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}if and only ifffis twice differentiable at allσ1​\(𝐗\),…,σd​\(𝐗\)\\sigma\_\{1\}\(\\mathbf\{X\}\),\\ldots,\\sigma\_\{d\}\(\\mathbf\{X\}\)\. In that case, if additionallyd1≤d2d\_\{1\}\\leq d\_\{2\}, the Hessian∇2F​\(𝐗\)\\nabla^\{2\}F\(\\mathbf\{X\}\)ofFFat𝐗\\mathbf\{X\}is given by

∇2F​\(𝐗\)​\(𝐙\)=𝐔𝐗​\[𝐇1∘S⁡\(𝐔𝐗⊤​𝐙𝐕𝐗,1\)\+𝐇2∘T⁡\(𝐔𝐗⊤​𝐙𝐕𝐗,1\)𝐇3∘\(𝐔𝐗⊤​𝐙𝐕𝐗,2\)\]​𝐕𝐗⊤\\nabla^\{2\}F\(\\mathbf\{X\}\)\(\\mathbf\{Z\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\begin\{bmatrix\}\\mathbf\{H\}\_\{1\}\\circ S\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\},1\}\)\+\\mathbf\{H\}\_\{2\}\\circ T\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\},1\}\)&\\mathbf\{H\}\_\{3\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\},2\}\)\\end\{bmatrix\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\(53\)for any𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, where𝐗\\mathbf\{X\}has the SVD𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}with𝐔𝐗∈ℝd1×d1\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\},𝐕𝐗=\[𝐕𝐗,1𝐕𝐗,2\]∈ℝd2×d2\\mathbf\{V\}\_\{\\mathbf\{X\}\}=\\begin\{bmatrix\}\\mathbf\{V\}\_\{\\mathbf\{X\},1\}&\\mathbf\{V\}\_\{\\mathbf\{X\},2\}\\end\{bmatrix\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\},𝐕𝐗,1∈ℝd2×d\\mathbf\{V\}\_\{\\mathbf\{X\},1\}\\in\\R^\{d\_\{2\}\\times d\},𝐕𝐗,2∈ℝd2×\(d2−d\)\\mathbf\{V\}\_\{\\mathbf\{X\},2\}\\in\\R^\{d\_\{2\}\\times\(d\_\{2\}\-d\)\},TTandSSare as in \([52](https://arxiv.org/html/2608.23765#A1.E52)\), and𝐇1∈ℝd×d\\mathbf\{H\}\_\{1\}\\in\\R^\{d\\times d\}is such that fori,j∈\[d\]i,j\\in\[d\],

\(𝐇1\)i​j=\{f′​\(σi\)−f′​\(σj\)σi−σjif​σi≠σj,f′′​\(σi\)if​σi=σj,\(\\mathbf\{H\}\_\{1\}\)\_\{ij\}=\\begin\{cases\}\\frac\{f^\{\\prime\}\(\\sigma\_\{i\}\)\-f^\{\\prime\}\(\\sigma\_\{j\}\)\}\{\\sigma\_\{i\}\-\\sigma\_\{j\}\}&\\text\{ if \}\\sigma\_\{i\}\\neq\\sigma\_\{j\},\\\\ f^\{\\prime\\prime\}\(\\sigma\_\{i\}\)&\\text\{ if \}\\sigma\_\{i\}=\\sigma\_\{j\},\\end\{cases\}the matrix𝐇2∈ℝd×d\\mathbf\{H\}\_\{2\}\\in\\R^\{d\\times d\}is such that fori,j∈\[d\]i,j\\in\[d\],

\(𝐇2\)i​j=\{f′​\(σi\)\+f′​\(σj\)σi\+σjif​σi\+σj≠0,f′′​\(σi\)if​σi=σj=0,\(\\mathbf\{H\}\_\{2\}\)\_\{ij\}=\\begin\{cases\}\\frac\{f^\{\\prime\}\(\\sigma\_\{i\}\)\+f^\{\\prime\}\(\\sigma\_\{j\}\)\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}&\\text\{ if \}\\sigma\_\{i\}\+\\sigma\_\{j\}\\neq 0,\\\\ f^\{\\prime\\prime\}\(\\sigma\_\{i\}\)&\\text\{ if \}\\sigma\_\{i\}=\\sigma\_\{j\}=0,\\end\{cases\}and𝐇3∈ℝd1×\(d2−d\)\\mathbf\{H\}\_\{3\}\\in\\R^\{d\_\{1\}\\times\(d\_\{2\}\-d\)\}is such that fori∈\[d1\],j∈\[d2−d\]i\\in\[d\_\{1\}\],j\\in\[d\_\{2\}\-d\],

\(𝐇3\)i​j=\{f′​\(σi\)/σiif​σi≠0,f′′​\(σi\)if​σi=0\.\(\\mathbf\{H\}\_\{3\}\)\_\{ij\}=\\begin\{cases\}f^\{\\prime\}\(\\sigma\_\{i\}\)/\\sigma\_\{i\}&\\text\{ if \}\\sigma\_\{i\}\\neq 0,\\\\ f^\{\\prime\\prime\}\(\\sigma\_\{i\}\)&\\text\{ if \}\\sigma\_\{i\}=0\.\\end\{cases\}

Proof of[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)Sinceffis differentiable, the spectral functionFFis differentiable with gradient

∇F​\(𝐗\)=𝐔​diag⁡\(∇f^​\(σ⁡\(𝐗\)\)\)​𝐕⊤=𝐔​diag⁡\(\(f′​\(σi\)\)i=1d\)​𝐕⊤\\nabla F\(\\mathbf\{X\}\)=\\mathbf\{U\}\\diag\\big\(\\nabla\\hat\{f\}\(\\sigma\(\\mathbf\{X\}\)\)\\big\)\\mathbf\{V\}^\{\\top\}=\\mathbf\{U\}\\diag\\left\(\\big\(f^\{\\prime\}\(\\sigma\_\{i\}\)\\big\)\_\{i=1\}^\{d\}\\right\)\\mathbf\{V\}^\{\\top\}at any𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\([Lewis and Sendov, 2005](https://arxiv.org/html/2608.23765#bib.bib84), Section 7\), asFFcan be written as the compositionf^∘σ\\hat\{f\}\\circ\\sigma, wheref^:ℝd→ℝ\\hat\{f\}:\\R^\{d\}\\to\\Rwithf^​\(σ\)=∑i=1df⁡\(σi\)\\hat\{f\}\(\\sigma\)=\\sum\_\{i=1\}^\{d\}f\(\\sigma\_\{i\}\)\. Maps of the form𝐗↦∇F​\(𝐗\)\\mathbf\{X\}\\mapsto\\nabla F\(\\mathbf\{X\}\)are also called*non\-Hermitian Loewner operators*\([Löwner, 1934](https://arxiv.org/html/2608.23765#bib.bib90);[Sun and Sun, 2008](https://arxiv.org/html/2608.23765#bib.bib91);[Ding et al\., 2018](https://arxiv.org/html/2608.23765#bib.bib92)\)or*generalized matrix functions*\([Hawkins and Ben\-Israel, 1973](https://arxiv.org/html/2608.23765#bib.bib93);[Noferini, 2017](https://arxiv.org/html/2608.23765#bib.bib94)\), and[Andersson et al\. \(2016, Theorem 1\.1\)](https://arxiv.org/html/2608.23765#bib.bib53)implies that sincef′f^\{\\prime\}isLL\-Lipschitz,𝐗↦∇F​\(𝐗\)\\mathbf\{X\}\\mapsto\\nabla F\(\\mathbf\{X\}\)isLL\-Lipschitz with respect to the Frobenius norm\. Rademacher’s theorem then implies that∇F\\nabla Fis almost everywhere differentiable \(with respect to the Lebesgue measure\)\.

Furthermore, it follows from[Yang \(2009, Theorem 2\.2\.6\)](https://arxiv.org/html/2608.23765#bib.bib95)thatffis twice differentiable atσ=σ⁡\(𝐗\)\\sigma=\\sigma\(\\mathbf\{X\}\)if and only ifFFis twice differentiable at𝐗\\mathbf\{X\}, and the formula for the Hessian[eq\.53](https://arxiv.org/html/2608.23765#A1.E53)at the points of twice differentiability𝐗\\mathbf\{X\}is due to[Yang \(2009, Theorem 2\.2\.6\)](https://arxiv.org/html/2608.23765#bib.bib95)and[Noferini \(2017, Corollary 3\.10\)](https://arxiv.org/html/2608.23765#bib.bib94)\. The machinery of[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)can now be used to state a second order condition that is necessary for majorization to hold\. The key observation is that such a condition can already be extracted from perturbations of𝐗\\mathbf\{X\}that are confined to a two\-dimensional singular block\. Along such perturbations, the smoothed nuclear norm objective𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}reduces, up to an additive constant, to a spectral function onℝ2×2\\R^\{2\\times 2\}, so that[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)only needs to be applied in dimension two and only the two singular values defining the block enter the argument\.

###### Lemma 9\(Second Order Necessary Condition for Majorization\)\.

Letε\>0\\varepsilon\>0and let𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}have the full singular value decomposition𝐗=𝐔𝐗​diag⁡\(𝛔\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}of[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2), with singular valuesσ1≥…≥σd≥0\\sigma\_\{1\}\\geq\\ldots\\geq\\sigma\_\{d\}\\geq 0\. LetQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)be the quadratic model

Qε​\(𝐗\+𝚫∣𝐗\)=𝒥ε​\(𝐗\)\+⟨∇𝒥ε​\(𝐗\),𝚫⟩F\+12​⟨𝚫,W𝐗,ε​\(𝚫\)⟩F,Q\_\{\\varepsilon\}\(\\mathbf\{X\}\+\\mathbf\{\\Delta\}\\mid\\mathbf\{X\}\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{\\Delta\}\\rangle\_\{F\}\+\\tfrac\{1\}\{2\}\\langle\\mathbf\{\\Delta\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{\\Delta\}\)\\rangle\_\{F\},whereW𝐗,εW\_\{\\mathbf\{X\},\\varepsilon\}is the weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)associated to a weight operator core matrix𝐇𝛔,ε∈ℝd1×d2\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}that is symmetric in the sense that\(𝐇𝛔,ε\)i​j=\(𝐇𝛔,ε\)j​i\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ji\}for alli,j∈\[d\]i,j\\in\[d\]\. Assume thatQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)majorizes𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}locally around𝐗\\mathbf\{X\}, i\.e\., there existsδ\>0\\delta\>0such that

𝒥ε​\(𝐗\+𝚫\)≤Qε​\(𝐗\+𝚫∣𝐗\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\+\\mathbf\{\\Delta\}\)\\leq Q\_\{\\varepsilon\}\(\\mathbf\{X\}\+\\mathbf\{\\Delta\}\\mid\\mathbf\{X\}\)\(54\)for all𝚫∈ℝd1×d2\\mathbf\{\\Delta\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}with‖𝚫‖F≤δ\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\_\{F\}\\leq\\delta\. Then for alli,j∈\[d\]i,j\\in\[d\]such thati≠ji\\neq j,σi\>ε\\sigma\_\{i\}\>\\varepsilonandσj\>ε\\sigma\_\{j\}\>\\varepsilon, it holds that

\(𝐇𝝈,ε\)i​j≥2σi\+σj\.\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}\\geq\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\.\(55\)No assumption is made on the remaining singular valuesσk\\sigma\_\{k\}withk∈\[d\]∖\{i,j\}k\\in\[d\]\\setminus\\\{i,j\\\}; in particular,𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}is not assumed to be twice differentiable at𝐗\\mathbf\{X\}\.

Proof of[Lemma9](https://arxiv.org/html/2608.23765#Thmlemma9)Leti,j∈\[d\]i,j\\in\[d\]satisfy the assumptions of the lemma\. Interchangingiiandjjif necessary \(which changes neither the assumptions nor the conclusion, as𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}is symmetric on\[d\]×\[d\]\[d\]\\times\[d\]\) we may assume thati<ji<j, so thatσi≥σj\\sigma\_\{i\}\\geq\\sigma\_\{j\}\.

*Step 1: Reduction to a2×22\\times 2singular block\.*Let𝐮1,…,𝐮d1\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{d\_\{1\}\}and𝐯1,…,𝐯d2\\mathbf\{v\}\_\{1\},\\ldots,\\mathbf\{v\}\_\{d\_\{2\}\}denote the columns of𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}and of𝐕𝐗\\mathbf\{V\}\_\{\\mathbf\{X\}\}, respectively, and define the perturbation direction

𝐙:=12​\(𝐮i​𝐯j⊤−𝐮j​𝐯i⊤\)∈ℝd1×d2,which satisfies​‖𝐙‖F=1\.\\mathbf\{Z\}:=\\tfrac\{1\}\{\\sqrt\{2\}\}\\left\(\\mathbf\{u\}\_\{i\}\\mathbf\{v\}\_\{j\}^\{\\top\}\-\\mathbf\{u\}\_\{j\}\\mathbf\{v\}\_\{i\}^\{\\top\}\\right\)\\in\\R^\{d\_\{1\}\\times d\_\{2\}\},\\qquad\\text\{which satisfies \}\\left\\lVert\\mathbf\{Z\}\\right\\rVert\_\{F\}=1\.\(56\)Then𝐙=𝐔𝐗​𝐄i​j​𝐕𝐗⊤\\mathbf\{Z\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{E\}\_\{ij\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}with𝐄i​j:=12​\(𝐞i​𝐞j⊤−𝐞j​𝐞i⊤\)∈ℝd1×d2\\mathbf\{E\}\_\{ij\}:=\\tfrac\{1\}\{\\sqrt\{2\}\}\(\\mathbf\{e\}\_\{i\}\\mathbf\{e\}\_\{j\}^\{\\top\}\-\\mathbf\{e\}\_\{j\}\\mathbf\{e\}\_\{i\}^\{\\top\}\)\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, where𝐞i∈ℝd1\\mathbf\{e\}\_\{i\}\\in\\R^\{d\_\{1\}\}and𝐞j∈ℝd2\\mathbf\{e\}\_\{j\}\\in\\R^\{d\_\{2\}\}in the first summand and𝐞j∈ℝd1\\mathbf\{e\}\_\{j\}\\in\\R^\{d\_\{1\}\}and𝐞i∈ℝd2\\mathbf\{e\}\_\{i\}\\in\\R^\{d\_\{2\}\}in the second summand, which is well defined sincei,j∈\[d\]=\[d\]i,j\\in\[d\]=\[d\]\. By orthogonality of𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}and𝐕𝐗\\mathbf\{V\}\_\{\\mathbf\{X\}\}, the matrix𝐗\+t​𝐙\\mathbf\{X\}\+t\\mathbf\{Z\}has, for eacht∈ℝt\\in\\R, the same singular values asdiag⁡\(𝝈\)\+t​𝐄i​j∈ℝd1×d2\\diag\(\\bm\{\\sigma\}\)\+t\\mathbf\{E\}\_\{ij\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. The latter matrix coincides withdiag⁡\(𝝈\)\\diag\(\\bm\{\\sigma\}\)outside the rows and columns with indicesiiandjj, and is therefore, after applying the permutations that move the indicesiiandjjto the positions11and22, block diagonal with blocks

𝐁\+t​𝐍,where​𝐁:=diag⁡\(σi,σj\)∈ℝ2×2⁡and​𝐍:=12​\(01−10\)∈ℝ2×2,\\mathbf\{B\}\+t\\mathbf\{N\},\\quad\\text\{ where \}\\mathbf\{B\}:=\\diag\(\\sigma\_\{i\},\\sigma\_\{j\}\)\\in\\R^\{2\\times 2\}\\text\{ and \}\\mathbf\{N\}:=\\tfrac\{1\}\{\\sqrt\{2\}\}\\begin\{pmatrix\}0&1\\\\ \-1&0\\end\{pmatrix\}\\in\\R^\{2\\times 2\},and the rectangular diagonal matrix carrying the remaining singular values\(σk\)k∈\[d\]∖\{i,j\}\(\\sigma\_\{k\}\)\_\{k\\in\[d\]\\setminus\\\{i,j\\\}\}\. As permutation matrices are orthogonal, the singular values of𝐗\+t​𝐙\\mathbf\{X\}\+t\\mathbf\{Z\}are, as a multiset, the union of\{σk:k∈\[d\]∖\{i,j\}\}\\\{\\sigma\_\{k\}:k\\in\[d\]\\setminus\\\{i,j\\\}\\\}and of the two singular values of𝐁\+t​𝐍\\mathbf\{B\}\+t\\mathbf\{N\}\. WritingF2:ℝ2×2→ℝF\_\{2\}:\\R^\{2\\times 2\}\\to\\R,F2​\(𝐀\):=jε​\(σ1​\(𝐀\)\)\+jε​\(σ2​\(𝐀\)\)F\_\{2\}\(\\mathbf\{A\}\):=j\_\{\\varepsilon\}\(\\sigma\_\{1\}\(\\mathbf\{A\}\)\)\+j\_\{\\varepsilon\}\(\\sigma\_\{2\}\(\\mathbf\{A\}\)\), for the spectral function ofjεj\_\{\\varepsilon\}in dimension two, we conclude that

𝒥ε​\(𝐗\+t​𝐙\)−𝒥ε​\(𝐗\)=F2​\(𝐁\+t​𝐍\)−F2​\(𝐁\)for all​t∈ℝ,\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\+t\\mathbf\{Z\}\)\-\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)=F\_\{2\}\(\\mathbf\{B\}\+t\\mathbf\{N\}\)\-F\_\{2\}\(\\mathbf\{B\}\)\\qquad\\text\{for all \}t\\in\\R,\(57\)since the summandsjε​\(σk\)j\_\{\\varepsilon\}\(\\sigma\_\{k\}\)withk∈\[d\]∖\{i,j\}k\\in\[d\]\\setminus\\\{i,j\\\}do not depend onttand cancel\. In particular, only the two singular valuesσi\\sigma\_\{i\}andσj\\sigma\_\{j\}enter the argument below\.

*Step 2: Application of[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)in dimension two\.*It is straightforward to check thatjεj\_\{\\varepsilon\}satisfies the assumptions of[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8): it is differentiable with1ε\\tfrac\{1\}\{\\varepsilon\}\-Lipschitz derivativejε′j\_\{\\varepsilon\}^\{\\prime\}, which is right differentiable at00and satisfiesjε′​\(0\)=0j\_\{\\varepsilon\}^\{\\prime\}\(0\)=0\. We apply[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)withf=jεf=j\_\{\\varepsilon\}to the spectral functionF=F2F=F\_\{2\}withd1=d2=d=2d\_\{1\}=d\_\{2\}=d=2at the point𝐁\\mathbf\{B\}, whose singular value decomposition is𝐁=𝐈2​diag⁡\(σi,σj\)​𝐈2⊤\\mathbf\{B\}=\\mathbf\{I\}\_\{2\}\\diag\(\\sigma\_\{i\},\\sigma\_\{j\}\)\\mathbf\{I\}\_\{2\}^\{\\top\}sinceσi≥σj≥0\\sigma\_\{i\}\\geq\\sigma\_\{j\}\\geq 0\. Asσi\>ε\\sigma\_\{i\}\>\\varepsilonandσj\>ε\\sigma\_\{j\}\>\\varepsilon, the functionjεj\_\{\\varepsilon\}is twice differentiable atσi\\sigma\_\{i\}and atσj\\sigma\_\{j\}withjε′​\(σi\)=jε′​\(σj\)=1j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{i\}\)=j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{j\}\)=1andjε′′​\(σi\)=jε′′​\(σj\)=0j\_\{\\varepsilon\}^\{\\prime\\prime\}\(\\sigma\_\{i\}\)=j\_\{\\varepsilon\}^\{\\prime\\prime\}\(\\sigma\_\{j\}\)=0, so that[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8)guarantees thatF2F\_\{2\}is twice differentiable at𝐁\\mathbf\{B\}\. Sinced1=d2d\_\{1\}=d\_\{2\}in this application, the block𝐕𝐁,2\\mathbf\{V\}\_\{\\mathbf\{B\},2\}is empty and the matrix𝐇3\\mathbf\{H\}\_\{3\}does not occur in[eq\.53](https://arxiv.org/html/2608.23765#A1.E53), while the off\-diagonal entries of𝐇1,𝐇2∈ℝ2×2\\mathbf\{H\}\_\{1\},\\mathbf\{H\}\_\{2\}\\in\\R^\{2\\times 2\}are given by

\(𝐇1\)12=\(𝐇1\)21\\displaystyle\\bigl\(\\mathbf\{H\}\_\{1\}\\bigr\)\_\{12\}=\\bigl\(\\mathbf\{H\}\_\{1\}\\bigr\)\_\{21\}=\{jε′​\(σi\)−jε′​\(σj\)σi−σj=0,if​σi≠σj,jε′′​\(σi\)=0,if​σi=σj,\\displaystyle=\\begin\{cases\}\\frac\{j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{i\}\)\-j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{j\}\)\}\{\\sigma\_\{i\}\-\\sigma\_\{j\}\}=0,&\\text\{ if \}\\sigma\_\{i\}\\neq\\sigma\_\{j\},\\\\ j\_\{\\varepsilon\}^\{\\prime\\prime\}\(\\sigma\_\{i\}\)=0,&\\text\{ if \}\\sigma\_\{i\}=\\sigma\_\{j\},\\end\{cases\}\(𝐇2\)12=\(𝐇2\)21\\displaystyle\\bigl\(\\mathbf\{H\}\_\{2\}\\bigr\)\_\{12\}=\\bigl\(\\mathbf\{H\}\_\{2\}\\bigr\)\_\{21\}=jε′​\(σi\)\+jε′​\(σj\)σi\+σj=2σi\+σj\.\\displaystyle=\\frac\{j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{i\}\)\+j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{j\}\)\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}=\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\.As𝐍\\mathbf\{N\}is skew\-symmetric, we haveS⁡\(𝐍\)=𝟎S\(\\mathbf\{N\}\)=\\mathbf\{0\}andT⁡\(𝐍\)=𝐍T\(\\mathbf\{N\}\)=\\mathbf\{N\}for the operators[eq\.52](https://arxiv.org/html/2608.23765#A1.E52), and therefore the Hessian formula[eq\.53](https://arxiv.org/html/2608.23765#A1.E53)yields that∇2F2​\(𝐁\)​\(𝐍\)=𝐇1∘S⁡\(𝐍\)\+𝐇2∘T⁡\(𝐍\)=𝐇2∘𝐍\\nabla^\{2\}F\_\{2\}\(\\mathbf\{B\}\)\(\\mathbf\{N\}\)=\\mathbf\{H\}\_\{1\}\\circ S\(\\mathbf\{N\}\)\+\\mathbf\{H\}\_\{2\}\\circ T\(\\mathbf\{N\}\)=\\mathbf\{H\}\_\{2\}\\circ\\mathbf\{N\}and, consequently, that

⟨𝐍,∇2F2​\(𝐁\)​\(𝐍\)⟩F=⟨𝐍,𝐇2∘𝐍⟩F=2σi\+σj​‖𝐍‖F2=2σi\+σj\.\\langle\\mathbf\{N\},\\nabla^\{2\}F\_\{2\}\(\\mathbf\{B\}\)\(\\mathbf\{N\}\)\\rangle\_\{F\}=\\langle\\mathbf\{N\},\\mathbf\{H\}\_\{2\}\\circ\\mathbf\{N\}\\rangle\_\{F\}=\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{F\}^\{2\}=\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\.\(58\)
*Step 3: First and second derivative of𝒢\\mathcal\{G\}at00\.*Define the one\-dimensional function

𝒢⁡\(t\):=𝒥ε​\(𝐗\+t​𝐙\)−Qε​\(𝐗\+t​𝐙∣𝐗\)\\mathcal\{G\}\(t\):=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\+t\\mathbf\{Z\}\)\-Q\_\{\\varepsilon\}\(\\mathbf\{X\}\+t\\mathbf\{Z\}\\mid\\mathbf\{X\}\)fort∈ℝt\\in\\R\. By definition ofQεQ\_\{\\varepsilon\}and by the reduction[eq\.57](https://arxiv.org/html/2608.23765#A1.E57), we have

𝒢⁡\(t\)\\displaystyle\\mathcal\{G\}\(t\)=F2​\(𝐁\+t​𝐍\)−F2​\(𝐁\)−t​⟨∇𝒥ε​\(𝐗\),𝐙⟩F−t22​⟨𝐙,W𝐗,ε​\(𝐙\)⟩F,\\displaystyle=F\_\{2\}\(\\mathbf\{B\}\+t\\mathbf\{N\}\)\-F\_\{2\}\(\\mathbf\{B\}\)\-t\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{Z\}\\rangle\_\{F\}\-\\frac\{t^\{2\}\}\{2\}\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\},\(59\)and, by construction, it holds that𝒢⁡\(0\)=0\\mathcal\{G\}\(0\)=0\. By[Lemma8](https://arxiv.org/html/2608.23765#Thmlemma8), the spectral functionF2F\_\{2\}is differentiable with gradient∇F2​\(𝐁\)=𝐈2​diag⁡\(jε′​\(σi\),jε′​\(σj\)\)​𝐈2⊤=𝐈2\\nabla F\_\{2\}\(\\mathbf\{B\}\)=\\mathbf\{I\}\_\{2\}\\diag\\bigl\(j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{i\}\),j\_\{\\varepsilon\}^\{\\prime\}\(\\sigma\_\{j\}\)\\bigr\)\\mathbf\{I\}\_\{2\}^\{\\top\}=\\mathbf\{I\}\_\{2\}\(cf\.[Lewis and Sendov, 2005](https://arxiv.org/html/2608.23765#bib.bib84), Section 7\), so thatt↦F2​\(𝐁\+t​𝐍\)t\\mapsto F\_\{2\}\(\\mathbf\{B\}\+t\\mathbf\{N\}\)is differentiable with derivative⟨∇F2​\(𝐁\+t​𝐍\),𝐍⟩F\\langle\\nabla F\_\{2\}\(\\mathbf\{B\}\+t\\mathbf\{N\}\),\\mathbf\{N\}\\rangle\_\{F\}\. Since𝐍\\mathbf\{N\}has a vanishing diagonal, this derivative equals⟨𝐈2,𝐍⟩F=0\\langle\\mathbf\{I\}\_\{2\},\\mathbf\{N\}\\rangle\_\{F\}=0att=0t=0\. Together with[eq\.57](https://arxiv.org/html/2608.23765#A1.E57)and the differentiability of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}\(see[SectionB\.2](https://arxiv.org/html/2608.23765#A2.SS2)\), this implies that

⟨∇𝒥ε​\(𝐗\),𝐙⟩F=dd​t\|t=0​𝒥ε​\(𝐗\+t​𝐙\)=0,\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{Z\}\\rangle\_\{F\}=\\frac\{d\}\{dt\}\\Big\|\_\{t=0\}\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\+t\\mathbf\{Z\}\)=0,and hence, by[eq\.59](https://arxiv.org/html/2608.23765#A1.E59), that𝒢′​\(0\)=0\\mathcal\{G\}^\{\\prime\}\(0\)=0\. Moreover, twice differentiability ofF2F\_\{2\}at𝐁\\mathbf\{B\}means that∇F2\\nabla F\_\{2\}is differentiable at𝐁\\mathbf\{B\}with derivative∇2F2​\(𝐁\)\\nabla^\{2\}F\_\{2\}\(\\mathbf\{B\}\), so thatt↦⟨∇F2​\(𝐁\+t​𝐍\),𝐍⟩Ft\\mapsto\\langle\\nabla F\_\{2\}\(\\mathbf\{B\}\+t\\mathbf\{N\}\),\\mathbf\{N\}\\rangle\_\{F\}is differentiable att=0t=0with derivative⟨∇2F2​\(𝐁\)​\(𝐍\),𝐍⟩F\\langle\\nabla^\{2\}F\_\{2\}\(\\mathbf\{B\}\)\(\\mathbf\{N\}\),\\mathbf\{N\}\\rangle\_\{F\}\. Therefore,𝒢\\mathcal\{G\}is twice differentiable att=0t=0, and[eq\.59](https://arxiv.org/html/2608.23765#A1.E59)and[eq\.58](https://arxiv.org/html/2608.23765#A1.E58)give

𝒢′′​\(0\)=⟨𝐍,∇2F2​\(𝐁\)​\(𝐍\)⟩F−⟨𝐙,W𝐗,ε​\(𝐙\)⟩F=2σi\+σj−⟨𝐙,W𝐗,ε​\(𝐙\)⟩F\.\\mathcal\{G\}^\{\\prime\\prime\}\(0\)=\\langle\\mathbf\{N\},\\nabla^\{2\}F\_\{2\}\(\\mathbf\{B\}\)\(\\mathbf\{N\}\)\\rangle\_\{F\}\-\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}=\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\-\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\.
*Step 4: Conclusion\.*Since‖t​𝐙‖F=\|t\|\\left\\lVert t\\mathbf\{Z\}\\right\\rVert\_\{F\}=\|t\|by[eq\.56](https://arxiv.org/html/2608.23765#A1.E56), the local majorization assumption[eq\.54](https://arxiv.org/html/2608.23765#A1.E54)implies that𝒢⁡\(t\)≤0=𝒢⁡\(0\)\\mathcal\{G\}\(t\)\\leq 0=\\mathcal\{G\}\(0\)for all\|t\|≤δ\|t\|\\leq\\delta\. Thus, the function𝒢\\mathcal\{G\}has a local maximum at00, and therefore𝒢′′​\(0\)≤0\\mathcal\{G\}^\{\\prime\\prime\}\(0\)\\leq 0\. By Step 3, this means that

⟨𝐙,W𝐗,ε​\(𝐙\)⟩F≥2σi\+σj\.\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\\geq\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\.\(60\)On the other hand, we have𝐔𝐗⊤​𝐙𝐕𝐗=𝐄i​j\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{Z\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}=\\mathbf\{E\}\_\{ij\}, so that the definition[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)of the weight operator and the symmetry of𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}yield

⟨𝐙,W𝐗,ε​\(𝐙\)⟩F=⟨𝐄i​j,𝐇𝝈,ε∘𝐄i​j⟩F=\(𝐇𝝈,ε\)i​j\+\(𝐇𝝈,ε\)j​i2=\(𝐇𝝈,ε\)i​j\.\\displaystyle\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}=\\langle\\mathbf\{E\}\_\{ij\},\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{E\}\_\{ij\}\\rangle\_\{F\}=\\frac\{\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}\+\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ji\}\}\{2\}=\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}\.Inserting this identity into[eq\.60](https://arxiv.org/html/2608.23765#A1.E60)gives[eq\.55](https://arxiv.org/html/2608.23765#A1.E55)\. This completes the proof\.

#### A\.2\.3Proof of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)

It now remains to prove[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), the statement about optimality of the harmonic\-mean weight operator as defined by[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)and[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)\.

Proof of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)We note that the statement thatQε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)majorizes𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}for anyq≥−1q\\geq\-1is a direct consequence of the harmonic mean majorization result[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)and the monotonicity of power mean weight operators \([Lemma1](https://arxiv.org/html/2608.23765#Thmlemma1)\)\. For the reverse direction, we need to show that global majorization of𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}is violated whenq<−1q<\-1\. Letq<−1q<\-1be arbitrary, and leti,j∈\[d\]i,j\\in\[d\]withi≠ji\\neq jbe indices such thatσi\>ε\\sigma\_\{i\}\>\\varepsilon,σj\>ε\\sigma\_\{j\}\>\\varepsilon, andσi≠σj\\sigma\_\{i\}\\neq\\sigma\_\{j\}, which exist by the assumption of[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2)\. Sinceσi\>ε\\sigma\_\{i\}\>\\varepsilonandσj\>ε\\sigma\_\{j\}\>\\varepsilon, we haveσ~i=σi−1\\widetilde\{\\sigma\}\_\{i\}=\\sigma\_\{i\}^\{\-1\}andσ~j=σj−1\\widetilde\{\\sigma\}\_\{j\}=\\sigma\_\{j\}^\{\-1\}in[eq\.21](https://arxiv.org/html/2608.23765#S3.E21), andσi≠σj\\sigma\_\{i\}\\neq\\sigma\_\{j\}implies thatσ~i≠σ~j\\widetilde\{\\sigma\}\_\{i\}\\neq\\widetilde\{\\sigma\}\_\{j\}\. Asq<−1q<\-1, we obtain by the definition[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)of𝐇𝝈,ε\(q\)\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}and by the monotonicity of power means, which is strict at distinct arguments\([Bullen, 2003](https://arxiv.org/html/2608.23765#bib.bib42), Section III\.3, Theorem 1\), that

\(𝐇𝝈,ε\(q\)\)i​j\\displaystyle\(\\mathbf\{H\}^\{\(q\)\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=ℳq​\(σi−1,σj−1\)<ℳ−1​\(σi−1,σj−1\)=2σi\+σj\.\\displaystyle=\\mathcal\{M\}\_\{q\}\\\!\\Big\(\\sigma\_\{i\}^\{\-1\},\\sigma\_\{j\}^\{\-1\}\\Big\)<\\mathcal\{M\}\_\{\-1\}\\\!\\Big\(\\sigma\_\{i\}^\{\-1\},\\sigma\_\{j\}^\{\-1\}\\Big\)=\\frac\{2\}\{\\sigma\_\{i\}\+\\sigma\_\{j\}\}\.Since power mean core matrices[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)are symmetric, this means that the necessary condition[eq\.55](https://arxiv.org/html/2608.23765#A1.E55)of[Lemma9](https://arxiv.org/html/2608.23765#Thmlemma9)is violated for the index pair\(i,j\)\(i,j\)\. By contraposition,[Lemma9](https://arxiv.org/html/2608.23765#Thmlemma9)therefore implies thatQε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)does not majorize𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}locally around𝐗\\mathbf\{X\}, i\.e\., for everyδ\>0\\delta\>0, there exists a𝚫∈ℝd1×d2\\mathbf\{\\Delta\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}with‖𝚫‖F≤δ\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\_\{F\}\\leq\\deltasuch that

𝒥ε​\(𝐗\+𝚫\)\>Qε\(q\)​\(𝐗\+𝚫∣𝐗\)\.\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\+\\mathbf\{\\Delta\}\)\>Q\_\{\\varepsilon\}^\{\(q\)\}\(\\mathbf\{X\}\+\\mathbf\{\\Delta\}\\mid\\mathbf\{X\}\)\.In particular,Qε\(q\)\(⋅∣𝐗\)Q\_\{\\varepsilon\}^\{\(q\)\}\(\\cdot\\mid\\mathbf\{X\}\)does not majorize𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}globally\. This completes the proof forq<−1q<\-1\.

### A\.3Proofs of[Theorems3](https://arxiv.org/html/2608.23765#Thmtheorem3)and[4](https://arxiv.org/html/2608.23765#Thmtheorem4)\(Global Linear Convergence\)

#### A\.3\.1Preliminaries and General Proof Strategy

We start by recalling the following lemma which states that the NSP induces a reverse triangle inequality\.

###### Lemma 10\([Fornasier et al\.](https://arxiv.org/html/2608.23765#bib.bib54),[2011](https://arxiv.org/html/2608.23765#bib.bib54), Lemma 6\.6\)\.

Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP of orderrrof[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)for someηr<1\\eta\_\{r\}<1\. Then for all𝐙,𝐗∈ℝd1×d2\\mathbf\{Z\},\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}such that𝒜⁡\(𝐙\)=𝒜⁡\(𝐗\)\\mathcal\{A\}\\left\(\\mathbf\{Z\}\\right\)=\\mathcal\{A\}\\left\(\\mathbf\{X\}\\right\)it holds that

‖𝐙−𝐗‖∗≤1\+ηr1−ηr​\(‖𝐗‖∗−‖𝐙‖∗\+2​βr​\(𝐙\)∗\)\.\\left\\lVert\\mathbf\{Z\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{Z\}\\right\\rVert\_\{\\ast\}\+2\\beta\_\{r\}\(\\mathbf\{Z\}\)\_\{\*\}\\right\)\.

In our proof we will need to relate several times the quantities𝒥ε​\(𝐗\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\},‖𝐗\(k\)−𝐗⋆‖∗\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}, andβr​\(𝐗\(k\)\)∗\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\. This will be achieved via the following inequality\.

###### Lemma 11\.

Let𝐗⋆,𝐗∈ℝd1×d2\\mathbf\{X\}\_\{\\star\},\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\. Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP of orderrrof[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)with constantηr<1\\eta\_\{r\}<1\. Furthermore, assume𝒜⁡\(𝐗⋆\)=𝒜⁡\(𝐗\)\\mathcal\{A\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)=\\mathcal\{A\}\\left\(\\mathbf\{X\}\\right\)and0≤ε≤βr​\(𝐗\)∗d0\\leq\\varepsilon\\leq\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}, where the boundary caseε=0\\varepsilon=0is understood with the convention𝒥0=‖⋅‖∗\\mathcal\{J\}\_\{0\}=\\left\\lVert\\cdot\\right\\rVert\_\{\\ast\}\. Then it holds that

1−ηr1\+ηr​‖𝐗−𝐗⋆‖∗−2​βr​\(𝐗⋆\)∗≤𝒥ε​\(𝐗\)−‖𝐗⋆‖∗≤\(32\+ηr\)​βr​\(𝐗\)∗\.\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-2\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\.\(61\)

The proof of[Lemma11](https://arxiv.org/html/2608.23765#Thmlemma11)is an improved version of the proof of[Kümmerle et al\. \(2021, Lemma B\.1\)](https://arxiv.org/html/2608.23765#bib.bib37), which is a corresponding lemma in theℓ1\\ell\_\{1\}\-minimization scenario\. For the sake of completeness, we have included a proof below\.

Proof of Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)It follows directly from the definition of𝒥ε​\(𝐗\)\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\), see[eq\.3](https://arxiv.org/html/2608.23765#S1.E3), that𝒥ε​\(𝐗\)≥‖𝐗‖∗\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\\geq\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}for any matrix𝐗\\mathbf\{X\}\. This yields that

𝒥ε​\(𝐗\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≥‖𝐗‖∗−‖𝐗⋆‖∗\\displaystyle\\geq\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≥1−ηr1\+ηr​‖𝐗−𝐗⋆‖∗−2​βr​\(𝐗⋆\)∗,\\displaystyle\\geq\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-2\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},where in the second inequality we applied Lemma[10](https://arxiv.org/html/2608.23765#Thmlemma10)\. This proves the first inequality in \([61](https://arxiv.org/html/2608.23765#A1.E61)\)\.

It remains to show the second inequality in[eq\.61](https://arxiv.org/html/2608.23765#A1.E61)\. Assume nowε\>0\\varepsilon\>0\. For that, we defineI:=\{i∈\[d\]:σi​\(𝐗\)\>ε\}I:=\\\{i\\in\[d\]:\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)\>\\varepsilon\\\}\. It follows that

𝒥ε​\(𝐗\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=∑i∈Iσi​\(𝐗\)\+12​∑i∈Ic\(σi​\(𝐗\)2ε\+ε\)−‖𝐗⋆‖∗\\displaystyle=\\sum\_\{i\\in I\}\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)\+\\frac\{1\}\{2\}\\sum\_\{i\\in I^\{c\}\}\\left\(\\frac\{\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)^\{2\}\}\{\\varepsilon\}\+\\varepsilon\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=‖𝐗‖∗\+∑i∈Ic\(\(ε−σi​\(𝐗\)\)22​ε\)−‖𝐗⋆‖∗\\displaystyle=\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\sum\_\{i\\in I^\{c\}\}\\left\(\\frac\{\\left\(\\varepsilon\-\\sigma\_\{i\}\(\\mathbf\{X\}\)\\right\)^\{2\}\}\{2\\varepsilon\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤‖𝐗‖∗\+\|Ic\|​ε2−‖𝐗⋆‖∗\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\|I^\{c\}\|\\varepsilon\}\{2\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(a\)​‖𝐗‖∗\+βr​\(𝐗\)∗2−‖𝐗⋆‖∗\.\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{2\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(62\)For inequality\(a\)\(a\)we used the assumptionε≤βr​\(𝐗\)∗d\\varepsilon\\leq\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}\. Now note that

\(1−ηr1\+ηr\+1\)​\(‖𝐗‖∗−‖𝐗⋆‖∗\)\\displaystyle\\left\(\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\+1\\right\)\\left\(\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)≤\(a\)​1−ηr1\+ηr​‖𝐗−𝐗⋆‖∗−\(‖𝐗⋆‖∗−‖𝐗‖∗\)\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\(\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\right\)≤\(b\)​\(‖𝐗⋆‖∗−‖𝐗‖∗\+2​βr​\(𝐗\)∗\)−\(‖𝐗⋆‖∗−‖𝐗‖∗\)\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\left\(\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+2\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\\right\)\-\\left\(\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\right\)≤2​βr​\(𝐗\)∗,\\displaystyle\\leq 2\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\},\(63\)where inequality\(a\)\(a\)is the reverse triangle inequality and\(b\)\(b\)is again due to[Lemma10](https://arxiv.org/html/2608.23765#Thmlemma10)\. By rearranging terms we obtain that

‖𝐗‖∗−‖𝐗⋆‖∗≤\(1\+ηr\)​βr​\(𝐗\)∗\.\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\.We insert this into[eq\.62](https://arxiv.org/html/2608.23765#A1.E62), which implies that

𝒥ε​\(𝐗\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\\left\(\\mathbf\{X\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(32\+ηr\)​βr​\(𝐗\)∗\.\\displaystyle\\leq\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\.This shows the second inequality in[eq\.61](https://arxiv.org/html/2608.23765#A1.E61), which finishes the proof, noting that the caseε=0\\varepsilon=0follows directly from[eq\.63](https://arxiv.org/html/2608.23765#A1.E63)\. With[Lemma11](https://arxiv.org/html/2608.23765#Thmlemma11)in place, we can outline the proof strategy for global linear convergence \([Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)and[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)\) and local linear convergence with a faster rate \([Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)\) below\. While our proofs follow the proof strategy presented by[Kümmerle et al\. \(2021\)](https://arxiv.org/html/2608.23765#bib.bib37), there are several important differences\. This stems from the fact that for low\-rank matrices there is no clear notion of support in contrast to sparse vectors\.

Now recall that our goal is to show that𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}converges linearly to zero\. First, we set𝐍\(k\):=𝐗⋆−𝐗\(k\)\\mathbf\{N\}^\{\(k\)\}:=\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\. Then, we note that for allt∈ℝt\\in\\Rwe have that

𝒥εk\+1​\(𝐗\(k\+1\)\)\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)≤Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)\\displaystyle\\leq Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)=𝒥εk​\(𝐗\(k\)\)\+t​⟨∇𝒥εk​\(𝐗\(k\)\),𝐍\(k\)⟩F⏟=\(a\)\+t22​⟨𝐍\(k\),W𝐗\(k\),εk​\(𝐍\(k\)\)⟩F⏟=\(b\)\\displaystyle=\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\+t\\,\\underset\{\{=\(a\)\}\}\{\\underbrace\{\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\),\\mathbf\{N\}^\{\(k\)\}\\rangle\_\{F\}\}\}\+\\frac\{t^\{2\}\}\{2\}\\underset\{\{=\(b\)\}\}\{\\underbrace\{\\langle\\mathbf\{N\}^\{\(k\)\},W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\(\\mathbf\{N\}^\{\(k\)\}\)\\rangle\_\{F\}\}\}\(64\)in the case that the quadratic model functionQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\)satisfies the majorization property of[eq\.16](https://arxiv.org/html/2608.23765#S3.E16), which we established for weight operators with weight operator core matrix𝐇𝝈,εk\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\_\{k\}\}that corresponds to harmonic[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)in[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1), and with power mean weights[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq∈\[−1,∞\]q\\in\[\-1,\\infty\]in[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), and which is well\-known in the literature for one\-sided weights[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5)\.

Thus, in these cases, in order to get an estimate for the decrease of𝒥εk\+1​\(𝐗\(k\+1\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)made in the\(k\+1\)\(k\+1\)th iteration, we can establish a \(negative\) upper bound for the \(negative\) term\(a\)\(a\)and a positive upper bound for term\(b\)\(b\)\. After having established these bounds, we can optimize overttsuch that the right\-hand side in \([64](https://arxiv.org/html/2608.23765#A1.E64)\) becomes minimal\. As it turns out, we will use the same estimate for term\(a\)\(a\)both for the global linear convergence proof and for the local linear convergence proof\. However, for term\(b\)\(b\)we will use different estimates, as𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}being in a neighborhood of𝐗⋆\\mathbf\{X\}\_\{\\star\}allows us to derive sharper estimates of \(b\) than if we have no further information on𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}—at least, if the weight operatorW𝐗\(k\),εk​\(⋅\)W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\(\\cdot\)is defined with a harmonic\-mean weight operator core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)\.

The next lemma below deals with term\(a\)\(a\)in \([64](https://arxiv.org/html/2608.23765#A1.E64)\)\.

###### Lemma 12\.

Recall thatβr​\(𝐗\)∗=∑i=r\+1dσi​\(𝐗\)\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}=\\sum\_\{i=r\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{X\}\)\. Assume that

𝒜⁡\(𝐗⋆\)=𝒜⁡\(𝐗\),\\mathcal\{A\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)=\\mathcal\{A\}\\left\(\\mathbf\{X\}\\right\),that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}has the NSP of orderrrfor someηr<1\\eta\_\{r\}<1and thatε≤βr​\(𝐗\)∗d\\varepsilon\\leq\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}\. Then it holds that

⟨∇𝒥ε​\(𝐗\),𝐗⋆−𝐗⟩F≤−\(1−ηr1\+ηr−1/4\)​‖𝐗−𝐗⋆‖∗\+94​βr​\(𝐗⋆\)∗\.\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}\\leq\-\\left\(\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\-1/4\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+\\frac\{9\}\{4\}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\.

ProofDenote the singular value decomposition of the matrix𝐗\\mathbf\{X\}by𝐗=𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\. We recall from[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)that the gradient∇𝒥ε​\(𝐗\)\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)at𝐗\\mathbf\{X\}satisfies

∇𝒥ε​\(𝐗\)=W𝐗,ε​\(𝐗\)=𝐔𝐗​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐗𝐕𝐗\)\]​𝐕𝐗⊤=𝐔𝐗​𝚺ε​𝐕𝐗⊤,\\displaystyle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)=W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\left\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\right\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},where𝚺ε:=diag⁡\(\(σi​\(𝐗\)max⁡\(σi​\(𝐗\),ε\)\)i=1d\)\\mathbf\{\\Sigma\}\_\{\\varepsilon\}:=\\diag\\bigg\(\\Big\(\\frac\{\\sigma\_\{i\}\(\\mathbf\{X\}\)\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}\\Big\)\_\{i=1\}^\{d\}\\bigg\)\. Thus, we observe that

⟨∇𝒥ε​\(𝐗\),𝐗⋆−𝐗⟩F=⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⋆−𝐗⟩F=⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⋆⟩F−⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⟩F\.\\displaystyle\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\_\{\\star\}\\rangle\_\{F\}\-\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\\rangle\_\{F\}\.To control the first summand, we apply Hölder’s inequality and obtain that

⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⋆⟩F≤‖𝚺ε‖​‖𝐗⋆‖∗≤‖𝐗⋆‖∗\.\\displaystyle\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\_\{\\star\}\\rangle\_\{F\}\\leq\\left\\lVert\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\right\\rVert\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(65\)For the second summand, we note first that

⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⟩F\\displaystyle\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\\rangle\_\{F\}=⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤⟩F\\displaystyle=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\rangle\_\{F\}=⟨𝚺ε,diag⁡\(𝝈\)⟩F\\displaystyle=\\langle\\mathbf\{\\Sigma\}\_\{\\varepsilon\},\\diag\(\\bm\{\\sigma\}\)\\rangle\_\{F\}=∑i=1dσi2​\(𝐗\)max⁡\(σi​\(𝐗\),ε\),\\displaystyle=\\sum\_\{i=1\}^\{d\}\\frac\{\\sigma^\{2\}\_\{i\}\(\\mathbf\{X\}\)\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\},using the definition of𝚺ε\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\. With the notationI:=\{i∈\[d\]:σi​\(𝐗\)\>ε\}I:=\\left\\\{i\\in\\left\[d\\right\]:\\sigma\_\{i\}\(\\mathbf\{X\}\)\>\\varepsilon\\right\\\}, we obtain that

⟨𝐔𝐗​𝚺ε​𝐕𝐗⊤,𝐗⟩F=∑i∈Iσi​\(𝐗\)\+∑i∈Icσi2​\(𝐗\)ε\.\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\mathbf\{\\Sigma\}\_\{\\varepsilon\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},\\mathbf\{X\}\\rangle\_\{F\}=\\sum\_\{i\\in I\}\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)\+\\sum\_\{i\\in I^\{c\}\}\\frac\{\\sigma\_\{i\}^\{2\}\\left\(\\mathbf\{X\}\\right\)\}\{\\varepsilon\}\.\(66\)By combining inequalities \([65](https://arxiv.org/html/2608.23765#A1.E65)\) and \([66](https://arxiv.org/html/2608.23765#A1.E66)\) we obtain that

⟨∇𝒥ε​\(𝐗\),𝐗⋆−𝐗⟩F\\displaystyle\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}≤‖𝐗⋆‖∗−∑i∈Iσi​\(𝐗\)−∑i∈Icσi​\(𝐗\)2ε\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\sum\_\{i\\in I\}\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)\-\\sum\_\{i\\in I^\{c\}\}\\frac\{\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)^\{2\}\}\{\\varepsilon\}=‖𝐗⋆‖∗−‖𝐗‖∗\+∑i∈Icσi​\(𝐗\)−∑i∈Icσi​\(𝐗\)2ε\\displaystyle=\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\sum\_\{i\\in I^\{c\}\}\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)\-\\sum\_\{i\\in I^\{c\}\}\\frac\{\\sigma\_\{i\}\\left\(\\mathbf\{X\}\\right\)^\{2\}\}\{\\varepsilon\}=‖𝐗⋆‖∗−‖𝐗‖∗\+‖σ​\(𝐗\)Ic‖1−‖σ​\(𝐗\)Ic‖22ε,\\displaystyle=\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\\|\\sigma\(\\mathbf\{X\}\)\_\{I^\{c\}\}\\\|\_\{1\}\-\\frac\{\\\|\\sigma\(\\mathbf\{X\}\)\_\{I^\{c\}\}\\\|\_\{2\}^\{2\}\}\{\\varepsilon\},whereσ​\(𝐗\)Ic\\sigma\(\\mathbf\{X\}\)\_\{I^\{c\}\}denotes the vector which contains the singular values\(σi​\(𝐗\)\)i∈Ic\(\\sigma\_\{i\}\(\\mathbf\{X\}\)\)\_\{i\\in I^\{c\}\}and∥⋅∥p\\\|\\cdot\\\|\_\{p\}denotes theℓp\\ell\_\{p\}\-norm of a vector\. Sinceσi​\(𝐗\)≤ε\\sigma\_\{i\}\(\\mathbf\{X\}\)\\leq\\varepsilonfor everyi∈Ici\\in I^\{c\}, we obtain that

∑i∈Ic\(σi​\(𝐗\)−σi​\(𝐗\)2ε\)≤\|Ic\|​ε4,\\sum\_\{i\\in I^\{c\}\}\\left\(\\sigma\_\{i\}\(\\mathbf\{X\}\)\-\\frac\{\\sigma\_\{i\}\(\\mathbf\{X\}\)^\{2\}\}\{\\varepsilon\}\\right\)\\leq\\frac\{\|I^\{c\}\|\\varepsilon\}\{4\},since the scalar functions↦s−s2/εs\\mapsto s\-s^\{2\}/\\varepsilonis maximized on\[0,ε\]\[0,\\varepsilon\]ats=ε/2s=\\varepsilon/2with valueε/4\\varepsilon/4\. Hence, we have shown that

⟨∇𝒥ε​\(𝐗\),𝐗⋆−𝐗⟩F\\displaystyle\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}≤‖𝐗⋆‖∗−‖𝐗‖∗\+ε​d4\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\varepsilon d\}\{4\}≤\(a\)​‖𝐗⋆‖∗−‖𝐗‖∗\+βr​\(𝐗\)∗4\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{4\}=‖𝐗⋆‖∗−‖𝐗‖∗\+‖𝐗−𝐗r‖∗4,\\displaystyle=\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{r\}\\right\\rVert\_\{\\ast\}\}\{4\},where𝐗r\\mathbf\{X\}\_\{r\}denotes the best rank\-rrapproximation of𝐗\\mathbf\{X\}and where in inequality\(a\)\(a\)we have used the assumptionε≤βr​\(𝐗\)∗d\\varepsilon\\leq\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}\. Denoting by𝐗⋆,r\\mathbf\{X\}\_\{\\star,r\}the best rank\-rrapproximation of𝐗⋆\\mathbf\{X\}\_\{\\star\}and using the Eckart–Young theorem we obtain that

⟨∇𝒥ε​\(𝐗\),𝐗⋆−𝐗⟩F\\displaystyle\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\rangle\_\{F\}≤‖𝐗⋆‖∗−‖𝐗‖∗\+‖𝐗−𝐗⋆,r‖∗4\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star,r\}\\right\\rVert\_\{\\ast\}\}\{4\}≤‖𝐗⋆‖∗−‖𝐗‖∗\+‖𝐗−𝐗⋆‖∗4\+βr​\(𝐗⋆\)∗4\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{4\}\+\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\}\{4\}≤\(a\)−\(1−ηr1\+ηr−1/4\)​‖𝐗−𝐗⋆‖∗\+94​βr​\(𝐗⋆\)∗,\\displaystyle\\overset\{\(a\)\}\{\\leq\}\-\\left\(\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\-1/4\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+\\frac\{9\}\{4\}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},where for inequality\(a\)\(a\)we applied the reverse triangle inequality, see Lemma[10](https://arxiv.org/html/2608.23765#Thmlemma10)\.

#### A\.3\.2Upper Bounds on Quadratic Forms Implied by Weight Operators

A key ingredient in the global rate proofs are norm bounds of Hadamard products𝐇𝝈,ε∘𝐁\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{B\}of the weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}with arbitrary matrices𝐁\\mathbf\{B\}, as the weight operator core matrix corresponds to the non\-isometric part of the action of the weight operator of[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)\. We state and prove auxiliary results involving such bounds in[Lemmas13](https://arxiv.org/html/2608.23765#Thmlemma13),[14](https://arxiv.org/html/2608.23765#Thmlemma14)and[15](https://arxiv.org/html/2608.23765#Thmlemma15)\. The norm bounds are then used in[Lemma16](https://arxiv.org/html/2608.23765#Thmlemma16)to upper bound the quadratic terms⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}, which will be useful to handle the terms\(b\)\(b\)in inequality \([64](https://arxiv.org/html/2608.23765#A1.E64)\)\.

The following lemma provides the key result to be used for power mean core matrices[eq\.21](https://arxiv.org/html/2608.23765#S3.E21), which include harmonic and arithmetic mean core matrices as special cases\.

###### Lemma 13\.

Letλ1,…,λd\\lambda\_\{1\},\\ldots,\\lambda\_\{d\}be a sequence of positive numbers\. Set𝛌:=\(λ1,…,λd\)\\bm\{\\lambda\}:=\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\}\)andλmax:=maxi∈\[d\]⁡λi\\lambda\_\{\\max\}:=\\max\_\{i\\in\[d\]\}\\lambda\_\{i\}\. Letq∈\[−∞,∞\]q\\in\[\-\\infty,\\infty\]and𝐇~𝛌\(q\)∈ℝd×d\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\in\\R^\{d\\times d\}be the matrix defined by

\(𝐇~𝝀\(q\)\)i​j:=ℳq​\(λi,λj\)for​i,j∈\[d\],\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}:=\\mathcal\{M\}\_\{q\}\(\\lambda\_\{i\},\\lambda\_\{j\}\)\\quad\\text\{ for \}i,j\\in\[d\],whereℳq\\mathcal\{M\}\_\{q\}is the power mean as in Definition[3](https://arxiv.org/html/2608.23765#Thmdefinition3)\. Recallcqc\_\{q\}from[eq\.27](https://arxiv.org/html/2608.23765#S4.E27), i\.e\., forq∈\[−∞,∞\]q\\in\[\-\\infty,\\infty\],

cq=\{1,if​q∈\[−∞,1\],22−1/q−1,if​q∈\(1,∞\),3,if​q=∞\.c\_\{q\}=\\begin\{cases\}1,&\\text\{ if \}q\\in\[\-\\infty,1\],\\\\ 2^\{2\-1/q\}\-1,&\\text\{ if \}q\\in\(1,\\infty\),\\\\ 3,&\\text\{ if \}q=\\infty\.\\end\{cases\}Then for any unitarily invariant norm\|\|\|⋅\|\|\|\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\cdot\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}onℝd×d\\R^\{d\\times d\}and any matrix𝐁∈ℝd×d\\mathbf\{B\}\\in\\R^\{d\\times d\}, it holds that

‖\|𝐇~𝝀\(q\)∘𝐁\|‖≤cq​λmax​‖\|𝐁\|‖≤\{λmax​‖\|𝐁\|‖,if​q∈\[−∞,1\],3​λmax​‖\|𝐁\|‖,if​q∈\(1,∞\]\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq c\_\{q\}\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\begin\{cases\}\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\},&\\text\{ if \}q\\in\[\-\\infty,1\],\\\\ 3\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\},&\\text\{ if \}q\\in\(1,\\infty\]\.\\end\{cases\}

An ingredient for showing[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)for a range of values ofqqis the following lemma, which is a consequence of an inequality by[Ando et al\. \(1987\)](https://arxiv.org/html/2608.23765#bib.bib43)\.

###### Lemma 14\.

Let𝐀∈ℝd×d\\mathbf\{A\}\\in\\R^\{d\\times d\}be a positive semidefinite matrix and let𝐁∈ℝd×d\\mathbf\{B\}\\in\\R^\{d\\times d\}be arbitrary\. Let\|\|\|⋅\|\|\|\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\cdot\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}be any unitarily invariant norm onℝd×d\\R^\{d\\times d\}\. Then it holds that

‖\|𝐀∘𝐁\|‖≤maxi∈\[d\]​𝐀i​i​‖\|𝐁\|‖\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{A\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\underset\{i\\in\\left\[d\\right\]\}\{\\max\}\\ \\mathbf\{A\}\_\{ii\}\\ \{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.

Proof of Lemma[14](https://arxiv.org/html/2608.23765#Thmlemma14)From[Horn \(1990, Equation \(3\.7\.15\)\)](https://arxiv.org/html/2608.23765#bib.bib61)\(see also[Ando et al\., 1987](https://arxiv.org/html/2608.23765#bib.bib43), p\. 363, eq\. \(35\)\), it follows that

‖\|𝐀∘𝐁\|‖≤c1​\(𝐗\)​c1​\(𝐘\)​‖\|𝐁\|‖\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{A\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq c\_\{1\}\(\\mathbf\{X\}\)c\_\{1\}\(\\mathbf\{Y\}\)\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}for square matrices𝐀,𝐁∈ℝd×d\\mathbf\{A\},\\mathbf\{B\}\\in\\R^\{d\\times d\}, where𝐗,𝐘∈ℝr×d\\mathbf\{X\},\\mathbf\{Y\}\\in\\R^\{r\\times d\}are such that𝐀=𝐗∗​𝐘\\mathbf\{A\}=\\mathbf\{X\}^\{\*\}\\mathbf\{Y\}andc1​\(𝐗\),c1​\(𝐘\)c\_\{1\}\(\\mathbf\{X\}\),c\_\{1\}\(\\mathbf\{Y\}\)are the maximum Euclidean norms among the columns of𝐗\\mathbf\{X\}and𝐘\\mathbf\{Y\}, respectively\. Since𝐀\\mathbf\{A\}is positive semidefinite, we can choose𝐘=𝐗=𝚺1/2​𝐔∗\\mathbf\{Y\}=\\mathbf\{X\}=\\mathbf\{\\Sigma\}^\{1/2\}\\mathbf\{U\}^\{\*\}where𝐀=𝐔​𝚺​𝐔∗\\mathbf\{A\}=\\mathbf\{U\}\\mathbf\{\\Sigma\}\\mathbf\{U\}^\{\*\}is an eigendecomposition of𝐀\\mathbf\{A\}\. Writing𝚺=diag⁡\(σ\)\\mathbf\{\\Sigma\}=\\diag\(\\sigma\), we observe that

c1\(𝐘\)=c1\(𝐗\)=maxi∈\[d\]∥𝐗:,i∥2=maxi∈\[d\]∑j=1rσj​𝐔i​j2=maxi∈\[d\]\(𝐔​𝚺​𝐔∗\)i​i=maxi∈\[d\]𝐀i​i\.c\_\{1\}\(\\mathbf\{Y\}\)=c\_\{1\}\(\\mathbf\{X\}\)=\\max\_\{i\\in\[d\]\}\\\|\\mathbf\{X\}\_\{:,i\}\\\|\_\{2\}=\\max\_\{i\\in\[d\]\}\\sqrt\{\\sum\_\{j=1\}^\{r\}\\sigma\_\{j\}\\mathbf\{U\}\_\{ij\}^\{2\}\}=\\max\_\{i\\in\[d\]\}\\sqrt\{\(\\mathbf\{U\}\\mathbf\{\\Sigma\}\\mathbf\{U\}^\{\*\}\)\_\{ii\}\}=\\max\_\{i\\in\[d\]\}\\sqrt\{\\mathbf\{A\}\_\{ii\}\}\.This finishes the proof of Lemma[14](https://arxiv.org/html/2608.23765#Thmlemma14)\. In the proofs below,𝐀\\mathbf\{A\}will be chosen as \(a square extension of the\) weight operator core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}of Definition[1](https://arxiv.org/html/2608.23765#Thmdefinition1)\.

For the case of the harmonic\-mean core matrix[eq\.6](https://arxiv.org/html/2608.23765#S3.E6), which corresponds to a−1\-1\-power meanℳ−1\\mathcal\{M\}\_\{\-1\}in[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13),[Lemma14](https://arxiv.org/html/2608.23765#Thmlemma14)can be quite directly applied to obtain the desired norm bound of[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)due to the following positive semidefiniteness result\.

###### Lemma 15\.

Let the matrix𝐀∈ℝd×d\\mathbf\{A\}\\in\\R^\{d\\times d\}be of the form𝐀i​j=1λi\+λj\\mathbf\{A\}\_\{ij\}=\\frac\{1\}\{\\lambda\_\{i\}\+\\lambda\_\{j\}\}, whereλi\>0\\lambda\_\{i\}\>0for alli∈\[d\]i\\in\\left\[d\\right\]\. Then𝐀\\mathbf\{A\}is positive semidefinite\.

For a proof of[Lemma15](https://arxiv.org/html/2608.23765#Thmlemma15)we refer to[Bhatia \(2007, Exercise 1\.6\.4\)](https://arxiv.org/html/2608.23765#bib.bib59)\. The key idea is to show that𝐀\\mathbf\{A\}is the Gram matrix of suitably chosen vectors in a Hilbert space and, thus,𝐀\\mathbf\{A\}is positive semidefinite\.

In order to establish[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)across all cases ofqq, a bit more work is needed\. We provide the general proof below\.

Proof of[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)We distinguish several cases\.

Caseq=−∞q=\-\\infty:In this case we have

\(𝐇~𝝀\(q\)\)i​j=\(𝐇~𝝀\(−∞\)\)i​j=min⁡\(λi,λj\)=∫0λmax1t≤λi​1t≤λj​𝑑t\.\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\-\\infty\)\}\)\_\{ij\}=\\min\(\\lambda\_\{i\},\\lambda\_\{j\}\)=\\int\_\{0\}^\{\\lambda\_\{\\max\}\}1\_\{t\\leq\\lambda\_\{i\}\}1\_\{t\\leq\\lambda\_\{j\}\}\\,dt\.Hence, for any𝐱∈ℝd\\mathbf\{x\}\\in\\R^\{d\},

𝐱⊤​𝐇~𝝀\(−∞\)​𝐱\\displaystyle\\mathbf\{x\}^\{\\top\}\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\-\\infty\)\}\\mathbf\{x\}=∫0λmax\(∑i=1dxi​1t≤λi\)2​𝑑t≥0\.\\displaystyle=\\int\_\{0\}^\{\\lambda\_\{\\max\}\}\\left\(\\sum\_\{i=1\}^\{d\}x\_\{i\}1\_\{t\\leq\\lambda\_\{i\}\}\\right\)^\{2\}dt\\geq 0\.Thus𝐇~𝝀\(−∞\)\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\-\\infty\)\}is positive semidefinite\. Since\(𝐇~𝝀\(−∞\)\)i​i=min⁡\(λi,λi\)=λi\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\-\\infty\)\}\)\_\{ii\}=\\min\(\\lambda\_\{i\},\\lambda\_\{i\}\)=\\lambda\_\{i\},[Lemma14](https://arxiv.org/html/2608.23765#Thmlemma14)yields

‖\|𝐇~𝝀\(−∞\)∘𝐁\|‖≤λmax​‖\|𝐁\|‖\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\-\\infty\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.
Caseq∈\(−∞,0\)q\\in\(\-\\infty,0\):We first note that for alli,j∈\[d\]i,j\\in\[d\],λi,λj\>0\\lambda\_\{i\},\\lambda\_\{j\}\>0, and

\(𝐇~𝝀\(q\)\)i​j=ℳq​\(λi,λj\)=\(λiq\+λjq2\)1/q=2−1q​\(λiq\+λjq\)1/q\.\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=\\mathcal\{M\}\_\{q\}\(\\lambda\_\{i\},\\lambda\_\{j\}\)=\\left\(\\frac\{\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\}\{2\}\\right\)^\{1/q\}=2^\{\-\\frac\{1\}\{q\}\}\\left\(\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\\right\)^\{1/q\}\.In order to proceed, we recall that the Gamma functionΓ\\Gammais defined byΓ⁡\(z\)=∫0∞tz−1​e−t​𝑑t\\Gamma\(z\)=\\int\_\{0\}^\{\\infty\}t^\{z\-1\}e^\{\-t\}dtforz∈ℂz\\in\\mathbb\{C\}withℜ​𝔢⁡\(z\)\>0\\operatorname\{\\mathfrak\{Re\}\}\(z\)\>0\. Now use the substitutionu:=t/xu:=t/xforx\>0x\>0\. Then we have thatd​t=x​d​udt=xdu\. Hence, we obtain that

Γ⁡\(z\)=∫0∞tz−1​e−t​𝑑t=∫0∞\(x​u\)z−1​e−x​u​x​𝑑u=xz​∫0∞uz−1​e−x​u​𝑑u\.\\Gamma\(z\)=\\int\_\{0\}^\{\\infty\}t^\{z\-1\}e^\{\-t\}dt=\\int\_\{0\}^\{\\infty\}\\left\(xu\\right\)^\{z\-1\}e^\{\-xu\}xdu=x^\{z\}\\int\_\{0\}^\{\\infty\}u^\{z\-1\}e^\{\-xu\}du\.Then we obtain the well\-known integral representation

x−z=1Γ⁡\(z\)​∫0∞uz−1​e−x​u​𝑑ux^\{\-z\}=\\frac\{1\}\{\\Gamma\(z\)\}\\int\_\{0\}^\{\\infty\}u^\{z\-1\}e^\{\-xu\}duforx\>0x\>0andz\>0z\>0\. Using this integral representation, we can write

\(λiq\+λjq\)1/q\\displaystyle\\left\(\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\\right\)^\{1/q\}=\(λiq\+λjq\)−1/\|q\|=1Γ⁡\(1/\|q\|\)∫0∞u1\|q\|−1e−u⁡\(λiq\+λjq\)du\.\\displaystyle=\\left\(\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\\right\)^\{\-1/\|q\|\}=\\frac\{1\}\{\\Gamma\(1/\|q\|\)\}\\int\_\{0\}^\{\\infty\}u^\{\\frac\{1\}\{\|q\|\}\-1\}e^\{\-u\\left\(\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\\right\)\}du\.Now let𝐱∈ℝd\\mathbf\{x\}\\in\\R^\{d\}be arbitrary\. Then we have that

𝐱⊤​𝐇~𝝀\(q\)​𝐱=\\displaystyle\\mathbf\{x\}^\{\\top\}\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\mathbf\{x\}=21/\|q\|Γ⁡\(1/\|q\|\)​∑i,j=1dxi​xj​∫0∞u1\|q\|−1​e−u⁡\(λiq\+λjq\)​𝑑u\\displaystyle\\frac\{2^\{1/\|q\|\}\}\{\\Gamma\(1/\|q\|\)\}\\sum\_\{i,j=1\}^\{d\}x\_\{i\}x\_\{j\}\\int\_\{0\}^\{\\infty\}u^\{\\frac\{1\}\{\|q\|\}\-1\}e^\{\-u\\left\(\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\\right\)\}du=\\displaystyle=21/\|q\|Γ⁡\(1/\|q\|\)​∫0∞u1\|q\|−1​\(∑i=1dxi​e−u​λiq\)2​𝑑u≥0\.\\displaystyle\\frac\{2^\{1/\|q\|\}\}\{\\Gamma\(1/\|q\|\)\}\\int\_\{0\}^\{\\infty\}u^\{\\frac\{1\}\{\|q\|\}\-1\}\\left\(\\sum\_\{i=1\}^\{d\}x\_\{i\}e^\{\-u\\lambda\_\{i\}^\{q\}\}\\right\)^\{2\}du\\geq 0\.It follows that𝐇~𝝀\(q\)\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}is positive semidefinite\. Since

\(𝐇~𝝀\(q\)\)i​i=ℳq​\(λi,λi\)=λi,\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ii\}=\\mathcal\{M\}\_\{q\}\(\\lambda\_\{i\},\\lambda\_\{i\}\)=\\lambda\_\{i\},Lemma[14](https://arxiv.org/html/2608.23765#Thmlemma14)yields‖\|𝐇~𝝀\(q\)∘𝐁\|‖≤λmax​‖\|𝐁\|‖\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.

Caseq=0q=0:In this case

\(𝐇~𝝀\(q\)\)i​j=\(𝐇~𝝀\(0\)\)i​j=λi​λj\.\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(0\)\}\)\_\{ij\}=\\sqrt\{\\lambda\_\{i\}\\lambda\_\{j\}\}\.Hence𝐇~𝝀\(0\)=𝐯𝐯⊤\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(0\)\}=\\mathbf\{v\}\\mathbf\{v\}^\{\\top\}with𝐯:=\(λ1,…,λd\)⊤\\mathbf\{v\}:=\(\\sqrt\{\\lambda\_\{1\}\},\\ldots,\\sqrt\{\\lambda\_\{d\}\}\)^\{\\top\}, so𝐇~𝝀\(0\)\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(0\)\}is positive semidefinite and\(𝐇~𝝀\(0\)\)i​i=λi\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(0\)\}\)\_\{ii\}=\\lambda\_\{i\}\. Therefore Lemma[14](https://arxiv.org/html/2608.23765#Thmlemma14)gives‖\|𝐇~𝝀∘𝐁\|‖≤λmax​‖\|𝐁\|‖\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.

Caseq∈\(0,∞\)q\\in\(0,\\infty\):In this case we have that

\(𝐇~𝝀\(q\)\)i​j=ℳq​\(λi,λj\)=\(λiq\+λjq2\)1/q\.\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=\\mathcal\{M\}\_\{q\}\(\\lambda\_\{i\},\\lambda\_\{j\}\)=\\left\(\\frac\{\\lambda\_\{i\}^\{q\}\+\\lambda\_\{j\}^\{q\}\}\{2\}\\right\)^\{1/q\}\.Now define the functionf\(x,y\)=2−1/q\(xq\+yq\)1/qf\(x,y\)=2^\{\-1/q\}\(x^\{q\}\+y^\{q\}\)^\{1/q\}forx,y≥0x,y\\geq 0\. Then the two\-dimensional fundamental theorem of calculus implies that

f⁡\(x,y\)\\displaystyle f\(x,y\)=f⁡\(x,0\)\+f⁡\(0,y\)−f⁡\(0,0\)\+∫0x∫0y∂2f∂x​∂y​\(s,t\)​𝑑t​𝑑s\\displaystyle=f\(x,0\)\+f\(0,y\)\-f\(0,0\)\+\\int\_\{0\}^\{x\}\\int\_\{0\}^\{y\}\\frac\{\\partial^\{2\}f\}\{\\partial x\\partial y\}\(s,t\)dtds=x21/q\+y21/q\+1−q21/q​∫0x∫0ysq−1​tq−1​\(sq\+tq\)1/q−2​𝑑t​𝑑s\.\\displaystyle=\\frac\{x\}\{2^\{1/q\}\}\+\\frac\{y\}\{2^\{1/q\}\}\+\\frac\{1\-q\}\{2^\{1/q\}\}\\int\_\{0\}^\{x\}\\int\_\{0\}^\{y\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}dtds\.It follows that

\(𝐇~𝝀\(q\)\)i​j\\displaystyle\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=λi21/q\+λj21/q\+1−q21/q​∫0λi∫0λjsq−1​tq−1​\(sq\+tq\)1/q−2​𝑑t​𝑑s\\displaystyle=\\frac\{\\lambda\_\{i\}\}\{2^\{1/q\}\}\+\\frac\{\\lambda\_\{j\}\}\{2^\{1/q\}\}\+\\frac\{1\-q\}\{2^\{1/q\}\}\\int\_\{0\}^\{\\lambda\_\{i\}\}\\int\_\{0\}^\{\\lambda\_\{j\}\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}dtds=λi21/q\+λj21/q\+1−q21/q​∫0∞∫0∞sq−1​tq−1​\(sq\+tq\)1/q−2​1s≤λi​1t≤λj​𝑑t​𝑑s\.\\displaystyle=\\frac\{\\lambda\_\{i\}\}\{2^\{1/q\}\}\+\\frac\{\\lambda\_\{j\}\}\{2^\{1/q\}\}\+\\frac\{1\-q\}\{2^\{1/q\}\}\\int\_\{0\}^\{\\infty\}\\int\_\{0\}^\{\\infty\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}1\_\{s\\leq\\lambda\_\{i\}\}1\_\{t\\leq\\lambda\_\{j\}\}dtds\.Denoting by𝐃:=diag⁡\(𝝀\)\\mathbf\{D\}:=\\diag\(\\bm\{\\lambda\}\)and by

𝐏s:=diag⁡\(1s≤λi\)i=1d,𝐏t:=diag⁡\(1t≤λi\)i=1d,\\mathbf\{P\}\_\{s\}:=\\diag\\big\(1\_\{s\\leq\\lambda\_\{i\}\}\\big\)\_\{i=1\}^\{d\},\\qquad\\mathbf\{P\}\_\{t\}:=\\diag\\big\(1\_\{t\\leq\\lambda\_\{i\}\}\\big\)\_\{i=1\}^\{d\},we obtain

𝐇~𝝀\(q\)∘𝐁=\\displaystyle\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}=121/q​𝐃𝐁\+121/q​𝐁𝐃\\displaystyle\\frac\{1\}\{2^\{1/q\}\}\\mathbf\{D\}\\mathbf\{B\}\+\\frac\{1\}\{2^\{1/q\}\}\\mathbf\{B\}\\mathbf\{D\}\+1−q21/q∫0∞∫0∞sq−1tq−1\(sq\+tq\)1/q−2𝐏s𝐁𝐏tdtds\.\\displaystyle\+\\frac\{1\-q\}\{2^\{1/q\}\}\\int\_\{0\}^\{\\infty\}\\int\_\{0\}^\{\\infty\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}\\mathbf\{P\}\_\{s\}\\mathbf\{B\}\\mathbf\{P\}\_\{t\}dtds\.Since‖𝐃‖=λmax\\\|\\mathbf\{D\}\\\|=\\lambda\_\{\\max\}and‖𝐏s‖,‖𝐏t‖≤1\\\|\\mathbf\{P\}\_\{s\}\\\|,\\\|\\mathbf\{P\}\_\{t\}\\\|\\leq 1, the ideal property of unitarily invariant norms yields

‖\|𝐇~𝝀\(q\)∘𝐁\|‖≤\\displaystyle\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq121/q​‖\|𝐃𝐁\|‖\+121/q​‖\|𝐁𝐃\|‖\\displaystyle\\frac\{1\}\{2^\{1/q\}\}\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{D\}\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\+\\frac\{1\}\{2^\{1/q\}\}\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\mathbf\{D\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\+\|1−q\|21/q∫0λmax∫0λmaxsq−1tq−1\(sq\+tq\)1/q−2\|\|\|𝐏s𝐁𝐏t\|\|\|dtds\\displaystyle\+\\frac\{\|1\-q\|\}\{2^\{1/q\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{P\}\_\{s\}\\mathbf\{B\}\\mathbf\{P\}\_\{t\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}dtds≤\\displaystyle\\leq21−1q​λmax​‖\|𝐁\|‖\\displaystyle 2^\{1\-\\frac\{1\}\{q\}\}\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\+\|1−q\|21/q\|\|\|𝐁\|\|\|∫0λmax∫0λmaxsq−1tq−1\(sq\+tq\)1/q−2dtds\.\\displaystyle\+\\frac\{\|1\-q\|\}\{2^\{1/q\}\}\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\int\_\{0\}^\{\\lambda\_\{\\max\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}dtds\.Ifq=1q=1, the integral term vanishes and therefore

‖\|𝐇~𝝀\(q\)∘𝐁\|‖≤λmax​‖\|𝐁\|‖\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.Assume now thatq≠1q\\neq 1\. Using the substitutionu=squ=s^\{q\}andv=tqv=t^\{q\}, we obtain

∫0λmax∫0λmaxsq−1​tq−1​\(sq\+tq\)1/q−2​𝑑t​𝑑s\\displaystyle\\int\_\{0\}^\{\\lambda\_\{\\max\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}\}s^\{q\-1\}t^\{q\-1\}\(s^\{q\}\+t^\{q\}\)^\{1/q\-2\}dtds=1q2​∫0λmaxq∫0λmaxq\(u\+v\)1/q−2​𝑑u​𝑑v\\displaystyle=\\frac\{1\}\{q^\{2\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}^\{q\}\}\\int\_\{0\}^\{\\lambda\_\{\\max\}^\{q\}\}\(u\+v\)^\{1/q\-2\}dudv=1q2​\(1/q−1\)​∫0λmaxq\[\(u\+v\)1/q−1\]u=0u=λmaxq​𝑑v\\displaystyle=\\frac\{1\}\{q^\{2\}\(1/q\-1\)\}\\int\_\{0\}^\{\\lambda\_\{\\max\}^\{q\}\}\\left\[\(u\+v\)^\{1/q\-1\}\\right\]\_\{u=0\}^\{u=\\lambda\_\{\\max\}^\{q\}\}dv=1q⁡\(1−q\)​∫0λmaxq\(\(λmaxq\+v\)1/q−1−v1/q−1\)​𝑑v\\displaystyle=\\frac\{1\}\{q\(1\-q\)\}\\int\_\{0\}^\{\\lambda\_\{\\max\}^\{q\}\}\\left\(\(\\lambda\_\{\\max\}^\{q\}\+v\)^\{1/q\-1\}\-v^\{1/q\-1\}\\right\)dv=11−q​\[\(λmaxq\+v\)1/q−v1/q\]v=0v=λmaxq\\displaystyle=\\frac\{1\}\{1\-q\}\\left\[\(\\lambda\_\{\\max\}^\{q\}\+v\)^\{1/q\}\-v^\{1/q\}\\right\]\_\{v=0\}^\{v=\\lambda\_\{\\max\}^\{q\}\}=λmax1−q​\(21/q−2\)\.\\displaystyle=\\frac\{\\lambda\_\{\\max\}\}\{1\-q\}\(2^\{1/q\}\-2\)\.Therefore

‖\|𝐇~𝝀\(q\)∘𝐁\|‖\\displaystyle\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}≤\(21−1q\+\|1−21−1/q\|\)​λmax​‖\|𝐁\|‖\.\\displaystyle\\leq\\left\(2^\{1\-\\frac\{1\}\{q\}\}\+\|1\-2^\{1\-1/q\}\|\\right\)\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.For0<q≤10<q\\leq 1, the constant in parentheses equals11\. For1<q<∞1<q<\\infty, it equals22−1/q−12^\{2\-1/q\}\-1\.

Caseq=∞q=\\infty:In this case

\(𝐇~𝝀\(q\)\)i​j=\(𝐇~𝝀\(∞\)\)i​j=max⁡\(λi,λj\)=λi\+λj−min⁡\(λi,λj\)\.\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(q\)\}\)\_\{ij\}=\(\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\\infty\)\}\)\_\{ij\}=\\max\(\\lambda\_\{i\},\\lambda\_\{j\}\)=\\lambda\_\{i\}\+\\lambda\_\{j\}\-\\min\(\\lambda\_\{i\},\\lambda\_\{j\}\)\.Let𝐌∈ℝd×d\\mathbf\{M\}\\in\\R^\{d\\times d\}be defined by𝐌i​j:=min⁡\(λi,λj\)\\mathbf\{M\}\_\{ij\}:=\\min\(\\lambda\_\{i\},\\lambda\_\{j\}\)for eachi,j∈\[d\]i,j\\in\[d\]\. Then

𝐇~𝝀\(∞\)∘𝐁=diag⁡\(𝝀\)​𝐁\+𝐁​diag⁡\(𝝀\)−𝐌∘𝐁\.\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\\infty\)\}\\circ\\mathbf\{B\}=\\diag\(\\bm\{\\lambda\}\)\\mathbf\{B\}\+\\mathbf\{B\}\\diag\(\\bm\{\\lambda\}\)\-\\mathbf\{M\}\\circ\\mathbf\{B\}\.By the ideal property of unitarily invariant norms,

‖\|diag⁡\(𝝀\)​𝐁\|‖,‖\|𝐁​diag⁡\(𝝀\)\|‖≤λmax​‖\|𝐁\|‖\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\diag\(\\bm\{\\lambda\}\)\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\},\\ \{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\diag\(\\bm\{\\lambda\}\)\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.Moreover, the matrix𝐌\\mathbf\{M\}is positive semidefinite by the argument from the caseq=−∞q=\-\\infty, and𝐌i​i=λi\\mathbf\{M\}\_\{ii\}=\\lambda\_\{i\}\. Hence Lemma[14](https://arxiv.org/html/2608.23765#Thmlemma14)yields‖\|𝐌∘𝐁\|‖≤λmax​‖\|𝐁\|‖\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{M\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\. Therefore

‖\|𝐇~𝝀\(∞\)∘𝐁\|‖≤\(λmax​‖\|𝐁\|‖\+λmax​‖\|𝐁\|‖\+λmax​‖\|𝐁\|‖\)=3​λmax​‖\|𝐁\|‖\.\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\widetilde\{\\mathbf\{H\}\}\_\{\\bm\{\\lambda\}\}^\{\(\\infty\)\}\\circ\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\leq\\left\(\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\+\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\+\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\\right\)=3\\lambda\_\{\\max\}\\,\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\mathbf\{B\}\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}\.
The bounds of[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)can now be used to upper bound the quadratic term\(b\)\(b\)in inequality \([64](https://arxiv.org/html/2608.23765#A1.E64)\) as follows\.

###### Lemma 16\.

Assume that

𝒜⁡\(𝐗⋆\)=𝒜⁡\(𝐗\)\\mathcal\{A\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)=\\mathcal\{A\}\\left\(\\mathbf\{X\}\\right\)and that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP of orderrrof[Definition4](https://arxiv.org/html/2608.23765#Thmdefinition4)for some constant0<ηr<10<\\eta\_\{r\}<1, and the NSP of order11with0<η1<10<\\eta\_\{1\}<1\. LetW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)be defined as in[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2)\. Then

⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F≤η1​‖𝐗⋆−𝐗‖∗2ε,\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\leq\\frac\{\\eta\_\{1\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\\varepsilon\},\(67\)if the weight operator core matrix𝐇𝛔,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}ofW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)corresponds to harmonic mean[eq\.6](https://arxiv.org/html/2608.23765#S3.E6), one\-sided weights[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5), or power mean weights[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq∈\[−∞,1\]q\\in\[\-\\infty,1\], and

⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F≤cq​η1​‖𝐗⋆−𝐗‖∗2ε≤3​η1​‖𝐗⋆−𝐗‖∗2ε,\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\leq\\frac\{c\_\{q\}\\eta\_\{1\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\\varepsilon\}\\leq\\frac\{3\\eta\_\{1\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\\varepsilon\},\(68\)if𝐇𝛔,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}corresponds to power mean weights[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq∈\(1,∞\]q\\in\(1,\\infty\], wherecqc\_\{q\}is the constant[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)\.

ProofLet𝐍:=𝐗⋆−𝐗\\mathbf\{N\}:=\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\. We may assume without loss of generality thatd1≤d2d\_\{1\}\\leq d\_\{2\}\. For a singular value decomposition𝐗=𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}of𝐗\\mathbf\{X\}, we set𝐌:=𝐔𝐗⊤​𝐍𝐕𝐗\\mathbf\{M\}:=\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{N\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\. First, we notice that

⟨𝐍,W𝐗,ε​\(𝐍\)⟩F\\displaystyle\\langle\\mathbf\{N\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{N\}\)\\rangle\_\{F\}≤‖𝐍‖∗​‖W𝐗,ε​\(𝐍\)‖\\displaystyle\\leq\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\\left\\lVert W\_\{\\mathbf\{X\},\\varepsilon\}\\left\(\\mathbf\{N\}\\right\)\\right\\rVert=‖𝐍‖∗​‖𝐔𝐗​\[𝐇𝝈,ε∘𝐌\]​𝐕𝐗⊤‖\\displaystyle=\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\right\\rVert≤‖𝐍‖∗​‖𝐇𝝈,ε∘𝐌‖\.\\displaystyle\\leq\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\\rVert\.\(69\)We now show that‖𝐇𝝈,ε∘𝐌‖≤cq​‖𝐍‖ε\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\\rVert\\leq\\frac\{c\_\{q\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\}\{\\varepsilon\}\.

One\-sided weights:For left\-sided weights,\(𝐇𝝈,ε\)i​j=1max⁡\(σi​\(𝐗\),ε\)\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ij\}=\\frac\{1\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}depends only on the row indexii\. Hence𝐇𝝈,ε∘𝐌=diag⁡\(h1,…,hd1\)​𝐌\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}=\\diag\(h\_\{1\},\\ldots,h\_\{d\_\{1\}\}\)\\mathbf\{M\}wherehi=1max⁡\(σi​\(𝐗\),ε\)h\_\{i\}=\\frac\{1\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}\. Sincemaxi⁡hi≤1/ε\\max\_\{i\}h\_\{i\}\\leq 1/\\varepsilon, we obtain

‖𝐇𝝈,ε∘𝐌‖≤‖𝐌‖ε=‖𝐍‖ε\.\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\\rVert\\leq\\frac\{\\left\\lVert\\mathbf\{M\}\\right\\rVert\}\{\\varepsilon\}=\\frac\{\\left\\lVert\\mathbf\{N\}\\right\\rVert\}\{\\varepsilon\}\.For right\-sided weights the argument is analogous with multiplication from the right\.

Power mean weights \(including harmonic mean\):Define thed2×d2d\_\{2\}\\times d\_\{2\}matrix𝐇^\\widehat\{\\mathbf\{H\}\}by

𝐇^i​j:=ℳq​\(1max⁡\(σ^i,ε\),1max⁡\(σ^j,ε\)\),\\widehat\{\\mathbf\{H\}\}\_\{ij\}:=\\mathcal\{M\}\_\{q\}\\\!\\left\(\\frac\{1\}\{\\max\(\\hat\{\\sigma\}\_\{i\},\\varepsilon\)\},\\,\\frac\{1\}\{\\max\(\\hat\{\\sigma\}\_\{j\},\\varepsilon\)\}\\right\),whereσ^i:=σi​\(𝐗\)\\hat\{\\sigma\}\_\{i\}:=\\sigma\_\{i\}\(\\mathbf\{X\}\)fori∈\[d1\]i\\in\[d\_\{1\}\]andσ^i:=0\\hat\{\\sigma\}\_\{i\}:=0fori∈\{d1\+1,…,d2\}i\\in\\\{d\_\{1\}\+1,\\ldots,d\_\{2\}\\\}\. Note thatmax⁡\(σ^i,ε\)=ε\\max\(\\hat\{\\sigma\}\_\{i\},\\varepsilon\)=\\varepsilonfori\>d1i\>d\_\{1\}, and therefore the firstd1d\_\{1\}rows of𝐇^\\widehat\{\\mathbf\{H\}\}coincide with𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\. Define𝐌^∈ℝd2×d2\\widehat\{\\mathbf\{M\}\}\\in\\R^\{d\_\{2\}\\times d\_\{2\}\}by setting𝐌^i​j=𝐌i​j\\widehat\{\\mathbf\{M\}\}\_\{ij\}=\\mathbf\{M\}\_\{ij\}fori∈\[d1\]i\\in\[d\_\{1\}\]and𝐌^i​j=0\\widehat\{\\mathbf\{M\}\}\_\{ij\}=0fori\>d1i\>d\_\{1\}\. Since the rows of𝐇^∘𝐌^\\widehat\{\\mathbf\{H\}\}\\circ\\widehat\{\\mathbf\{M\}\}with indexi\>d1i\>d\_\{1\}are all zero, it follows that‖𝐇𝝈,ε∘𝐌‖=‖𝐇^∘𝐌^‖\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\\rVert=\\left\\lVert\\widehat\{\\mathbf\{H\}\}\\circ\\widehat\{\\mathbf\{M\}\}\\right\\rVert\. We can now invoke[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)for𝐇^\\widehat\{\\mathbf\{H\}\}and𝐌^\\widehat\{\\mathbf\{M\}\}with the positive numbersλi:=1max⁡\(σ^i,ε\)\\lambda\_\{i\}:=\\frac\{1\}\{\\max\(\\hat\{\\sigma\}\_\{i\},\\varepsilon\)\},i∈\[d2\]i\\in\[d\_\{2\}\], and for the spectral norm\|\|\|⋅\|\|\|=∥⋅∥\{\\left\|\\kern\-1\.07639pt\\left\|\\kern\-1\.07639pt\\left\|\\cdot\\right\|\\kern\-1\.07639pt\\right\|\\kern\-1\.07639pt\\right\|\}=\\\|\\cdot\\\|, which is unitarily invariant\. Sinceλmax=maxi∈\[d2\]⁡λi≤1ε\\lambda\_\{\\max\}=\\max\_\{i\\in\[d\_\{2\}\]\}\\lambda\_\{i\}\\leq\\frac\{1\}\{\\varepsilon\}, this yields for everyq∈\[−∞,∞\]q\\in\[\-\\infty,\\infty\]that

‖𝐇^∘𝐌^‖≤cq​λmax​‖𝐌^‖≤cq​‖𝐌^‖ε=cq​‖𝐍‖ε,\\left\\lVert\\widehat\{\\mathbf\{H\}\}\\circ\\widehat\{\\mathbf\{M\}\}\\right\\rVert\\leq c\_\{q\}\\,\\lambda\_\{\\max\}\\left\\lVert\\widehat\{\\mathbf\{M\}\}\\right\\rVert\\leq\\frac\{c\_\{q\}\\left\\lVert\\widehat\{\\mathbf\{M\}\}\\right\\rVert\}\{\\varepsilon\}=\\frac\{c\_\{q\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\}\{\\varepsilon\},wherecqc\_\{q\}is the constant[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)and where we used in the last equality that

‖𝐌^‖=‖𝐌‖=‖𝐔𝐗⊤​𝐍𝐕𝐗‖=‖𝐍‖,\\left\\lVert\\widehat\{\\mathbf\{M\}\}\\right\\rVert=\\left\\lVert\\mathbf\{M\}\\right\\rVert=\\left\\lVert\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{N\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\right\\rVert=\\left\\lVert\\mathbf\{N\}\\right\\rVert,which follows from the unitary invariance of the spectral norm\.

In all cases, we have thus shown that‖𝐇𝝈,ε∘𝐌‖≤cq​‖𝐍‖ε\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathbf\{M\}\\right\\rVert\\leq\\frac\{c\_\{q\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\}\{\\varepsilon\}, which combined with \([69](https://arxiv.org/html/2608.23765#A1.E69)\) gives

⟨𝐍,W𝐗,ε​\(𝐍\)⟩F≤cq​‖𝐍‖​‖𝐍‖∗ε\.\\langle\\mathbf\{N\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{N\}\)\\rangle\_\{F\}\\leq\\frac\{c\_\{q\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\}\{\\varepsilon\}\.This implies that

⟨𝐍,W𝐗,ε​\(𝐍\)⟩F≤cq​η1​‖𝐍‖∗2ε\.\\langle\\mathbf\{N\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{N\}\)\\rangle\_\{F\}\\leq\\frac\{c\_\{q\}\\,\\eta\_\{1\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\\varepsilon\}\.by the order\-one NSP\. Sincecq=1c\_\{q\}=1forq∈\[−∞,1\]q\\in\[\-\\infty,1\], this proves[eq\.67](https://arxiv.org/html/2608.23765#A1.E67), including for the harmonic\-mean case ofq=−1q=\-1, while1<cq≤31<c\_\{q\}\\leq 3forq∈\(1,∞\]q\\in\(1,\\infty\]proves[eq\.68](https://arxiv.org/html/2608.23765#A1.E68)\.

#### A\.3\.3Proof of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)

Having derived estimates for terms\(a\)\(a\)and\(b\)\(b\)in inequality \([64](https://arxiv.org/html/2608.23765#A1.E64)\), see Lemma[12](https://arxiv.org/html/2608.23765#Thmlemma12)and Lemma[16](https://arxiv.org/html/2608.23765#Thmlemma16), the following proposition quantifies the decrease of𝒥εk​\(𝐗\(k\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)in each iteration\.

###### Proposition 2\.

Let𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP[eq\.24](https://arxiv.org/html/2608.23765#S4.E24)of orderrrfor some0<ηr<10<\\eta\_\{r\}<1and that𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\. Let the IRLS iterates\{𝐗\(k\)\}k\\left\\\{\\mathbf\{X\}^\{\(k\)\}\\right\\\}\_\{k\}and\{εk\}k\\left\\\{\\varepsilon\_\{k\}\\right\\\}\_\{k\}be defined by[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)and[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)with arbitrary positive definite initial weight operatorW\(0\)W^\{\(0\)\}and rank estimater~=r\\widetilde\{r\}=r, and assume that[Algorithm1](https://arxiv.org/html/2608.23765#alg1)does not return in iterationkk, i\.e\., thatεk\>0\\varepsilon\_\{k\}\>0, so that the iterate𝐗\(k\+1\)\\mathbf\{X\}^\{\(k\+1\)\}is defined\. Assume that the weight operatorW𝐗\(k\),εk​\(⋅\)W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\(\\cdot\)used in each iteration of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)is defined as in[Definition1](https://arxiv.org/html/2608.23765#Thmdefinition1)is admissible in the sense of[Definition5](https://arxiv.org/html/2608.23765#Thmdefinition5), letcqc\_\{q\}be the weight operator\-dependent constant[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)of[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13)\. Set

γ:=\{3/4if​βr​\(𝐗⋆\)∗=0,1/2otherwise\.\\gamma:=\\begin\{cases\}3/4\\quad&\\text\{if \}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0,\\\\ 1/2\\quad&\\text\{otherwise\}\.\\end\{cases\}If

βr​\(𝐗⋆\)∗≤19​‖𝐗⋆−𝐗\(k\)‖∗andγ\>2​ηr1\+ηr,\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\\quad\\text\{and\}\\quad\\gamma\>\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\},then it holds that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤\(1−Cηr,γcq​η1​d\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\-\\frac\{C\_\{\\eta\_\{r\},\\gamma\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)where the constantCηrC\_\{\\eta\_\{r\}\}is defined by

Cηr,γ:=\(γ−2​ηr1\+ηr\)2\(3\+2​ηr\)\.C\_\{\\eta\_\{r\},\\gamma\}:=\\frac\{\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{\\left\(3\+2\\eta\_\{r\}\\right\)\}\.

ProofSet𝐍\(k\):=𝐗⋆−𝐗\(k\)\\mathbf\{N\}^\{\(k\)\}:=\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\. It follows from the majorization of𝒥εk​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\cdot\)by the quadratic model functionQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\)\(see[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1),[Theorem2](https://arxiv.org/html/2608.23765#Thmtheorem2), and[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)\) for weight operators in question that for anyt∈ℝt\\in\\mathbb\{R\}, it holds that

𝒥εk\+1​\(𝐗\(k\+1\)\)≤Qεk​\(𝐗\(k\+1\)∣𝐗\(k\)\)≤Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\),\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\\leq Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\\leq Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\),\(70\)where we used the optimality of𝐗\(k\+1\)\\mathbf\{X\}^\{\(k\+1\)\}in[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)in the second inequality\. Moreover, by the definition of the quadratic objectiveQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\), see[eq\.8](https://arxiv.org/html/2608.23765#S3.E8), it holds that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)=t​⟨∇𝒥εk​\(𝐗\(k\)\),𝐍\(k\)⟩F\+t22​⟨𝐍\(k\),W𝐗\(k\),εk​\(𝐍\(k\)\)⟩F\.\\begin\{split\}&Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\\\ =&t\\,\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\),\\mathbf\{N\}^\{\(k\)\}\\rangle\_\{F\}\+\\frac\{t^\{2\}\}\{2\}\\langle\\mathbf\{N\}^\{\(k\)\},W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\(\\mathbf\{N\}^\{\(k\)\}\)\\rangle\_\{F\}\.\\end\{split\}Our goal is to minimize the differenceQεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)<0Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)<0by choosingt≥0t\\geq 0accordingly\. By using Lemma[12](https://arxiv.org/html/2608.23765#Thmlemma12)as well as Lemma[16](https://arxiv.org/html/2608.23765#Thmlemma16), it follows that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)\\displaystyle Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)≤\\displaystyle\\leq−t⁡\(1−ηr1\+ηr−1/4\)​‖𝐍\(k\)‖∗\+94​t​βr​\(𝐗⋆\)∗\+t2​cq​η1​‖𝐍\(k\)‖∗22​εk\\displaystyle\-t\\left\(\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\-1/4\\right\)\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\+\\frac\{9\}\{4\}t\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\+t^\{2\}\\frac\{c\_\{q\}\\eta\_\{1\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{2\\varepsilon\_\{k\}\}=\\displaystyle=−t⁡\(34−2​ηr1\+ηr\)​‖𝐍\(k\)‖∗\+94​t​βr​\(𝐗⋆\)∗\+t2​cq​η1​‖𝐍\(k\)‖∗22​εk,\\displaystyle\-t\\left\(\\frac\{3\}\{4\}\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\+\\frac\{9\}\{4\}t\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\+t^\{2\}\\frac\{c\_\{q\}\\eta\_\{1\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{2\\varepsilon\_\{k\}\},wherecqc\_\{q\}is the constant[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)\. Consequently, we obtain that

Qεk\(𝐗\(k\)\+t𝐍\(k\)∣𝐗\(k\)\)−𝒥εk\(𝐗\(k\)\)≤−t⋅\(γ−2​ηr1\+ηr\)​‖𝐍\(k\)‖∗⏟=:b\+t2⋅cq​η1​‖𝐍\(k\)‖∗22​εk⏟=:a,Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\-t\\cdot\\underset\{\{=:b\}\}\{\\underbrace\{\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\}\}\+t^\{2\}\\cdot\\underset\{\{=:a\}\}\{\\underbrace\{\\frac\{c\_\{q\}\\eta\_\{1\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{2\\varepsilon\_\{k\}\}\}\},where we have used the assumptionβr​\(𝐗⋆\)∗≤19​‖𝐍\(k\)‖∗\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\. The right\-hand side is minimized byt:=b2​at:=\\frac\{b\}\{2a\}\. We obtain that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)≤−b24​a=−\(γ−2​ηr1\+ηr\)2​εk2​cq​η1\.\\displaystyle Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\\frac\{\-b^\{2\}\}\{4a\}=\\frac\{\-\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\\varepsilon\_\{k\}\}\{2c\_\{q\}\\eta\_\{1\}\}\.Together with inequality \([70](https://arxiv.org/html/2608.23765#A1.E70)\) this yields that

𝒥εk\+1​\(𝐗\(k\+1\)\)−𝒥εk​\(𝐗\(k\)\)≤−\(γ−2​ηr1\+ηr\)2​εk2​cq​η1\.\\begin\{split\}\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\\frac\{\-\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\\varepsilon\_\{k\}\}\{2c\_\{q\}\\eta\_\{1\}\}\.\\end\{split\}In particular, we obtain that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗−\(γ−2​ηr1\+ηr\)22​cq​η1​εk\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\frac\{\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{2c\_\{q\}\\eta\_\{1\}\}\\varepsilon\_\{k\}\.\(71\)In order to proceed, we need to boundεk\\varepsilon\_\{k\}from below\. For that, we note that

εk=min⁡\(εk−1,βr​\(𝐗\(k\)\)∗d\)=βr​\(𝐗\(ℓ\)\)∗d\\varepsilon\_\{k\}=\\min\\left\(\\varepsilon\_\{k\-1\},\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\}\{d\}\\right\)=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\ell\)\}\)\_\{\*\}\}\{d\}for someℓ≤k\\ell\\leq k\. Using[Lemma11](https://arxiv.org/html/2608.23765#Thmlemma11), we obtain the inequality chain

d​εk=βr​\(𝐗\(ℓ\)\)∗\\displaystyle d\\varepsilon\_\{k\}=\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\ell\)\}\)\_\{\*\}≥13/2\+ηr​\(𝒥εℓ​\(𝐗\(ℓ\)\)−‖𝐗⋆‖∗\)≥13/2\+ηr​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\),\\displaystyle\\geq\\frac\{1\}\{3/2\+\\eta\_\{r\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{\\ell\}\}\\left\(\\mathbf\{X\}^\{\(\\ell\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\\geq\\frac\{1\}\{3/2\+\\eta\_\{r\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\),where in the second inequality we have used that𝒥εk​\(𝐗\(k\)\)≤𝒥εℓ​\(𝐗\(ℓ\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\\leq\\mathcal\{J\}\_\{\\varepsilon\_\{\\ell\}\}\\left\(\\mathbf\{X\}^\{\(\\ell\)\}\\right\), which follows from the monotonicity of𝒥εk​\(𝐗\(k\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)inkk, see \([15](https://arxiv.org/html/2608.23765#S3.E15)\)\. Plugging this into[eq\.71](https://arxiv.org/html/2608.23765#A1.E71)leads to

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(1−\(γ−2​ηr1\+ηr\)22​cq​\(3/2\+ηr\)​η1​d\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)\\displaystyle\\leq\\left\(1\-\\frac\{\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{2c\_\{q\}\(3/2\+\\eta\_\{r\}\)\\eta\_\{1\}d\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)=\(1−\(γ−2​ηr1\+ηr\)2cq​\(3\+2​ηr\)​η1​d\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)\.\\displaystyle=\\left\(1\-\\frac\{\\left\(\\gamma\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{c\_\{q\}\(3\+2\\eta\_\{r\}\)\\eta\_\{1\}d\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\.The constantcqc\_\{q\}of[eq\.27](https://arxiv.org/html/2608.23765#S4.E27)is equal to11for one\-sided, harmonic\-mean and power mean weights withq∈\[−1,1\]q\\in\[\-1,1\], and satisfies1<cq≤31<c\_\{q\}\\leq 3for power mean weights withq∈\(1,∞\]q\\in\(1,\\infty\], which finishes the proof of[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)\.

Now, using[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2), we can prove the main result concerning global linear convergence of IRLS in the case that𝐗⋆\\mathbf\{X\}\_\{\\star\}is exactly low\-rank, which was stated as[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)\.

Proof of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)Set

Cηr:=Cηr,34=\(34−2​ηr1\+ηr\)23\+2​ηr\.C\_\{\\eta\_\{r\}\}:=C\_\{\\eta\_\{r\},\\frac\{3\}\{4\}\}=\\frac\{\\left\(\\frac\{3\}\{4\}\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{3\+2\\eta\_\{r\}\}\.Sincekkis an iteration carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1), we haveεj\>0\\varepsilon\_\{j\}\>0in each of the preceding iterationsj=0,…,k−1j=0,\\ldots,k\-1, so that[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)is applicable in these iterations\. Chaining its assertion, inequality[eq\.28](https://arxiv.org/html/2608.23765#S4.E28)follows asηr<3/5\\eta\_\{r\}<3/5if and only if3/4\>2​ηr1\+ηr3/4\>\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\. Next, we show inequality \([29](https://arxiv.org/html/2608.23765#S4.E29)\)\. We note that withcqc\_\{q\}as in[eq\.27](https://arxiv.org/html/2608.23765#S4.E27), we have that

1−ηr1\+ηr​‖𝐗\(k\)−𝐗⋆‖∗\\displaystyle\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(a\)​𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(b\)​\(1−Cηrcq​η1​d\)k​\(𝒥ε0​\(𝐗\(0\)\)−‖𝐗⋆‖∗\)\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\left\(1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{k\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{0\}\}\\left\(\\mathbf\{X\}^\{\(0\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)≤\(c\)​\(32\+ηr\)​\(1−Cηrcq​η1​d\)k​βr​\(𝐗\(0\)\)∗\\displaystyle\\overset\{\(c\)\}\{\\leq\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{k\}\\beta\_\{r\}\(\\mathbf\{X\}^\{\(0\)\}\)\_\{\*\}≤\(d\)​\(32\+ηr\)​\(1−Cηrcq​η1​d\)k​‖𝐗\(0\)−𝐗⋆‖∗,\\displaystyle\\overset\{\(d\)\}\{\\leq\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\-\\frac\{C\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{k\}\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\},where in inequalities\(a\)\(a\)and\(c\)\(c\)we used Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)\. Inequality\(b\)\(b\)follows from inequality \([28](https://arxiv.org/html/2608.23765#S4.E28)\) and in inequality\(d\)\(d\)we used the Eckart–Young theorem\. Multiplying both sides by1\+ηr1−ηr\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}yields \([29](https://arxiv.org/html/2608.23765#S4.E29)\)\.

#### A\.3\.4Proof of[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)

In order to show[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4), the global linear convergence of[Algorithm1](https://arxiv.org/html/2608.23765#alg1)under the assumption that𝐗⋆\\mathbf\{X\}\_\{\\star\}is approximately low\-rank, we need a slightly more involved argument compared to the proof of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)\. We show the details below\.

Proof of[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)Recall from the statement of[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)that

k^=min⁡\{k∈ℕ0:βr​\(𝐗⋆\)∗\>19​‖𝐗⋆−𝐗\(k\)‖∗\},\\hat\{k\}=\\min\\left\\\{k\\in\\mathbb\{N\}\_\{0\}:\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\>\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\\right\\\},with the conventionmin⁡∅=∞\\min\\varnothing=\\infty\(sok^=∞\\hat\{k\}=\\inftywheneverβr​\(𝐗⋆\)∗=0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0\)\. As in the proof of[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3), all indiceskkconsidered below are iterations carried out by[Algorithm1](https://arxiv.org/html/2608.23765#alg1), so thatεj\>0\\varepsilon\_\{j\}\>0in each preceding iterationj<kj<kand[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)is applicable in these iterations\.

Sinceηr<1/3\\eta\_\{r\}<1/3, we have12\>2​ηr1\+ηr\\frac\{1\}\{2\}\>\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}, so that the constant

C~ηr:=Cηr,1/2=\(12−2​ηr1\+ηr\)23\+2​ηr\\widetilde\{C\}\_\{\\eta\_\{r\}\}:=C\_\{\\eta\_\{r\},1/2\}=\\frac\{\\left\(\\frac\{1\}\{2\}\-\\frac\{2\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\right\)^\{2\}\}\{3\+2\\eta\_\{r\}\}is larger than00\. Abbreviatingμ:=1−C~ηrcq​η1​d∈\[0,1\)\\mu:=1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\in\[0,1\), wherecqc\_\{q\}is the weight operator dependent constant[eq\.27](https://arxiv.org/html/2608.23765#S4.E27), we claim that

𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗≤μmin⁡\(k,k^\)​\(𝒥ε0​\(𝐗\(0\)\)−‖𝐗⋆‖∗\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\mu^\{\\min\(k,\\hat\{k\}\)\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{0\}\}\(\\mathbf\{X\}^\{\(0\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\(72\)for allk∈ℕ0k\\in\\mathbb\{N\}\_\{0\}\. Indeed, by the minimality in the definition ofk^\\hat\{k\}, the conditionβr​\(𝐗⋆\)∗≤19​‖𝐗⋆−𝐗\(j\)‖∗\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(j\)\}\\right\\rVert\_\{\\ast\}of[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)is satisfied for everyj<k^j<\\hat\{k\}, so that[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2), applied withγ=1/2\\gamma=1/2, yields

𝒥εj\+1​\(𝐗\(j\+1\)\)−‖𝐗⋆‖∗≤μ⁡\(𝒥εj​\(𝐗\(j\)\)−‖𝐗⋆‖∗\)for all​j<k^\.\\mathcal\{J\}\_\{\\varepsilon\_\{j\+1\}\}\(\\mathbf\{X\}^\{\(j\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\mu\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{j\}\}\(\\mathbf\{X\}^\{\(j\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\\qquad\\text\{for all \}j<\\hat\{k\}\.\(73\)\(In the caseβr​\(𝐗⋆\)∗=0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0, we havek^=∞\\hat\{k\}=\\inftyand[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)applies withγ=3/4\\gamma=3/4; sinceCηr,3/4≥C~ηrC\_\{\\eta\_\{r\},3/4\}\\geq\\widetilde\{C\}\_\{\\eta\_\{r\}\}, inequality[eq\.73](https://arxiv.org/html/2608.23765#A1.E73)holds for allj∈ℕ0j\\in\\mathbb\{N\}\_\{0\}a fortiori\.\) Fork≤k^k\\leq\\hat\{k\}, inequality[eq\.72](https://arxiv.org/html/2608.23765#A1.E72)now follows by chaining[eq\.73](https://arxiv.org/html/2608.23765#A1.E73)forj=0,…,k−1j=0,\\ldots,k\-1, whereas fork\>k^k\>\\hat\{k\}, it follows from the casek=k^k=\\hat\{k\}together with𝒥εk​\(𝐗\(k\)\)≤𝒥εk^​\(𝐗\(k^\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\\mathcal\{J\}\_\{\\varepsilon\_\{\\hat\{k\}\}\}\(\\mathbf\{X\}^\{\(\\hat\{k\}\)\}\), which is a consequence of the monotonicity of the sequence\{𝒥εk​\(𝐗\(k\)\)\}k\\left\\\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\\right\\\}\_\{k\}, see[eq\.15](https://arxiv.org/html/2608.23765#S3.E15)\. Hence, we have shown inequality[eq\.32](https://arxiv.org/html/2608.23765#S4.E32)\.

In order to show inequality \([33](https://arxiv.org/html/2608.23765#S4.E33)\), we note first that for allk∈ℕ0k\\in\\mathbb\{N\}\_\{0\}

1−ηr1\+ηr​‖𝐗\(k\)−𝐗⋆‖∗−2​βr​\(𝐗⋆\)∗\\displaystyle\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-2\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}≤\(a\)​𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(b\)​\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​\(𝒥ε0​\(𝐗\(0\)\)−‖𝐗⋆‖∗\)\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{0\}\}\(\\mathbf\{X\}^\{\(0\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)≤\(c\)​\(32\+ηr\)​\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​βr​\(𝐗\(0\)\)∗,\\displaystyle\\overset\{\(c\)\}\{\\leq\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\beta\_\{r\}\(\\mathbf\{X\}^\{\(0\)\}\)\_\{\*\},where inequalities\(a\)\(a\)and\(c\)\(c\)follow from Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)and inequality\(b\)\(b\)follows from inequality[eq\.72](https://arxiv.org/html/2608.23765#A1.E72)\. By rearranging terms, it follows that

‖𝐗\(k\)−𝐗⋆‖∗\\displaystyle\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤Aηr​\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​βr​\(𝐗\(0\)\)∗\+2​\(1\+ηr\)1−ηr​βr​\(𝐗⋆\)∗,\\displaystyle\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\beta\_\{r\}\(\\mathbf\{X\}^\{\(0\)\}\)\_\{\*\}\+\\frac\{2\(1\+\\eta\_\{r\}\)\}\{1\-\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},\(74\)whereAηrA\_\{\\eta\_\{r\}\}is as in \([26](https://arxiv.org/html/2608.23765#S4.E26)\)\. Denote by𝐗⋆,r\\mathbf\{X\}\_\{\\star,r\}the best rank\-rrapproximation of the matrix𝐗⋆\\mathbf\{X\}\_\{\\star\}\. Then it follows from the Eckart–Young theorem that

βr​\(𝐗\(0\)\)∗≤‖𝐗\(0\)−𝐗⋆,r‖∗≤‖𝐗\(0\)−𝐗⋆‖∗\+βr​\(𝐗⋆\)∗,\\beta\_\{r\}\(\\mathbf\{X\}^\{\(0\)\}\)\_\{\*\}\\leq\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star,r\}\\right\\rVert\_\{\\ast\}\\leq\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},\(75\)where in the second inequality we used the triangle inequality\. Combining inequalities \([74](https://arxiv.org/html/2608.23765#A1.E74)\) and \([75](https://arxiv.org/html/2608.23765#A1.E75)\) shows that

‖𝐗\(k\)−𝐗⋆‖∗≤Aηr​\(1−C~ηrcq​η1​d\)min⁡\(k,k^\)​‖𝐗\(0\)−𝐗⋆‖∗\+Bηr​βr​\(𝐗⋆\)∗,\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\min\(k,\\hat\{k\}\)\}\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+B\_\{\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},with constantBηr=\(72\+ηr\)​\(1\+ηr\)1−ηrB\_\{\\eta\_\{r\}\}=\\frac\{\\left\(\\frac\{7\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\+\\eta\_\{r\}\\right\)\}\{1\-\\eta\_\{r\}\}as in \([30](https://arxiv.org/html/2608.23765#S4.E30)\), which corresponds to inequality[eq\.33](https://arxiv.org/html/2608.23765#S4.E33)\. In order to finish the proof, it remains to show inequality \([34](https://arxiv.org/html/2608.23765#S4.E34)\)\. Fork≥k^k\\geq\\hat\{k\}we can compute that

𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(a\)​𝒥εk^​\(𝐗\(k^\)\)−‖𝐗⋆‖∗\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\mathcal\{J\}\_\{\\varepsilon\_\{\\hat\{k\}\}\}\(\\mathbf\{X\}^\{\(\\hat\{k\}\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(b\)​\(32\+ηr\)​βr​\(𝐗\(k^\)\)∗\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\hat\{k\}\)\}\)\_\{\*\}≤\(c\)​\(32\+ηr\)​‖𝐗\(k^\)−𝐗⋆‖∗\+\(32\+ηr\)​βr​\(𝐗⋆\)∗\\displaystyle\\overset\{\(c\)\}\{\\leq\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\\lVert\\mathbf\{X\}^\{\(\\hat\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\+\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}≤\(d\)​10​\(32\+ηr\)​βr​\(𝐗⋆\)∗\.\\displaystyle\\overset\{\(d\)\}\{\\leq\}10\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\.Inequality\(a\)\(a\)is due to the monotonicity of the sequence\{𝒥εk​\(𝐗\(k\)\)\}k\\left\\\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\\right\\\}\_\{k\}and inequality\(b\)\(b\)follows from Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)\. Furthermore, inequality\(c\)\(c\)can be obtained by arguing as in \([75](https://arxiv.org/html/2608.23765#A1.E75)\) and inequality\(d\)\(d\)is a direct consequence of the definition ofk^\\hat\{k\}, which implies that‖𝐗\(k^\)−𝐗⋆‖∗≤9​βr​\(𝐗⋆\)∗\\left\\lVert\\mathbf\{X\}^\{\(\\hat\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq 9\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\. Next, Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)combined with the above inequality chain implies that

1−ηr1\+ηr​‖𝐗\(k\)−𝐗⋆‖∗−2​βr​\(𝐗⋆\)∗≤10​\(32\+ηr\)​βr​\(𝐗⋆\)∗\.\\displaystyle\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-2\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq 10\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\.By rearranging terms, we obtain

‖𝐗\(k\)−𝐗⋆‖∗≤1\+ηr1−ηr​\(10​\(32\+ηr\)\+2\)​βr​\(𝐗⋆\)∗≤41​βr​\(𝐗⋆\)∗,\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(10\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\+2\\right\)\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq 41\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\},using the assumption thatηr<1/3\\eta\_\{r\}<1/3in the last inequality, which corresponds to inequality[eq\.34](https://arxiv.org/html/2608.23765#S4.E34)fork≥k^k\\geq\\hat\{k\}\.

It remains to establish the upper bound onk^\\hat\{k\}stated after the theorem\. Ifβr​\(𝐗⋆\)∗=0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}=0, thenk^=∞\\hat\{k\}=\\inftyand there is nothing to prove, so assumeβr​\(𝐗⋆\)∗\>0\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\>0\. The bound holds trivially ifk^=0\\hat\{k\}=0, so assumek^≥1\\hat\{k\}\\geq 1\. By definition ofk^\\hat\{k\}, the strict inequalityβr​\(𝐗⋆\)∗\>19​‖𝐗\(k\)−𝐗⋆‖∗\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\>\\frac\{1\}\{9\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}fails for everyk<k^k<\\hat\{k\}, so in particular

9​βr​\(𝐗⋆\)∗≤‖𝐗\(k^−1\)−𝐗⋆‖∗\.9\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq\\left\\lVert\\mathbf\{X\}^\{\(\\hat\{k\}\-1\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.Applying inequality[eq\.33](https://arxiv.org/html/2608.23765#S4.E33)atk=k^−1k=\\hat\{k\}\-1\(so thatmin⁡\(k,k^\)=k^−1\\min\(k,\\hat\{k\}\)=\\hat\{k\}\-1\) and combining the two inequalities yields

\(9−Bηr\)​βr​\(𝐗⋆\)∗≤Aηr​\(1−C~ηrcq​η1​d\)k^−1​‖𝐗\(0\)−𝐗⋆‖∗\.\(9\-B\_\{\\eta\_\{r\}\}\)\\,\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-\\frac\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\{c\_\{q\}\\eta\_\{1\}d\}\\right\)^\{\\hat\{k\}\-1\}\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.The assumptionηr<1/3\\eta\_\{r\}<1/3impliesBηr<23/3<9B\_\{\\eta\_\{r\}\}<23/3<9, so the left\-hand side is positive\. Taking logarithms and using−log⁡\(1−x\)≥x\-\\log\(1\-x\)\\geq xforx∈\(0,1\)x\\in\(0,1\)gives

k^≤1\+cq​η1​dC~ηr​log\+⁡\(Aηr9−Bηr⋅‖𝐗\(0\)−𝐗⋆‖∗βr​\(𝐗⋆\)∗\),\\hat\{k\}\\leq 1\+\\frac\{c\_\{q\}\\eta\_\{1\}d\}\{\\widetilde\{C\}\_\{\\eta\_\{r\}\}\}\\log\_\{\+\}\\left\(\\frac\{A\_\{\\eta\_\{r\}\}\}\{9\-B\_\{\\eta\_\{r\}\}\}\\cdot\\frac\{\\left\\lVert\\mathbf\{X\}^\{\(0\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\beta\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\_\{\*\}\}\\right\),as claimed, wherelog\+⁡\(x\)=max⁡\(0,log⁡\(x\)\)\\log\_\{\+\}\(x\)=\\max\(0,\\log\(x\)\)\. This completes the proof of[Theorem4](https://arxiv.org/html/2608.23765#Thmtheorem4)\.

### A\.4Proofs of[Theorems5](https://arxiv.org/html/2608.23765#Thmtheorem5)and[6](https://arxiv.org/html/2608.23765#Thmtheorem6)\(Local Linear Convergence and Counterexample\)

A limitation of the global linear convergence rate proofs is that they rely on estimates for the quadratic term⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}of the quadratic model mismatch such as[Lemma16](https://arxiv.org/html/2608.23765#Thmlemma16), which scale with1/ε1/\\varepsilon\. In this section, we overcome this limitation in the case of harmonic\-mean \(and relatedqq\-power mean\) weight operators and obtain a faster linear convergence rate by using a more precise estimate for the quadratic term in the case that𝐗\\mathbf\{X\}is close enough to the ground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}in[SectionA\.4\.1](https://arxiv.org/html/2608.23765#A1.SS4.SSS1), which leads to the proof of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)in[SectionA\.4\.2](https://arxiv.org/html/2608.23765#A1.SS4.SSS2)\. On the other hand, we provide in[SectionA\.4\.3](https://arxiv.org/html/2608.23765#A1.SS4.SSS3)a counterexample exhibiting that this is not possible for one\-sided weight operators, such as those with left\-sided or right\-sided core matrices[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5), even within arbitrarily smaller local neighborhoods of𝐗⋆\\mathbf\{X\}\_\{\\star\}than the one defined by[eq\.35](https://arxiv.org/html/2608.23765#S4.E35), which establishes[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)\.

#### A\.4\.1Tight Upper Bounds on Harmonic\-Mean Quadratic Forms

We proceed with a technical result on a tight upper bound on the quadratic form⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}implied byqq\-power mean weight operators withq∈\[−1,0\)q\\in\[\-1,0\), which applies if the iterate𝐗\\mathbf\{X\}on which the weight operator is based is close enough to the rank\-rrground truth𝐗⋆\\mathbf\{X\}\_\{\\star\}\.

###### Lemma 17\.

Assume that the linear measurement operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\rightarrow\\mathbb\{R\}^\{m\}satisfies the NSP of orderrrwith constantηr≤3/5\\eta\_\{r\}\\leq 3/5, and that for some𝐗⋆\\mathbf\{X\}\_\{\\star\}of rankrr, it holds that𝒜⁡\(𝐗\)=𝒜⁡\(𝐗⋆\)\\mathcal\{A\}\(\\mathbf\{X\}\)=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\)for some𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\. Assume that the weight operatorW𝐗,εW\_\{\\mathbf\{X\},\\varepsilon\}is aqq\-power mean weight operator[eq\.21](https://arxiv.org/html/2608.23765#S3.E21)withq∈\[−1,0\)q\\in\[\-1,0\)444This includes the harmonic\-mean weight operator as special case withq=−1q=\-1\., that

‖𝐗−𝐗⋆‖∗≤σr​\(𝐗⋆\)max⁡\(η1​d,2\),\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\}\{\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\},\(76\)where0<η1≤ηr0<\\eta\_\{1\}\\leq\\eta\_\{r\}denotes again the order\-one NSP constant of𝒜\\mathcal\{A\}, and that

ε≥ϑ​βr​\(𝐗\)∗d\\varepsilon\\geq\\vartheta\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}for someϑ\>0\\vartheta\>0\. Then for the constantDηr=3\+ηr1−ηrD\_\{\\eta\_\{r\}\}=\\frac\{3\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}, it holds that

⟨𝐗⋆−𝐗,W𝐗,ε\(𝐗⋆−𝐗\)⟩F≤\(2\+2−1/q\+Dηr2ϑ\)‖𝐗−𝐗⋆‖∗\.\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\leq\\left\(2\+2^\{\-1/q\}\+\\frac\{D\_\{\\eta\_\{r\}\}^\{2\}\}\{\\vartheta\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.Ifϑ=1−ηr\(1\+ηr\)​\(32\+ηr\)\\vartheta=\\frac\{1\-\\eta\_\{r\}\}\{\(1\+\\eta\_\{r\}\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}, the bound reduces to

⟨𝐗⋆−𝐗,W𝐗,ε\(𝐗⋆−𝐗\)⟩F≤\(2\+2−1/q\+Eηr\)‖𝐗−𝐗⋆‖∗\.\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}\\leq\\left\(2\+2^\{\-1/q\}\+E\_\{\\eta\_\{r\}\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(77\)withEηr:=\(3\+ηr\)2​\(1\+ηr\)​\(32\+ηr\)\(1−ηr\)3E\_\{\\eta\_\{r\}\}:=\\frac\{\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\{\(1\-\\eta\_\{r\}\)^\{3\}\}\.

Note that the right\-hand side in inequality \([77](https://arxiv.org/html/2608.23765#A1.E77)\) is up to constants a factor ofη1​‖𝐗−𝐗⋆‖∗ε\\frac\{\\eta\_\{1\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\varepsilon\}smaller than the right\-hand side of inequality \([67](https://arxiv.org/html/2608.23765#A1.E67)\), the corresponding inequality in the global linear convergence proof\. In particular, since there is noε\\varepsilon\-dependence anymore, we can improve the convergence rate by a factor ofddcompared to[Theorem3](https://arxiv.org/html/2608.23765#Thmtheorem3)\.

Forq=−1q=\-1, which corresponds to the harmonic\-mean weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7), and an NSP constant ofηr=1/10\\eta\_\{r\}=1/10, the right hand side of[eq\.77](https://arxiv.org/html/2608.23765#A1.E77)amounts to≈27\.20​‖𝐗−𝐗⋆‖∗\\approx 27\.20\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}, whereas forηr=1/2\\eta\_\{r\}=1/2, it amounts to298​‖𝐗−𝐗⋆‖∗298\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.

For proving[Lemma17](https://arxiv.org/html/2608.23765#Thmlemma17), we use the following elementary inequality about power means \(recall[Definition3](https://arxiv.org/html/2608.23765#Thmdefinition3)\), as well as an elementary norm bound for matrices in the null space of a measurement operator equipped with the NSP \([Lemma18](https://arxiv.org/html/2608.23765#Thmlemma18)\)\.

###### Proposition 3\.

Fora,b\>0a,b\>0andq∈\[−1,0\)q\\in\[\-1,0\), it holds that

min\(a,b\)≤ℳq\(a,b\)≤2−1/qmin\(a,b\)\.\\min\(a,b\)\\leq\\mathcal\{M\}\_\{q\}\(a,b\)\\leq 2^\{\-1/q\}\\min\(a,b\)\.

Proof of[Proposition3](https://arxiv.org/html/2608.23765#Thmproposition3)The first inequality follows from the simple internality property\([Bullen, 2003](https://arxiv.org/html/2608.23765#bib.bib42), Section III\.1, Theorem 2\(a\)\)\. For the second inequality, we note that for anyq∈\[−1,0\)q\\in\[\-1,0\)anda,b\>0a,b\>0, it holds that

ℳq​\(a,b\)=\(aq\+bq2\)1/q=\(1\+\(max⁡\(a,b\)min⁡\(a,b\)\)q2\)1/q​min⁡\(a,b\)=\(21\+\(max⁡\(a,b\)min⁡\(a,b\)\)q\)−1/qmin\(a,b\)≤2−1/qmin\(a,b\),\\begin\{split\}\\mathcal\{M\}\_\{q\}\(a,b\)=\\left\(\\frac\{a^\{q\}\+b^\{q\}\}\{2\}\\right\)^\{1/q\}=\\left\(\\frac\{1\+\\left\(\\frac\{\\max\(a,b\)\}\{\\min\(a,b\)\}\\right\)^\{q\}\}\{2\}\\right\)^\{1/q\}\\min\(a,b\)&=\\left\(\\frac\{2\}\{1\+\\left\(\\frac\{\\max\(a,b\)\}\{\\min\(a,b\)\}\\right\)^\{q\}\}\\right\)^\{\-1/q\}\\min\(a,b\)\\\\ &\\leq 2^\{\-1/q\}\\min\(a,b\),\\end\{split\}using that−1/q\>0\-1/q\>0and thatmax⁡\(a,b\)min⁡\(a,b\)≥1\\frac\{\\max\(a,b\)\}\{\\min\(a,b\)\}\\geq 1\.

###### Lemma 18\.

Assume that the linear measurement operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\rightarrow\\mathbb\{R\}^\{m\}satisfies the NSP of order11with constant0<η1<10<\\eta\_\{1\}<1\. Then it holds that for any𝐍∈ker⁡\(𝒜\)\\mathbf\{N\}\\in\\ker\(\\mathcal\{A\}\),

‖𝐍‖≤η11\+η1​‖𝐍‖∗\.\\left\\lVert\\mathbf\{N\}\\right\\rVert\\leq\\frac\{\\eta\_\{1\}\}\{1\+\\eta\_\{1\}\}\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\.\(78\)

Proof of[Lemma18](https://arxiv.org/html/2608.23765#Thmlemma18)From the NSP inequality[eq\.24](https://arxiv.org/html/2608.23765#S4.E24)of order11, it follows that

‖𝐍‖=σ1​\(𝐍\)≤η1​∑i=2dσi​\(𝐍\)=η1​\(‖𝐍‖∗−‖𝐍‖\),\\left\\lVert\\mathbf\{N\}\\right\\rVert=\\sigma\_\{1\}\(\\mathbf\{N\}\)\\leq\\eta\_\{1\}\\sum\_\{i=2\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{N\}\)=\\eta\_\{1\}\\left\(\\left\\lVert\\mathbf\{N\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{N\}\\right\\rVert\\right\),which is equivalent to the desired inequality \([78](https://arxiv.org/html/2608.23765#A1.E78)\)\.

Proof of[Lemma17](https://arxiv.org/html/2608.23765#Thmlemma17)We first observe that from Weyl’s inequality and assumption[eq\.76](https://arxiv.org/html/2608.23765#A1.E76), it follows that

σr​\(𝐗\)≥σr​\(𝐗⋆\)−‖𝐗−𝐗⋆‖≥σr​\(𝐗⋆\)−‖𝐗−𝐗⋆‖∗≥σr​\(𝐗⋆\)​\(max⁡\(η1​d,2\)−1max⁡\(η1​d,2\)\)\.\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\\geq\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\-\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\\geq\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\-\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\geq\\sigma\_\{r\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\\left\(\\frac\{\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\-1\}\{\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\}\\right\)\.Denote by𝐗=𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}a full singular value decomposition of𝐗\\mathbf\{X\}\. From the definition of the weight operatorW𝐗,ε​\(⋅\):=W𝐗,ε\(q\)​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\):=W\_\{\\mathbf\{X\},\\varepsilon\}^\{\(q\)\}\(\\cdot\)[eq\.22](https://arxiv.org/html/2608.23765#S3.E22)and[eq\.21](https://arxiv.org/html/2608.23765#S3.E21), it follows that

⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\displaystyle\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}=⟨𝐗−𝐗⋆,W𝐗,ε​\(𝐗−𝐗⋆\)⟩F\\displaystyle=\\langle\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\rangle\_\{F\}=⟨𝐗−𝐗⋆,𝐔𝐗​\(𝐇𝝈,ε\(q\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)\)​𝐕𝐗⊤⟩F\\displaystyle=\\langle\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\},\\mathbf\{U\}\_\{\\mathbf\{X\}\}\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\\rangle\_\{F\}=⟨𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗,𝐇𝝈,ε\(q\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)⟩F\\displaystyle=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\},\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\rangle\_\{F\}=⟨\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\.\\displaystyle=\\langle\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\.Define the set of entries

S:=\{\(i,j\)∈\[d1\]×\[d2\]:1≤i≤r​or​1≤j≤r\}S:=\\left\\\{\(i,j\)\\in\[d\_\{1\}\]\\times\[d\_\{2\}\]:1\\leq i\\leq r\\text\{ or \}1\\leq j\\leq r\\right\\\}and its complementSc:=\[d1\]×\[d2\]∖SS^\{c\}:=\[d\_\{1\}\]\\times\[d\_\{2\}\]\\setminus S\. Denote by𝒫S,𝒫Sc:ℝd1×d2→ℝd1×d2\\mathcal\{P\}\_\{S\},\\mathcal\{P\}\_\{S^\{c\}\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\rightarrow\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}the orthogonal projections which sets all entries not belonging toSSorScS^\{c\}to zero, respectively\. It follows that

⟨\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\\displaystyle=⟨𝒫S​\(\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S\}\\left\(\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\+⟨𝒫Sc​\(\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\+\\langle\\mathcal\{P\}\_\{S^\{c\}\}\\left\(\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\(a\)\\displaystyle\\overset\{\(a\)\}\{=\}⟨𝒫S​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘𝒫S​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F⏟=:\(I\)\\displaystyle\\underset\{\{=:\(\\mathrm\{I\}\)\}\}\{\\underbrace\{\\langle\\mathcal\{P\}\_\{S\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\}\}\+⟨𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F⏟=:\(II\)\.\\displaystyle\+\\underset\{\{=:\(\\mathrm\{II\}\)\}\}\{\\underbrace\{\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\}\}\.\(79\)In equality\(a\)\(a\), we used that𝒫S​\(𝐀∘𝐁\)=𝒫S​\(𝐀\)∘𝒫S​\(𝐁\)\\mathcal\{P\}\_\{S\}\(\\mathbf\{A\}\\circ\\mathbf\{B\}\)=\\mathcal\{P\}\_\{S\}\(\\mathbf\{A\}\)\\circ\\mathcal\{P\}\_\{S\}\(\\mathbf\{B\}\)for all matrices𝐀,𝐁∈ℝd1×d2\\mathbf\{A\},\\mathbf\{B\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\. We bound the two summands individually\.

For bounding the summand\(I\)\(\\mathrm\{I\}\), we note that[Proposition3](https://arxiv.org/html/2608.23765#Thmproposition3)implies that for any\(i,j\)∈S\(i,j\)\\in S, it holds that

\(𝐇𝝈,ε\(q\)\)i,j≤2−1/qmin\(max\(σi\(𝐗\),ε\)−1,max\(σj\(𝐗\),ε\)−1\)≤2−1/qσr\(𝐗\)−1\.\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\)\_\{i,j\}\\leq 2^\{\-1/q\}\\min\\left\(\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)^\{\-1\},\\max\(\\sigma\_\{j\}\(\\mathbf\{X\}\),\\varepsilon\)^\{\-1\}\\right\)\\leq 2^\{\-1/q\}\\sigma\_\{r\}\(\\mathbf\{X\}\)^\{\-1\}\.\(80\)From[eq\.80](https://arxiv.org/html/2608.23765#A1.E80), it follows that

\(I\)=\\displaystyle\(\\mathrm\{I\}\)=⟨𝒫S​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘𝒫S​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\\displaystyle=∑\(i,j\)∈S\(𝐇𝝈,ε\(q\)\)i,j​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)i,j2\\displaystyle\\sum\_\{\(i,j\)\\in S\}\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\)\_\{i,j\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}^\{2\}≤[eq\.80](https://arxiv.org/html/2608.23765#A1.E80)\\displaystyle\\overset\{\\lx@cref\{creftype~refnum\}\{ineq:powermeanineq\}\}\{\\leq\}2−1/qσr​\(𝐗\)​∑\(i,j\)∈S\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)i,j2\\displaystyle\\frac\{2^\{\-1/q\}\}\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\}\\sum\_\{\(i,j\)\\in S\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}^\{2\}≤\\displaystyle\\leq2−1/qσr​\(𝐗\)​∑i=1d1∑j=1d2\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)i,j2\\displaystyle\\frac\{2^\{\-1/q\}\}\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\}\\sum\_\{i=1\}^\{d\_\{1\}\}\\sum\_\{j=1\}^\{d\_\{2\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\_\{i,j\}^\{2\}=\\displaystyle=2−1/qσr​\(𝐗\)​⟨𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗,𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗⟩F\\displaystyle\\frac\{2^\{\-1/q\}\}\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\}\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\},\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\rangle\_\{F\}≤\\displaystyle\\leq2−1/qσr​\(𝐗\)​‖𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗‖​‖𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗‖∗\\displaystyle\\frac\{2^\{\-1/q\}\}\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\}\\left\\lVert\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\right\\rVert\\left\\lVert\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\right\\rVert\_\{\\ast\}=\\displaystyle=2−1/qσr​\(𝐗\)​‖𝐗−𝐗⋆‖​‖𝐗−𝐗⋆‖∗\\displaystyle\\frac\{2^\{\-1/q\}\}\{\\sigma\_\{r\}\\left\(\\mathbf\{X\}\\right\)\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}Now, note that as𝒜\\mathcal\{A\}satisfies the NSP of orderrrwith0<ηr≤3/50<\\eta\_\{r\}\\leq 3/5, it also satisfies the NSP of order11with0<η1≤3/50<\\eta\_\{1\}\\leq 3/5\. Thus, due to Weyl’s inequality and[Lemma18](https://arxiv.org/html/2608.23765#Thmlemma18), we note that

‖𝐗−𝐗⋆‖σr​\(𝐗\)≤‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)−‖𝐗−𝐗⋆‖≤η1\(1\+η1\)​max⁡\(η1​d,2\)​σr​\(𝐗⋆\)σr​\(𝐗⋆\)−η1\(1\+η1\)​max⁡\(η1​d,2\)​σr​\(𝐗⋆\)=η1\(1\+η1\)​max⁡\(η1​d,2\)−η1≤1max⁡\(η1​d,2\),\\begin\{split\}\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\)\}&\\leq\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\-\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\\leq\\frac\{\\frac\{\\eta\_\{1\}\}\{\(1\+\\eta\_\{1\}\)\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\}\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\-\\frac\{\\eta\_\{1\}\}\{\(1\+\\eta\_\{1\}\)\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\}\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\\\\ &=\\frac\{\\eta\_\{1\}\}\{\(1\+\\eta\_\{1\}\)\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\-\\eta\_\{1\}\}\\leq\\frac\{1\}\{\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\},\\end\{split\}\(81\)using also the assumption[eq\.76](https://arxiv.org/html/2608.23765#A1.E76)in the second inequality, while the last inequality follows straightforwardly from the fact that it is equivalent to

η1max\(η1d,2\)≤\(1\+η1\)max\(η1d,2\)−η1⟺0≤max\(η1d,2\)−η1\.\\eta\_\{1\}\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\\leq\(1\+\\eta\_\{1\}\)\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\-\\eta\_\{1\}\\quad\\Longleftrightarrow\\quad 0\\leq\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\-\\eta\_\{1\}\.
In order to deal with summand\(II\)\(\\mathrm\{II\}\)of[eq\.79](https://arxiv.org/html/2608.23765#A1.E79), we compute that

⟨𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\\displaystyle=⟨𝒫Sc​\(𝐔𝐗⊤​𝐗𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F−2​⟨𝒫Sc​\(𝐔𝐗⊤​𝐗𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\-2\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\+⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\+\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\\displaystyle=⟨−𝒫Sc\(𝐔𝐗⊤𝐗𝐕𝐗\)∘𝒫Sc\(𝐔𝐗⊤𝐗𝐕𝐗\)−2𝒫Sc\(𝐔𝐗⊤𝐗𝐕𝐗\)∘𝒫Sc\(𝐔𝐗⊤\(𝐗⋆−𝐗\)𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\\langle\-\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\-2\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\+⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\+\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}≤\\displaystyle\\leq−⟨2​𝒫Sc​\(𝐔𝐗⊤​𝐗𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗⋆−𝐗\)​𝐕𝐗\)\+𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\\displaystyle\-\\langle 2\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\+\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}=\\displaystyle=2⟨𝒫Sc​\(diag⁡\(𝝈\)\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F⏟=:\(i\)\\displaystyle 2\\underset\{\{=:\(i\)\}\}\{\\underbrace\{\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\}\}\+⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F⏟=:\(i​i\)\.\\displaystyle\+\\underset\{\{=:\(ii\)\}\}\{\\underbrace\{\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\}\}\.\(82\)We estimate the two terms individually\. For the first term\(i\)\(i\), we obtain that

\|⟨𝒫Sc​\(diag⁡\(𝝈\)\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε\(q\)⟩F\|\\displaystyle\|\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\rangle\_\{F\}\|≤\\displaystyle\\leq\|⟨𝐇𝝈,ε\(q\)∘𝒫Sc​\(diag⁡\(𝝈\)\),𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)⟩F\|\\displaystyle\|\\langle\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\),\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\rangle\_\{F\}\|≤\\displaystyle\\leq‖𝐇𝝈,ε\(q\)∘𝒫Sc​\(diag⁡\(𝝈\)\)‖​‖𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)‖∗\\displaystyle\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\right\\rVert\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert\_\{\\ast\}≤\\displaystyle\\leq‖𝐇𝝈,ε\(q\)∘𝒫Sc​\(diag⁡\(𝝈\)\)‖​‖𝐗−𝐗⋆‖∗\.\\displaystyle\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\right\\rVert\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.Since𝐇𝝈,ε\(q\)∘𝒫Sc​\(diag⁡\(𝝈\)\)\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)has only non\-zero entries on its diagonal it follows that

‖𝐇𝝈,ε\(q\)∘𝒫Sc​\(diag⁡\(𝝈\)\)‖=maxi∈\[d\]∖\[r\]​\(𝐇𝝈,ε\(q\)\)\(i,i\)​σi​\(𝐗\)=maxi∈\[d\]∖\[r\]​\(σi​\(𝐗\)max⁡\(σi​\(𝐗\),ε\)\)≤1\.\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\right\\rVert=\\underset\{i\\in\[d\]\\setminus\[r\]\}\{\\max\}\\left\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}^\{\(q\)\}\\right\)\_\{\(i,i\)\}\\sigma\_\{i\}\(\\mathbf\{X\}\)\\\\ =\\underset\{i\\in\[d\]\\setminus\[r\]\}\{\\max\}\\left\(\\frac\{\\sigma\_\{i\}\(\\mathbf\{X\}\)\}\{\\max\\left\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\\right\)\}\\right\)\\\\ \\leq 1\.Thus, we have shown that

\|\(i\)\|=\|⟨𝒫Sc​\(diag⁡\(𝝈\)\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε⟩F\|≤‖𝐗−𝐗⋆‖∗\.\|\(i\)\|=\|\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\diag\(\\bm\{\\sigma\}\)\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\rangle\_\{F\}\|\\leq\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(83\)For the second summand\(i​i\)\(ii\)in \([82](https://arxiv.org/html/2608.23765#A1.E82)\), we note that

⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\rangle\_\{F\}=\\displaystyle=⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\rangle\_\{F\}≤\\displaystyle\\leq‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖∗​‖𝐇𝝈,ε∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖\\displaystyle\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert=\\displaystyle=‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖∗​‖𝒫Sc​\(𝐇𝝈,ε\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖\\displaystyle\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert≤\(a\)\\displaystyle\\overset\{\(a\)\}\{\\leq\}\(maxi∈\{r\+1,…,d\}​max⁡\(σi​\(𝐗\),ε\)−1\)​‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖∗​‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖\\displaystyle\\left\(\\underset\{i\\in\\\{r\+1,\\ldots,d\\\}\}\{\\max\}\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)^\{\-1\}\\right\)\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert≤\\displaystyle\\leq1ε​‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖∗​‖𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)‖\\displaystyle\\frac\{1\}\{\\varepsilon\}\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert\_\{\\ast\}\\left\\lVert\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\\rVert=\(b\)\\displaystyle\\overset\{\(b\)\}\{=\}1ε‖\(𝐔𝐗,r\+1:d1\)⊤𝐔⋆𝚺⋆𝐕⋆⊤𝐕𝐗,r\+1:d2‖∗‖\(𝐔𝐗,r\+1:d1\)⊤𝐔⋆𝚺⋆𝐕⋆⊤𝐕𝐗,r\+1:d2‖\\displaystyle\\frac\{1\}\{\\varepsilon\}\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\mathbf\{V\}\_\{\\star\}^\{\\top\}\\mathbf\{V\}\_\{\\mathbf\{X\},r\+1:d\_\{2\}\}\\right\\rVert\_\{\\ast\}\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\mathbf\{V\}\_\{\\star\}^\{\\top\}\\mathbf\{V\}\_\{\\mathbf\{X\},r\+1:d\_\{2\}\}\\right\\rVert≤\\displaystyle\\leq1ε‖\(𝐔𝐗,r\+1:d1\)⊤𝐔⋆𝚺⋆‖∗‖\(𝐔𝐗,r\+1:d1\)⊤𝐔⋆𝚺⋆‖‖𝐕⋆⊤𝐕𝐗,r\+1:d2‖2\.\\displaystyle\\frac\{1\}\{\\varepsilon\}\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\\left\\lVert\\mathbf\{V\}\_\{\\star\}^\{\\top\}\\mathbf\{V\}\_\{\\mathbf\{X\},r\+1:d\_\{2\}\}\\right\\rVert^\{2\}\.\(84\)Inequality\(a\)\(a\)follows from[Lemma13](https://arxiv.org/html/2608.23765#Thmlemma13), and in equation\(b\)\(b\), we used the decomposition𝐔=\[𝐔𝐗,1:r𝐔𝐗,r\+1:d1\]\\mathbf\{U\}=\[\\mathbf\{U\}\_\{\\mathbf\{X\},1:r\}\\ \\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\]and𝐕=\[𝐕𝐗,1:r𝐕𝐗,r\+1:d2\]\\mathbf\{V\}=\[\\mathbf\{V\}\_\{\\mathbf\{X\},1:r\}\\ \\mathbf\{V\}\_\{\\mathbf\{X\},r\+1:d\_\{2\}\}\]\. In order to proceed we denote by𝐗r\\mathbf\{X\}\_\{r\}the best rank\-rrapproximation of𝐗\\mathbf\{X\}\. It follows that

‖\(𝐔𝐗,r\+1:d1\)⊤𝐔⋆𝚺⋆‖∗\\displaystyle\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=‖\(𝐔𝐗,r\+1:d1\)⊤𝐗⋆‖∗=‖\(𝐔𝐗,r\+1:d1\)⊤\(𝐗⋆−𝐗r\)‖∗\\displaystyle=\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=\\left\\lVert\(\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\)^\{\\top\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\_\{r\}\)\\right\\rVert\_\{\\ast\}≤‖𝐗⋆−𝐗r‖∗≤‖𝐗⋆−𝐗‖∗\+βr​\(𝐗\)∗,\\displaystyle\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\_\{r\}\\right\\rVert\_\{\\ast\}\\leq\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\},where we recall that by definitionβr​\(𝐗\)∗=‖𝐗−𝐗r‖∗\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}=\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{r\}\\right\\rVert\_\{\\ast\}\. Note that due to the reverse triangle inequality, see Lemma[10](https://arxiv.org/html/2608.23765#Thmlemma10), it follows that

‖𝐗⋆−𝐗‖∗≤1\+ηr1−ηr​\(‖𝐗⋆‖∗−‖𝐗‖∗\+2​βr​\(𝐗\)∗\)≤2​\(1\+ηr\)1−ηr​βr​\(𝐗\)∗,\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}\+2\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\\right\)\\leq\\frac\{2\(1\+\\eta\_\{r\}\)\}\{1\-\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\},where in the second inequality we have used that‖𝐗⋆‖∗≤‖𝐗‖∗\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\\lVert\\mathbf\{X\}\\right\\rVert\_\{\\ast\}which follows from the fact that𝐗⋆\\mathbf\{X\}\_\{\\star\}is the nuclear norm minimizer\. Combining the last two inequality chains we obtain that

‖𝐔⟂⊤​𝐔⋆​𝚺⋆‖∗≤Dηr​βr​\(𝐗\)∗,\\left\\lVert\\mathbf\{U\}\_\{\\perp\}^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq D\_\{\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\},using the definition of𝐔⟂=𝐔𝐗,r\+1:d1\\mathbf\{U\}\_\{\\perp\}=\\mathbf\{U\}\_\{\\mathbf\{X\},r\+1:d\_\{1\}\}\. This inequality also implies that

‖𝐔⟂⊤​𝐔⋆​𝚺⋆‖≤‖𝐔⟂⊤​𝐔⋆​𝚺⋆‖∗≤Dηr​βr​\(𝐗\)∗≤Dηr​‖𝐗−𝐗⋆‖∗,\\left\\lVert\\mathbf\{U\}\_\{\\perp\}^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\\leq\\left\\lVert\\mathbf\{U\}\_\{\\perp\}^\{\\top\}\\mathbf\{U\}\_\{\\star\}\\mathbf\{\\Sigma\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq D\_\{\\eta\_\{r\}\}\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\\ast\}\\leq D\_\{\\eta\_\{r\}\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\},where in the last inequality we used the Eckart–Young theorem\. Moreover, it follows from Wedin’ssin⁡θ\\sin\\thetatheorem\([Wedin, 1972](https://arxiv.org/html/2608.23765#bib.bib44);[Chen et al\., 2021](https://arxiv.org/html/2608.23765#bib.bib82), see, e\.g\.,\)that the inner product matrix of𝐕⋆\\mathbf\{V\}\_\{\\star\}with𝐕⟂=𝐕𝐗,r\+1:d2\\mathbf\{V\}\_\{\\perp\}=\\mathbf\{V\}\_\{\\mathbf\{X\},r\+1:d\_\{2\}\}can be bounded in spectral norm such that

‖𝐕⋆⊤​𝐕⟂‖\\displaystyle\\left\\lVert\\mathbf\{V\}\_\{\\star\}^\{\\top\}\\mathbf\{V\}\_\{\\perp\}\\right\\rVert≤‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)−‖𝐗−𝐗⋆‖\\displaystyle\\leq\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\-\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}≤\(a\)​2​\(1\+η1\)2\+η1​‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\frac\{2\(1\+\\eta\_\{1\}\)\}\{2\+\\eta\_\{1\}\}\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}≤\(b\)​2​η12\+η1​‖𝐗−𝐗⋆‖∗σr​\(𝐗⋆\)\.\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\frac\{2\\eta\_\{1\}\}\{2\+\\eta\_\{1\}\}\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\.For inequality\(a\)\(a\), set

α:=‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)\.\\alpha:=\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\.Since𝐗−𝐗⋆∈ker⁡\(𝒜\)\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\in\\ker\(\\mathcal\{A\}\),[Lemma18](https://arxiv.org/html/2608.23765#Thmlemma18)and assumption \([76](https://arxiv.org/html/2608.23765#A1.E76)\) imply

α≤η1\(1\+η1\)​max⁡\(η1​d,2\)≤η12​\(1\+η1\)\.\\alpha\\leq\\frac\{\\eta\_\{1\}\}\{\(1\+\\eta\_\{1\}\)\\max\(\\eta\_\{1\}\\sqrt\{d\},2\)\}\\leq\\frac\{\\eta\_\{1\}\}\{2\(1\+\\eta\_\{1\}\)\}\.Therefore,

11−α≤11−η12​\(1\+η1\)=2​\(1\+η1\)2\+η1,\\frac\{1\}\{1\-\\alpha\}\\leq\\frac\{1\}\{1\-\\frac\{\\eta\_\{1\}\}\{2\(1\+\\eta\_\{1\}\)\}\}=\\frac\{2\(1\+\\eta\_\{1\}\)\}\{2\+\\eta\_\{1\}\},and consequently

‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)−‖𝐗−𝐗⋆‖=α1−α≤2​\(1\+η1\)2\+η1​‖𝐗−𝐗⋆‖σr​\(𝐗⋆\)\.\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\-\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}=\\frac\{\\alpha\}\{1\-\\alpha\}\\leq\\frac\{2\(1\+\\eta\_\{1\}\)\}\{2\+\\eta\_\{1\}\}\\frac\{\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\.This proves\(a\)\(a\)\. Inequality\(b\)\(b\)follows again from[Lemma18](https://arxiv.org/html/2608.23765#Thmlemma18)\. Inserting the last three inequalities into \([84](https://arxiv.org/html/2608.23765#A1.E84)\) we obtain that

⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε⟩F\\displaystyle\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\rangle\_\{F\}≤4​Dηr2​η12​βr​\(𝐗\)∗2​‖𝐗−𝐗⋆‖∗2\(2\+η1\)2​ε⋅σr​\(𝐗⋆\)2\\displaystyle\\leq\\frac\{4D\_\{\\eta\_\{r\}\}^\{2\}\\eta\_\{1\}^\{2\}\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}^\{2\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\(2\+\\eta\_\{1\}\)^\{2\}\\varepsilon\\cdot\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)^\{2\}\}≤\(a\)​4​Dηr2​d​η12​‖𝐗−𝐗⋆‖∗3\(2\+η1\)2​ϑ​σr​\(𝐗⋆\)2,\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\frac\{4D\_\{\\eta\_\{r\}\}^\{2\}d\\eta\_\{1\}^\{2\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}^\{3\}\}\{\(2\+\\eta\_\{1\}\)^\{2\}\\vartheta\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)^\{2\}\},where we used that by assumptionε≥ϑ​βr​\(𝐗\)∗/d\\varepsilon\\geq\\vartheta\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}/dand the Eckart–Young theorem in the last inequality\. Using assumption \([76](https://arxiv.org/html/2608.23765#A1.E76)\) we have

d​η12​‖𝐗−𝐗⋆‖∗2σr​\(𝐗⋆\)2≤1,\\frac\{d\\eta\_\{1\}^\{2\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}^\{2\}\}\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)^\{2\}\}\\leq 1,because eitherη1​d≥2\\eta\_\{1\}\\sqrt\{d\}\\geq 2, in which case‖𝐗−𝐗⋆‖∗≤σr​\(𝐗⋆\)/\(η1​d\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)/\(\\eta\_\{1\}\\sqrt\{d\}\), orη1​d<2\\eta\_\{1\}\\sqrt\{d\}<2, in which cased​η12<4d\\eta\_\{1\}^\{2\}<4and‖𝐗−𝐗⋆‖∗≤σr​\(𝐗⋆\)/2\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)/2\. Hence, we obtain that

\(i​i\)=⟨𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​𝐗⋆​𝐕𝐗\),𝐇𝝈,ε⟩F≤4​Dηr2\(2\+η1\)2​ϑ​‖𝐗−𝐗⋆‖∗\.\(ii\)=\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\_\{\\star\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\rangle\_\{F\}\\leq\\frac\{4D\_\{\\eta\_\{r\}\}^\{2\}\}\{\(2\+\\eta\_\{1\}\)^\{2\}\\vartheta\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(85\)Inserting inequalities \([83](https://arxiv.org/html/2608.23765#A1.E83)\) and \([85](https://arxiv.org/html/2608.23765#A1.E85)\) into inequality \([82](https://arxiv.org/html/2608.23765#A1.E82)\), we obtain for summand\(II\)\(\\mathrm\{II\}\)that

⟨𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\)∘𝒫Sc​\(𝐔𝐗⊤​\(𝐗−𝐗⋆\)​𝐕𝐗\),𝐇𝝈,ε⟩F≤\(2\+Dηr2ϑ\)​‖𝐗−𝐗⋆‖∗\.\\langle\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\circ\\mathcal\{P\}\_\{S^\{c\}\}\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\rangle\_\{F\}\\leq\\left\(2\+\\frac\{D\_\{\\eta\_\{r\}\}^\{2\}\}\{\\vartheta\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.\(86\)Inserting our estimates for summand\(I\)\(\\mathrm\{I\}\)and summand\(II\)\(\\mathrm\{II\}\), see inequalities \([81](https://arxiv.org/html/2608.23765#A1.E81)\) and \([86](https://arxiv.org/html/2608.23765#A1.E86)\), into \([79](https://arxiv.org/html/2608.23765#A1.E79)\), we obtain that

⟨𝐗−𝐗⋆,W𝐗,ε\(𝐗−𝐗⋆\)⟩F≤\(2\+2−1/q\+Dηr2ϑ\)‖𝐗−𝐗⋆‖∗\.\\langle\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\rangle\_\{F\}\\leq\\left\(2\+2^\{\-1/q\}\+\\frac\{D\_\{\\eta\_\{r\}\}^\{2\}\}\{\\vartheta\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.This completes the proof of the lemma\. Recall that by definition ofεk\\varepsilon\_\{k\}we have thatεk≤βr​\(𝐗\(k\)\)∗d\\varepsilon\_\{k\}\\leq\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\}\{d\}\. The following technical lemma gives an explicit lower bound forεk\\varepsilon\_\{k\}\. In particular,[Lemma19](https://arxiv.org/html/2608.23765#Thmlemma19)verifies the condition onεk\\varepsilon\_\{k\}in[Lemma17](https://arxiv.org/html/2608.23765#Thmlemma17)with an explicitηr\\eta\_\{r\}\-dependent constant as used in the bound[eq\.77](https://arxiv.org/html/2608.23765#A1.E77)\.

###### Lemma 19\.

Assume that the linear measurement operator𝒜:ℝd1×d2→ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\rightarrow\\mathbb\{R\}^\{m\}satisfies the NSP of orderrrwith constantηr<1\\eta\_\{r\}<1\. Moreover, assume that𝐗⋆\\mathbf\{X\}\_\{\\star\}has rankrr\. Let\(𝐗\(k\)\)k≥0\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\_\{k\\geq 0\}and\(εk\)k≥0\\left\(\\varepsilon\_\{k\}\\right\)\_\{k\\geq 0\}be the iterates and smoothing parameters ofMatrixIRLSwith input𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\), arbitrary initial weight operator𝐖\(0\)\\mathbf\{W\}^\{\(0\)\},r~=r\\widetilde\{r\}=r, based onqq\-power mean weight operators withq∈\[−1,∞\]q\\in\[\-1,\\infty\]or on one\-sided weight operatorsW𝐗\(k\),εk​\(⋅\)W\_\{\\mathbf\{X\}^\{\(k\)\},\\varepsilon\_\{k\}\}\(\\cdot\)\. Then, for all natural numbersk≥1k\\geq 1, it holds that

εk≥ϑ​βr​\(𝐗\(k\)\)∗d,\\varepsilon\_\{k\}\\geq\\vartheta\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\}\{d\},where

ϑ:=1−ηr\(1\+ηr\)​\(32\+ηr\)\.\\vartheta:=\\frac\{1\-\\eta\_\{r\}\}\{\\left\(1\+\\eta\_\{r\}\\right\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\.

ProofRecall that by definitionεk=min⁡\(εk−1,βr​\(𝐗\(k\)\)∗d\)\\varepsilon\_\{k\}=\\min\\left\(\\varepsilon\_\{k\-1\},\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\}\{d\}\\right\)\. Chooseℓ≤k\\ell\\leq ksuch thatεk=εℓ=βr​\(𝐗\(ℓ\)\)∗d\\varepsilon\_\{k\}=\\varepsilon\_\{\\ell\}=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\ell\)\}\)\_\{\*\}\}\{d\}\. Then we obtain that

εk\\displaystyle\\varepsilon\_\{k\}=βr​\(𝐗\(ℓ\)\)∗d\\displaystyle=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\ell\)\}\)\_\{\*\}\}\{d\}≥\(a\)​𝒥εℓ​\(𝐗\(ℓ\)\)−‖𝐗⋆‖∗\(32\+ηr\)​d\\displaystyle\\overset\{\(a\)\}\{\\geq\}\\frac\{\\mathcal\{J\}\_\{\\varepsilon\_\{\\ell\}\}\(\\mathbf\{X\}^\{\(\\ell\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)d\}≥\(b\)​𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\(32\+ηr\)​d\\displaystyle\\overset\{\(b\)\}\{\\geq\}\\frac\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)d\}≥\(c\)​1−ηr\(1\+ηr\)​\(32\+ηr\)​d​‖𝐗\(k\)−𝐗⋆‖∗\\displaystyle\\overset\{\(c\)\}\{\\geq\}\\frac\{1\-\\eta\_\{r\}\}\{\\left\(1\+\\eta\_\{r\}\\right\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)d\}\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≥\(d\)​1−ηr\(1\+ηr\)​\(32\+ηr\)​d​βr​\(𝐗\(k\)\)∗\.\\displaystyle\\overset\{\(d\)\}\{\\geq\}\\frac\{1\-\\eta\_\{r\}\}\{\\left\(1\+\\eta\_\{r\}\\right\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)d\}\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\.Inequality\(a\)\(a\)follows from Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)and inequality\(b\)\(b\)follows from the monotonicity of the sequence\{𝒥εk​\(𝐗\(k\)\)\}k≥1\\left\\\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\\right\\\}\_\{k\\geq 1\}\. For inequality\(c\)\(c\)we again used Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)\. Inequality\(d\)\(d\)follows from the Eckart–Young theorem since𝐗⋆\\mathbf\{X\}\_\{\\star\}has rankrr\.

#### A\.4\.2Proof of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)\(Dimension\-Free Fast Linear Rate ofMatrixIRLS\)

Now we have all ingredients in place to prove the key result[Proposition4](https://arxiv.org/html/2608.23765#Thmproposition4), which shows that𝒥εk​\(𝐗\(k\)\)\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)decreases linearly at a dimension\-free rate when the iterate𝐗\(k\)\\mathbf\{X\}^\{\(k\)\}is close enough to the true solution𝐗⋆\\mathbf\{X\}\_\{\\star\}, before providing the complete proof of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)\. As discussed in[Section4\.3](https://arxiv.org/html/2608.23765#S4.SS3), we can prove suitable statements for harmonic\-mean weight operators and forqq\-power mean weight operators withq∈\[−1,0\)q\\in\[\-1,0\)\.

###### Proposition 4\(One\-Step Local Contraction\)\.

Let𝐗⋆∈ℝd1×d2\\mathbf\{X\}\_\{\\star\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}have rankrr\. Assume that the measurement operator𝒜:ℝd1×d2⟶ℝm\\mathcal\{A\}:\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}\\longrightarrow\\R^\{m\}satisfies the NSP of orderrrwith constant0<ηr<3/50<\\eta\_\{r\}<3/5and that𝐲=𝒜⁡\(𝐗⋆\)\\mathbf\{y\}=\\mathcal\{A\}\\left\(\\mathbf\{X\}\_\{\\star\}\\right\)\. Let the IRLS iterates\{𝐗\(k\)\}k\\left\\\{\\mathbf\{X\}^\{\(k\)\}\\right\\\}\_\{k\}and\{εk\}k\\left\\\{\\varepsilon\_\{k\}\\right\\\}\_\{k\}be defined by[eq\.12](https://arxiv.org/html/2608.23765#S3.E12)and[eq\.13](https://arxiv.org/html/2608.23765#S3.E13)with rank estimater~=r\\widetilde\{r\}=rand with fixedqq\-power mean weight operators for someq∈\[−1,0\)q\\in\[\-1,0\), and assume that[Algorithm1](https://arxiv.org/html/2608.23765#alg1)does not return in iterationkk, i\.e\., thatεk\>0\\varepsilon\_\{k\}\>0, so that the iterate𝐗\(k\+1\)\\mathbf\{X\}^\{\(k\+1\)\}is defined\. Assume that

‖𝐗\(k\)−𝐗⋆‖∗≤σr​\(𝐗⋆\)max⁡\(η1​d,2\)\.\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\}\.\(87\)where0<η1≤ηr0<\\eta\_\{1\}\\leq\\eta\_\{r\}denotes again the order\-one NSP constant of𝒜\\mathcal\{A\}\. Set

Kηr,q:=2\+2−1/q\+\(3\+ηr\)2​\(1\+ηr\)​\(32\+ηr\)\(1−ηr\)3K\_\{\\eta\_\{r\},q\}:=2\+2^\{\-1/q\}\+\\frac\{\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\{\(1\-\\eta\_\{r\}\)^\{3\}\}and

c^ηr,q:=\(3−5​ηr\)216​\(1\+ηr\)2​\(3\+2​ηr\)​Kηr,q\.\\widehat\{c\}\_\{\\eta\_\{r\},q\}:=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\}\{16\(1\+\\eta\_\{r\}\)^\{2\}\(3\+2\\eta\_\{r\}\)K\_\{\\eta\_\{r\},q\}\}\.Then it holds that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤\(1−c^ηr,q\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\-\\widehat\{c\}\_\{\\eta\_\{r\},q\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\.\(88\)In particular, for the harmonic\-mean weight operator, i\.e\.,q=−1q=\-1, this implies the same estimate with

cηr=\(3−5​ηr\)2​\(1−ηr\)38​\(1\+ηr\)2​\(3\+2​ηr\)​\[8​\(1−ηr\)3\+\(3\+ηr\)2​\(1\+ηr\)​\(3\+2​ηr\)\]c\_\{\\eta\_\{r\}\}=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\(1\-\\eta\_\{r\}\)^\{3\}\}\{8\(1\+\\eta\_\{r\}\)^\{2\}\(3\+2\\eta\_\{r\}\)\\Bigl\[8\(1\-\\eta\_\{r\}\)^\{3\}\+\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\(3\+2\\eta\_\{r\}\)\\Bigr\]\}in place ofc^ηr,−1\\widehat\{c\}\_\{\\eta\_\{r\},\-1\}\.

ProofThe proof of this proposition is structurally similar to the proof of[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2)\. Again, we define𝐍\(k\):=𝐗⋆−𝐗\(k\)\\mathbf\{N\}^\{\(k\)\}:=\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\. By the monotonicity ofε↦𝒥ε​\(𝐗\(k\+1\)\)\\varepsilon\\mapsto\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}^\{\(k\+1\)\}\), the majorization property forqq\-power mean weights withq≥−1q\\geq\-1, and the optimality of𝐗\(k\+1\)\\mathbf\{X\}^\{\(k\+1\)\}in[eq\.12](https://arxiv.org/html/2608.23765#S3.E12), for anyt∈ℝt\\in\\mathbb\{R\}we have that

𝒥εk\+1​\(𝐗\(k\+1\)\)≤Qεk​\(𝐗\(k\+1\)∣𝐗\(k\)\)≤Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\\leq Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\\leq Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\.\(89\)Moreover, by the definition of the quadratic objectiveQεk\(⋅∣𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\cdot\\mid\\mathbf\{X\}^\{\(k\)\}\), see[eq\.8](https://arxiv.org/html/2608.23765#S3.E8), it holds that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\),𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)=t​⟨∇𝒥εk​\(𝐗\(k\)\),𝐍\(k\)⟩F\\begin\{split\}&Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\},\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\\\ =&t\\,\\langle\\nabla\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\),\\mathbf\{N\}^\{\(k\)\}\\rangle\_\{F\}\\end\{split\}As in the proof of[Proposition2](https://arxiv.org/html/2608.23765#Thmproposition2), our goal is to minimize the difference

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)by choosingttaccordingly\. Note that the assumptions of Lemma[17](https://arxiv.org/html/2608.23765#Thmlemma17)are fulfilled due to assumption \([87](https://arxiv.org/html/2608.23765#A1.E87)\) and Lemma[19](https://arxiv.org/html/2608.23765#Thmlemma19), with the choice

ϑηr:=1−ηr\(1\+ηr\)​\(32\+ηr\)\.\\vartheta\_\{\\eta\_\{r\}\}:=\\frac\{1\-\\eta\_\{r\}\}\{\\left\(1\+\\eta\_\{r\}\\right\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\.Set

δηr:=1−ηr1\+ηr−14\.\\delta\_\{\\eta\_\{r\}\}:=\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\-\\frac\{1\}\{4\}\.Thus, we apply Lemma[12](https://arxiv.org/html/2608.23765#Thmlemma12)and Lemma[17](https://arxiv.org/html/2608.23765#Thmlemma17)and obtain that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)\\displaystyle Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)≤−tδηr​‖𝐍\(k\)‖∗⏟=:b\+Kηr,q2​‖𝐍\(k\)‖∗⏟=:a⋅t2\.\\displaystyle\\leq\-t\\underset\{\{=:b\}\}\{\\underbrace\{\\delta\_\{\\eta\_\{r\}\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\}\}\+\\underset\{\{=:a\}\}\{\\underbrace\{\\frac\{K\_\{\\eta\_\{r\},q\}\}\{2\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\}\}\\cdot t^\{2\}\.The right\-hand side is minimized by settingt:=b2​at:=\\frac\{b\}\{2a\}\. This yields that

Qεk​\(𝐗\(k\)\+t​𝐍\(k\)∣𝐗\(k\)\)−𝒥εk​\(𝐗\(k\)\)≤−b24​a=−δηr2​‖𝐍\(k\)‖∗2​Kηr,q\.\\displaystyle Q\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\+t\\mathbf\{N\}^\{\(k\)\}\\mid\\mathbf\{X\}^\{\(k\)\}\)\-\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\\leq\\frac\{\-b^\{2\}\}\{4a\}=\\frac\{\-\\delta\_\{\\eta\_\{r\}\}^\{2\}\\left\\lVert\\mathbf\{N\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\}\{2K\_\{\\eta\_\{r\},q\}\}\.Combining this estimate with inequality \([89](https://arxiv.org/html/2608.23765#A1.E89)\) it follows that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗−δηr2​‖𝐗⋆−𝐗\(k\)‖∗2​Kηr,q\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\-\\frac\{\\delta\_\{\\eta\_\{r\}\}^\{2\}\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\}\{2K\_\{\\eta\_\{r\},q\}\}\.Now note that Lemma[11](https://arxiv.org/html/2608.23765#Thmlemma11)and the fact that𝐗⋆\\mathbf\{X\}\_\{\\star\}has rankrrimply

𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗≤\(32\+ηr\)​βr​\(𝐗\(k\)\)∗≤\(32\+ηr\)​‖𝐗\(k\)−𝐗⋆‖∗\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\beta\_\{r\}\(\\mathbf\{X\}^\{\(k\)\}\)\_\{\*\}\\leq\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.Equivalently,

‖𝐗⋆−𝐗\(k\)‖∗≥𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗32\+ηr\.\\left\\lVert\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}^\{\(k\)\}\\right\\rVert\_\{\\ast\}\\geq\\frac\{\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\}\{\\frac\{3\}\{2\}\+\\eta\_\{r\}\}\.Inserting this into the above inequality, we obtain that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤\(1−δηr22​Kηr,q​\(32\+ηr\)\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)\.\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq\\left\(1\-\\frac\{\\delta\_\{\\eta\_\{r\}\}^\{2\}\}\{2K\_\{\\eta\_\{r\},q\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\(\\mathbf\{X\}^\{\(k\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\.It remains to verify that the bracketed contraction factor equals1−c^ηr,q1\-\\widehat\{c\}\_\{\\eta\_\{r\},q\}\. Combining the two fractions insideδηr\\delta\_\{\\eta\_\{r\}\},

δηr=1−ηr1\+ηr−14=4​\(1−ηr\)−\(1\+ηr\)4​\(1\+ηr\)=3−5​ηr4​\(1\+ηr\),\\delta\_\{\\eta\_\{r\}\}=\\frac\{1\-\\eta\_\{r\}\}\{1\+\\eta\_\{r\}\}\-\\frac\{1\}\{4\}=\\frac\{4\(1\-\\eta\_\{r\}\)\-\(1\+\\eta\_\{r\}\)\}\{4\(1\+\\eta\_\{r\}\)\}=\\frac\{3\-5\\eta\_\{r\}\}\{4\(1\+\\eta\_\{r\}\)\},so thatδηr2=\(3−5​ηr\)216​\(1\+ηr\)2\\delta\_\{\\eta\_\{r\}\}^\{2\}=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\}\{16\(1\+\\eta\_\{r\}\)^\{2\}\}\. Writing32\+ηr=3\+2​ηr2\\frac\{3\}\{2\}\+\\eta\_\{r\}=\\frac\{3\+2\\eta\_\{r\}\}\{2\}cancels the factor22in front ofKηr,qK\_\{\\eta\_\{r\},q\}, and we arrive at

δηr22​Kηr,q​\(32\+ηr\)=\(3−5​ηr\)216​\(1\+ηr\)2​\(3\+2​ηr\)​Kηr,q=c^ηr,q,\\frac\{\\delta\_\{\\eta\_\{r\}\}^\{2\}\}\{2K\_\{\\eta\_\{r\},q\}\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\}\{16\(1\+\\eta\_\{r\}\)^\{2\}\(3\+2\\eta\_\{r\}\)\\,K\_\{\\eta\_\{r\},q\}\}=\\widehat\{c\}\_\{\\eta\_\{r\},q\},which establishes the asserted contraction\.

For the harmonic\-mean weight operatorq=−1q=\-1, we have2−1/q=22^\{\-1/q\}=2and hence

Kηr,−1=4\+\(3\+ηr\)2​\(1\+ηr\)​\(32\+ηr\)\(1−ηr\)3=8​\(1−ηr\)3\+\(3\+ηr\)2​\(1\+ηr\)​\(3\+2​ηr\)2​\(1−ηr\)3,K\_\{\\eta\_\{r\},\-1\}=4\+\\frac\{\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\}\{\(1\-\\eta\_\{r\}\)^\{3\}\}=\\frac\{8\(1\-\\eta\_\{r\}\)^\{3\}\+\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\(3\+2\\eta\_\{r\}\)\}\{2\(1\-\\eta\_\{r\}\)^\{3\}\},where the second equality follows by putting both summands over the common denominator\(1−ηr\)3\(1\-\\eta\_\{r\}\)^\{3\}and using32\+ηr=3\+2​ηr2\\frac\{3\}\{2\}\+\\eta\_\{r\}=\\frac\{3\+2\\eta\_\{r\}\}\{2\}\. Inserting this intoc^ηr,−1\\widehat\{c\}\_\{\\eta\_\{r\},\-1\}yields

c^ηr,−1=\(3−5​ηr\)2​\(1−ηr\)38​\(1\+ηr\)2​\(3\+2​ηr\)​\[8​\(1−ηr\)3\+\(3\+ηr\)2​\(1\+ηr\)​\(3\+2​ηr\)\]=cηr,\\widehat\{c\}\_\{\\eta\_\{r\},\-1\}=\\frac\{\(3\-5\\eta\_\{r\}\)^\{2\}\\,\(1\-\\eta\_\{r\}\)^\{3\}\}\{8\\,\(1\+\\eta\_\{r\}\)^\{2\}\\,\(3\+2\\eta\_\{r\}\)\\bigl\[8\(1\-\\eta\_\{r\}\)^\{3\}\+\(3\+\\eta\_\{r\}\)^\{2\}\(1\+\\eta\_\{r\}\)\(3\+2\\eta\_\{r\}\)\\bigr\]\}=c\_\{\\eta\_\{r\}\},which is the displayed harmonic\-mean form\. This completes the proof\.

With[Proposition4](https://arxiv.org/html/2608.23765#Thmproposition4)in place, we can prove the main result regarding local linear convergence,[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5), with a dimension\-free rate, forMatrixIRLSusing harmonic\-mean weight operators\.

Proof of[Theorem5](https://arxiv.org/html/2608.23765#Thmtheorem5)We will prove this theorem by induction\. Recall that

Aηr:=\(32\+ηr\)​\(1\+ηr\)1−ηr\.A\_\{\\eta\_\{r\}\}:=\\frac\{\\left\(\\frac\{3\}\{2\}\+\\eta\_\{r\}\\right\)\\left\(1\+\\eta\_\{r\}\\right\)\}\{1\-\\eta\_\{r\}\}\.Sinceηr<3/5\\eta\_\{r\}<3/5, we have

Aηr<\(32\+35\)​\(1\+35\)1−35=425<9\.A\_\{\\eta\_\{r\}\}<\\frac\{\\left\(\\frac\{3\}\{2\}\+\\frac\{3\}\{5\}\\right\)\\left\(1\+\\frac\{3\}\{5\}\\right\)\}\{1\-\\frac\{3\}\{5\}\}=\\frac\{42\}\{5\}<9\.\(90\)Moreover, the displayed formula forcηrc\_\{\\eta\_\{r\}\}gives0<cηr<10<c\_\{\\eta\_\{r\}\}<1\. We immediately observe that in the base casek=k~k=\\tilde\{k\}both inequalities \([36](https://arxiv.org/html/2608.23765#S4.E36)\) and \([37](https://arxiv.org/html/2608.23765#S4.E37)\) hold\.

Now assume that the statement holds for somek≥k~k\\geq\\tilde\{k\}for which[Algorithm1](https://arxiv.org/html/2608.23765#alg1)carries out iterationk\+1k\+1, i\.e\., for whichεk\>0\\varepsilon\_\{k\}\>0\. To show the induction step we first note that due to assumption \([35](https://arxiv.org/html/2608.23765#S4.E35)\), inequality \([37](https://arxiv.org/html/2608.23765#S4.E37)\), \([90](https://arxiv.org/html/2608.23765#A1.E90)\), and\(1−cηr\)k−k~≤1\(1\-c\_\{\\eta\_\{r\}\}\)^\{k\-\\tilde\{k\}\}\\leq 1, we have that

‖𝐗\(k\)−𝐗⋆‖∗≤Aηr​\(1−cηr\)k−k~​σr​\(𝐗⋆\)9​max⁡\(η1​d,2\)≤σr​\(𝐗⋆\)max⁡\(η1​d,2\)\.\\left\\lVert\\mathbf\{X\}^\{\(k\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\leq A\_\{\\eta\_\{r\}\}\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\-\\tilde\{k\}\}\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{9\\max\\big\(\\eta\_\{1\}\\sqrt\{d\},2\\big\)\}\\leq\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{\\max\\big\(\\eta\_\{1\}\\sqrt\{d\},2\\big\)\}\.Thus, we can apply[Proposition4](https://arxiv.org/html/2608.23765#Thmproposition4)withq=−1q=\-1and obtain that

𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗≤\(1−cηr\)​\(𝒥εk​\(𝐗\(k\)\)−‖𝐗⋆‖∗\)≤\(1−cηr\)k\+1−k~​\(𝒥εk~​\(𝐗\(k~\)\)−‖𝐗⋆‖∗\)\\begin\{split\}\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}&\\leq\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\}\}\\left\(\\mathbf\{X\}^\{\(k\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\\\\ &\\leq\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\+1\-\\tilde\{k\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{\\tilde\{k\}\}\}\\left\(\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\\right\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)\\end\{split\}\(91\)In the second inequality we used the induction hypothesis\. This proves inequality \([36](https://arxiv.org/html/2608.23765#S4.E36)\) fork\+1k\+1\. It remains to prove inequality \([37](https://arxiv.org/html/2608.23765#S4.E37)\)\. For that, we note that

‖𝐗\(k\+1\)−𝐗⋆‖∗\\displaystyle\\left\\lVert\\mathbf\{X\}^\{\(k\+1\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}≤\(a\)​1\+ηr1−ηr​\(𝒥εk\+1​\(𝐗\(k\+1\)\)−‖𝐗⋆‖∗\)\\displaystyle\\overset\{\(a\)\}\{\\leq\}\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{k\+1\}\}\(\\mathbf\{X\}^\{\(k\+1\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)≤\(b\)​1\+ηr1−ηr​\(1−cηr\)k\+1−k~​\(𝒥εk~​\(𝐗\(k~\)\)−‖𝐗⋆‖∗\)\\displaystyle\\overset\{\(b\)\}\{\\leq\}\\frac\{1\+\\eta\_\{r\}\}\{1\-\\eta\_\{r\}\}\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\+1\-\\tilde\{k\}\}\\left\(\\mathcal\{J\}\_\{\\varepsilon\_\{\\tilde\{k\}\}\}\(\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\)\-\\left\\lVert\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\\right\)≤\(c\)​Aηr​\(1−cηr\)k\+1−k~​βr​\(𝐗\(k~\)\)∗\\displaystyle\\overset\{\(c\)\}\{\\leq\}A\_\{\\eta\_\{r\}\}\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\+1\-\\tilde\{k\}\}\\beta\_\{r\}\(\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\)\_\{\*\}≤\(d\)​Aηr​\(1−cηr\)k\+1−k~​‖𝐗\(k~\)−𝐗⋆‖∗\.\\displaystyle\\overset\{\(d\)\}\{\\leq\}A\_\{\\eta\_\{r\}\}\\left\(1\-c\_\{\\eta\_\{r\}\}\\right\)^\{k\+1\-\\tilde\{k\}\}\\left\\lVert\\mathbf\{X\}^\{\(\\tilde\{k\}\)\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.For inequality\(a\)\(a\)and inequality\(c\)\(c\), we used[Lemma11](https://arxiv.org/html/2608.23765#Thmlemma11), whereas inequality\(b\)\(b\)is due to \([91](https://arxiv.org/html/2608.23765#A1.E91)\)\. Finally, inequality\(d\)\(d\)is a consequence of the Eckart–Young theorem and the fact that the matrix𝐗⋆\\mathbf\{X\}\_\{\\star\}has rankrr\. This proves inequality \([37](https://arxiv.org/html/2608.23765#S4.E37)\) fork\+1k\+1\. Thus, the proof is complete\.

#### A\.4\.3Proof of[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)\(Counterexample for One\-Sided IRLS\)

In this section, we provide an explicit counterexample substantiating[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6), which shows that it is in general not possible to obtain a dimension\-independent upper bound on⟨𝐗⋆−𝐗,W𝐗,ε​\(𝐗⋆−𝐗\)⟩F\\langle\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\_\{\\star\}\-\\mathbf\{X\}\)\\rangle\_\{F\}if a one\-sided weight operator such as[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5)is used\. Leveraging a lower bound on⟨𝐗−𝐗⋆,W𝐗,ε​\(𝐗−𝐗⋆\)⟩F\\langle\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\rangle\_\{F\}, we show that in this example, the nuclear norm error‖𝐗\+−𝐗⋆‖∗\\left\\lVert\\mathbf\{X\}^\{\+\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}of the next IRLS iterate𝐗\+\\mathbf\{X\}^\{\+\}cannot decrease by more than a factor of\(1−c​r/d\)\(1\-cr/d\)for some constantc\>0c\>0compared to the nuclear norm error before the IRLS step\.

We provide the argument for left\-sided weight operators and note that it can be easily adapted to right\-sided weight operators by transposition of the underlying matrices\.

Proof of[Theorem6](https://arxiv.org/html/2608.23765#Thmtheorem6)We give an explicit construction\. Letd≥220​rd\\geq 220rand seth:=10​rh:=10r\. Let\{𝐞1,…,𝐞d\}\\left\\\{\\mathbf\{e\}\_\{1\},\\ldots,\\mathbf\{e\}\_\{d\}\\right\\\}and\{𝐟1,…,𝐟d\}\\left\\\{\\mathbf\{f\}\_\{1\},\\ldots,\\mathbf\{f\}\_\{d\}\\right\\\}be orthonormal bases ofℝd\\R^\{d\}\. Fixα\>0\\alpha\>0\. We choose an angleθ\>0\\theta\>0satisfying

0<θ≤min⁡\(π2,111​r​max⁡\(d11​r−1,2\)\)\.0<\\theta\\leq\\min\\left\(\\frac\{\\pi\}\{2\},\\frac\{1\}\{11r\\max\\left\(\\frac\{\\sqrt\{d\}\}\{11r\-1\},2\\right\)\}\\right\)\.\(92\)Fori=1,…,ri=1,\\ldots,r, define

𝐮i:=cos⁡\(θ\)​𝐞i\+sin⁡\(θ\)​𝐞r\+i,𝐩i:=−sin⁡\(θ\)​𝐞i\+cos⁡\(θ\)​𝐞r\+i\.\\mathbf\{u\}\_\{i\}:=\\cos\(\\theta\)\\mathbf\{e\}\_\{i\}\+\\sin\(\\theta\)\\mathbf\{e\}\_\{r\+i\},\\qquad\\mathbf\{p\}\_\{i\}:=\-\\sin\(\\theta\)\\mathbf\{e\}\_\{i\}\+\\cos\(\\theta\)\\mathbf\{e\}\_\{r\+i\}\.Seta:=2​α​sin⁡\(θ/2\)a:=2\\alpha\\sin\(\\theta/2\)and define

𝐗⋆:=α​∑i=1r𝐞i​𝐟i⊤,𝐗:=α​∑i=1r𝐮i​𝐟i⊤\+a​∑j=1h𝐞2​r\+j​𝐟r\+j⊤\.\\mathbf\{X\}\_\{\\star\}:=\\alpha\\sum\_\{i=1\}^\{r\}\\mathbf\{e\}\_\{i\}\\mathbf\{f\}\_\{i\}^\{\\top\},\\qquad\\mathbf\{X\}:=\\alpha\\sum\_\{i=1\}^\{r\}\\mathbf\{u\}\_\{i\}\\mathbf\{f\}\_\{i\}^\{\\top\}\+a\\sum\_\{j=1\}^\{h\}\\mathbf\{e\}\_\{2r\+j\}\\mathbf\{f\}\_\{r\+j\}^\{\\top\}\.Sinced≥220​rd\\geq 220r, all indices used in this definition are at mostdd\. Moreover, \([92](https://arxiv.org/html/2608.23765#A1.E92)\) impliesa≤α​θ≤α/\(22​r\)<αa\\leq\\alpha\\theta\\leq\\alpha/\(22r\)<\\alpha, so the nonzero singular values of𝐗\\mathbf\{X\}areα\\alpharepeatedrrtimes andaarepeatedh=10​rh=10rtimes\. Thus

βr​\(𝐗\)∗=10​r​a,ε=βr​\(𝐗\)∗d=10​r​ad\.\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}=10ra,\\qquad\\varepsilon=\\frac\{\\beta\_\{r\}\(\\mathbf\{X\}\)\_\{\*\}\}\{d\}=\\frac\{10ra\}\{d\}\.\(93\)
Let𝚫:=𝐗−𝐗⋆\\mathbf\{\\Delta\}:=\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\. Then

𝚫=α​∑i=1r\(𝐮i−𝐞i\)​𝐟i⊤\+a​∑j=1h𝐞2​r\+j​𝐟r\+j⊤\.\\mathbf\{\\Delta\}=\\alpha\\sum\_\{i=1\}^\{r\}\\left\(\\mathbf\{u\}\_\{i\}\-\\mathbf\{e\}\_\{i\}\\right\)\\mathbf\{f\}\_\{i\}^\{\\top\}\+a\\sum\_\{j=1\}^\{h\}\\mathbf\{e\}\_\{2r\+j\}\\mathbf\{f\}\_\{r\+j\}^\{\\top\}\.The rank\-one terms in this decomposition are mutually orthogonal on both the left and the right, and

‖α⁡\(𝐮i−𝐞i\)‖2=2​α​sin⁡\(θ/2\)=a\.\\left\\\|\\alpha\\left\(\\mathbf\{u\}\_\{i\}\-\\mathbf\{e\}\_\{i\}\\right\)\\right\\\|\_\{2\}=2\\alpha\\sin\(\\theta/2\)=a\.Consequently, the singular values of𝚫\\mathbf\{\\Delta\}areaarepeatedr\+h=11​rr\+h=11rtimes, and hence

∑i=1rσi​\(𝚫\)=r​a,∑i=r\+1dσi​\(𝚫\)=10​r​a,‖𝚫‖∗=11​r​a\.\\sum\_\{i=1\}^\{r\}\\sigma\_\{i\}\(\\mathbf\{\\Delta\}\)=ra,\\qquad\\sum\_\{i=r\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{\\Delta\}\)=10ra,\\qquad\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\_\{\\ast\}=11ra\.\(94\)
We now define the measurement operator\. Let𝐆1,…,𝐆d2−1\\mathbf\{G\}\_\{1\},\\ldots,\\mathbf\{G\}\_\{d^\{2\}\-1\}be an orthonormal basis of the Frobenius\-orthogonal complement of𝚫\\mathbf\{\\Delta\}inℝd×d\\R^\{d\\times d\}, and set

𝒜⁡\(𝐙\):=\(⟨𝐙,𝐆ℓ⟩F\)ℓ=1d2−1\.\\mathcal\{A\}\(\\mathbf\{Z\}\):=\\left\(\\langle\\mathbf\{Z\},\\mathbf\{G\}\_\{\\ell\}\\rangle\_\{F\}\\right\)\_\{\\ell=1\}^\{d^\{2\}\-1\}\.Thenker⁡\(𝒜\)=span⁡\{𝚫\}\\ker\(\\mathcal\{A\}\)=\\operatorname\{span\}\\\{\\mathbf\{\\Delta\}\\\}and, in particular,𝐗−𝐗⋆=𝚫∈ker⁡\(𝒜\)\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}=\\mathbf\{\\Delta\}\\in\\ker\(\\mathcal\{A\}\)\. For every nonzero𝐍∈ker⁡\(𝒜\)\\mathbf\{N\}\\in\\ker\(\\mathcal\{A\}\), we have𝐍=t​𝚫\\mathbf\{N\}=t\\mathbf\{\\Delta\}for somet≠0t\\neq 0\. Thus \([94](https://arxiv.org/html/2608.23765#A1.E94)\) gives

∑i=1rσi​\(𝐍\)=110​∑i=r\+1dσi​\(𝐍\),\\sum\_\{i=1\}^\{r\}\\sigma\_\{i\}\(\\mathbf\{N\}\)=\\frac\{1\}\{10\}\\sum\_\{i=r\+1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{N\}\),so𝒜\\mathcal\{A\}satisfies the NSP of orderrrwith constantηr=1/10\\eta\_\{r\}=1/10\. The same singular value calculation shows that𝒜\\mathcal\{A\}satisfies the NSP of order11with constant

η1=111​r−1,\\eta\_\{1\}=\\frac\{1\}\{11r\-1\},becauseσ1​\(𝚫\)=a\\sigma\_\{1\}\(\\mathbf\{\\Delta\}\)=aand∑i=2dσi​\(𝚫\)=\(11​r−1\)​a\\sum\_\{i=2\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{\\Delta\}\)=\(11r\-1\)a\.

It remains to check the local neighborhood condition\. Sinceσr​\(𝐗⋆\)=α\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)=\\alpha, \([92](https://arxiv.org/html/2608.23765#A1.E92)\) implies

‖𝐗−𝐗⋆‖∗=11​r​a≤11​r​α​θ≤αmax⁡\(d11​r−1,2\)=σr​\(𝐗⋆\)max⁡\(η1​d,2\)\.\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=11ra\\leq 11r\\alpha\\theta\\leq\\frac\{\\alpha\}\{\\max\\left\(\\frac\{\\sqrt\{d\}\}\{11r\-1\},2\\right\)\}=\\frac\{\\sigma\_\{r\}\(\\mathbf\{X\}\_\{\\star\}\)\}\{\\max\\left\(\\eta\_\{1\}\\sqrt\{d\},2\\right\)\}\.
Lower bound on the left\-sided quadratic form\.LetW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)be the weight operator with the left\-sided core matrix \([4](https://arxiv.org/html/2608.23765#S3.E4)\)\. The vectors𝐩1,…,𝐩r\\mathbf\{p\}\_\{1\},\\ldots,\\mathbf\{p\}\_\{r\}are orthogonal to the column space of𝐗\\mathbf\{X\}; hence they can be chosen as part of the zero left\-singular\-vector block in a full SVD of𝐗\\mathbf\{X\}\. On these rows the left\-sided weight is equal to1/ε1/\\varepsilon\. Furthermore,

𝐩i⊤​𝚫​𝐟i=α​𝐩i⊤​\(𝐮i−𝐞i\)=α​sin⁡θ=a​cos⁡\(θ/2\)\.\\mathbf\{p\}\_\{i\}^\{\\top\}\\mathbf\{\\Delta\}\\mathbf\{f\}\_\{i\}=\\alpha\\mathbf\{p\}\_\{i\}^\{\\top\}\\left\(\\mathbf\{u\}\_\{i\}\-\\mathbf\{e\}\_\{i\}\\right\)=\\alpha\\sin\\theta=a\\cos\(\\theta/2\)\.Writing𝐗=𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}and𝐌:=𝐔𝐗⊤​𝚫​𝐕𝐗\\mathbf\{M\}:=\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{\\Delta\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}, the left\-sided definition gives

⟨𝚫,W𝐗,ε​\(𝚫\)⟩F=∑k,ℓ\|Mk​ℓ\|2max⁡\(σk​\(𝐗\),ε\)\.\\langle\\mathbf\{\\Delta\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{\\Delta\}\)\\rangle\_\{F\}=\\sum\_\{k,\\ell\}\\frac\{\\left\|M\_\{k\\ell\}\\right\|^\{2\}\}\{\\max\(\\sigma\_\{k\}\(\\mathbf\{X\}\),\\varepsilon\)\}\.All summands are nonnegative\. Keeping only the entries corresponding to the rows𝐩i\\mathbf\{p\}\_\{i\}and columns𝐟i\\mathbf\{f\}\_\{i\}yields

⟨𝚫,W𝐗,ε​\(𝚫\)⟩F≥1ε​∑i=1r\|𝐩i⊤​𝚫​𝐟i\|2=r​a2​cos2⁡\(θ/2\)ε\.\\langle\\mathbf\{\\Delta\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{\\Delta\}\)\\rangle\_\{F\}\\geq\\frac\{1\}\{\\varepsilon\}\\sum\_\{i=1\}^\{r\}\\left\|\\mathbf\{p\}\_\{i\}^\{\\top\}\\mathbf\{\\Delta\}\\mathbf\{f\}\_\{i\}\\right\|^\{2\}=\\frac\{ra^\{2\}\\cos^\{2\}\(\\theta/2\)\}\{\\varepsilon\}\.Using \([93](https://arxiv.org/html/2608.23765#A1.E93)\) and \([94](https://arxiv.org/html/2608.23765#A1.E94)\), we obtain

⟨𝚫,W𝐗,ε​\(𝚫\)⟩F≥d​a​cos2⁡\(θ/2\)10=d110​r​cos2⁡\(θ/2\)​‖𝚫‖∗\.\\langle\\mathbf\{\\Delta\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{\\Delta\}\)\\rangle\_\{F\}\\geq\\frac\{d\\,a\\,\\cos^\{2\}\(\\theta/2\)\}\{10\}=\\frac\{d\}\{110r\}\\cos^\{2\}\(\\theta/2\)\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\_\{\\ast\}\.Sinceθ≤π/2\\theta\\leq\\pi/2,cos2⁡\(θ/2\)≥1/2\\cos^\{2\}\(\\theta/2\)\\geq 1/2, and therefore

⟨𝐗−𝐗⋆,W𝐗,ε​\(𝐗−𝐗⋆\)⟩F≥d220​r​‖𝐗−𝐗⋆‖∗\.\\langle\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\)\\rangle\_\{F\}\\geq\\frac\{d\}\{220r\}\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}\.
One\-step lower bound\.The feasible set of the weighted least\-squares problem is the affine line

\{𝐙:𝒜⁡\(𝐙\)=𝒜⁡\(𝐗⋆\)\}=\{𝐗⋆\+λ​𝚫:λ∈ℝ\}\.\\\{\\mathbf\{Z\}:\\mathcal\{A\}\(\\mathbf\{Z\}\)=\\mathcal\{A\}\(\\mathbf\{X\}\_\{\\star\}\)\\\}=\\\{\\mathbf\{X\}\_\{\\star\}\+\\lambda\\mathbf\{\\Delta\}:\\lambda\\in\\R\\\}\.Set

A𝚫:=⟨𝚫,W𝐗,ε​\(𝚫\)⟩F,b𝚫:=⟨W𝐗,ε​\(𝐗\),𝚫⟩F\.A\_\{\\mathbf\{\\Delta\}\}:=\\langle\\mathbf\{\\Delta\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{\\Delta\}\)\\rangle\_\{F\},\\qquad b\_\{\\mathbf\{\\Delta\}\}:=\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{\\Delta\}\\rangle\_\{F\}\.The minimizer on the affine line is

𝐗\+=𝐗⋆\+λ\+​𝚫,λ\+=1−b𝚫A𝚫\.\\mathbf\{X\}^\{\+\}=\\mathbf\{X\}\_\{\\star\}\+\\lambda\_\{\+\}\\mathbf\{\\Delta\},\\qquad\\lambda\_\{\+\}=1\-\\frac\{b\_\{\\mathbf\{\\Delta\}\}\}\{A\_\{\\mathbf\{\\Delta\}\}\}\.Since all nonzero singular values of𝐗\\mathbf\{X\}are larger thanε\\varepsilon,

W𝐗,ε​\(𝐗\)=∑i=1r𝐮i​𝐟i⊤\+∑j=1h𝐞2​r\+j​𝐟r\+j⊤\.W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)=\\sum\_\{i=1\}^\{r\}\\mathbf\{u\}\_\{i\}\\mathbf\{f\}\_\{i\}^\{\\top\}\+\\sum\_\{j=1\}^\{h\}\\mathbf\{e\}\_\{2r\+j\}\\mathbf\{f\}\_\{r\+j\}^\{\\top\}\.Consequently,

b𝚫=r​α​\(1−cos⁡θ\)\+h​a=r​a​sin⁡\(θ/2\)\+10​r​a≤11​r​a\.b\_\{\\mathbf\{\\Delta\}\}=r\\alpha\(1\-\\cos\\theta\)\+ha=ra\\sin\(\\theta/2\)\+10ra\\leq 11ra\.Combining this withA𝚫≥d​a​cos2⁡\(θ/2\)/10A\_\{\\mathbf\{\\Delta\}\}\\geq d\\,a\\cos^\{2\}\(\\theta/2\)/10gives

1−λ\+=b𝚫A𝚫≤110​rd​cos2⁡\(θ/2\)≤220​rd\.1\-\\lambda\_\{\+\}=\\frac\{b\_\{\\mathbf\{\\Delta\}\}\}\{A\_\{\\mathbf\{\\Delta\}\}\}\\leq\\frac\{110r\}\{d\\cos^\{2\}\(\\theta/2\)\}\\leq\\frac\{220r\}\{d\}\.Sinced≥220​rd\\geq 220r, this showsλ\+∈\[0,1\]\\lambda\_\{\+\}\\in\[0,1\]\. Hence

‖𝐗\+−𝐗⋆‖∗=λ\+​‖𝚫‖∗≥\(1−220​rd\)​‖𝐗−𝐗⋆‖∗,\\left\\lVert\\mathbf\{X\}^\{\+\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\}=\\lambda\_\{\+\}\\left\\lVert\\mathbf\{\\Delta\}\\right\\rVert\_\{\\ast\}\\geq\\left\(1\-\\frac\{220r\}\{d\}\\right\)\\left\\lVert\\mathbf\{X\}\-\\mathbf\{X\}\_\{\\star\}\\right\\rVert\_\{\\ast\},which proves the stated lower bound for the subsequent IRLS iterate\.

## Appendix BComplementary Proofs

In this section, we provide proofs for statements that are known in the literature \(explicitly or implicitly\) and point out an incorrect proof in the literature\. In particular, we recall a proof for the properties of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)of the IRLS quadratic model in[SectionB\.1](https://arxiv.org/html/2608.23765#A2.SS1), establish Lipschitz continuity of∇𝒥ε​\(⋅\)\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)in[SectionB\.2](https://arxiv.org/html/2608.23765#A2.SS2), and provide existing proofs for majorization of one\-sided IRLS quadratics in[SectionB\.3](https://arxiv.org/html/2608.23765#A2.SS3)\. In[SectionB\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4), we discuss to what extent the majorization result has remained elusive for previous works, in particular, by pointing out a key inaccuracy in the corresponding arguments of[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)\.

### B\.1Proof of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)\(Properties of the IRLS Quadratic Model\)

As mentioned in[Section3\.1](https://arxiv.org/html/2608.23765#S3.SS1), the proof of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)is rather straightforward, and applies for the quadratic model functionsQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)corresponding to any of the considered weight operator notions considered in this paper, including one\-sided weight operators, the harmonic\-mean weight operator, and allqq\-mean weight operators\. Two ingredients are used: \(i\) the diagonal elements of the core matrix𝐇𝝈,ε∈ℝd1×d2\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}satisfy\(𝐇𝝈,ε\)i​i=max⁡\(σi,ε\)−1\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=\\max\(\\sigma\_\{i\},\\varepsilon\)^\{\-1\}for alli∈\[d\]i\\in\[d\], which yields the gradient condition and can be easily verified for all considered weight operator notions; and \(ii\) the Hadamard form[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)of[Definition2](https://arxiv.org/html/2608.23765#Thmdefinition2), which implies thatW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is self\-adjoint and thus yields the simplified expression[eq\.11](https://arxiv.org/html/2608.23765#S3.E11)\.

Proof of[Proposition1](https://arxiv.org/html/2608.23765#Thmproposition1)Let us start by computing the gradient∇𝒥ε​\(𝐗\)\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)of the smoothed nuclear norm surrogate𝒥ε​\(𝐗\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)defined in[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)\. Note that𝒥ε​\(𝐗\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)is a convex spectral function of𝐗\\mathbf\{X\}, and thus by\([Lewis and Sendov, 2005](https://arxiv.org/html/2608.23765#bib.bib84), Proposition 6\.2\), the gradient itself is given by the spectral formula

∇𝒥ε​\(𝐗\)=𝐔𝐗​diag⁡\(jε′​\(σi​\(𝐗\)\)\)i=1d​𝐕𝐗⊤,\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\\big\(j^\{\\prime\}\_\{\\varepsilon\}\(\\sigma\_\{i\}\(\\mathbf\{X\}\)\)\\big\)\_\{i=1\}^\{d\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\},where𝐗=𝐔𝐗​diag⁡\(𝝈\)​𝐕𝐗⊤\\mathbf\{X\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\bm\{\\sigma\}\)\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}is the singular value decomposition of𝐗\\mathbf\{X\}\. We note that

jε′​\(σ\)=\{1,σ\>ε,σ/ε,σ≤ε\.=σmax⁡\(σ,ε\),j^\{\\prime\}\_\{\\varepsilon\}\(\\sigma\)=\\begin\{cases\}1,&\\sigma\>\\varepsilon,\\\\\[4\.0pt\] \\sigma/\\varepsilon,&\\sigma\\leq\\varepsilon\.\\end\{cases\}=\\frac\{\\sigma\}\{\\max\(\\sigma,\\varepsilon\)\},It follows that

∇𝒥ε​\(𝐗\)=𝐔𝐗​dg⁡\(σi​\(𝐗\)max⁡\(σi​\(𝐗\),ε\)\)i=1d​𝐕𝐗⊤\.\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\dg\\left\(\\frac\{\\sigma\_\{i\}\(\\mathbf\{X\}\)\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}\\right\)\_\{i=1\}^\{d\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}\.\(95\)Note that, regardless whether harmonic\-mean weights or one\-sided weights are used, the diagonal elements of𝐇𝝈,ε∈ℝd1×d2\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}are given by\(𝐇𝝈,ε\)i​i=max⁡\(σi,ε\)−1\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=\\max\(\\sigma\_\{i\},\\varepsilon\)^\{\-1\}\. Then the claim follows by simply inserting the definition of the weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\), see[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)\. Indeed, we have that

W𝐗,ε​\(𝐗\)\\displaystyle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)=𝐔𝐗​\[𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐗𝐕𝐗\)\]​𝐕𝐗⊤=𝐔𝐗​\[𝐇𝝈,ε∘dg⁡\(𝝈\)\]​𝐕𝐗⊤\\displaystyle=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{X\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\left\[\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\\dg\(\\bm\{\\sigma\}\)\\right\]\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}=𝐔𝐗​dg⁡\(σi​\(𝐗\)max⁡\(σi​\(𝐗\),ε\)\)i=1d​𝐕𝐗⊤=∇𝒥ε​\(𝐗\)\.\\displaystyle=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\dg\\left\(\\frac\{\\sigma\_\{i\}\(\\mathbf\{X\}\)\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}\\right\)\_\{i=1\}^\{d\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}^\{\\top\}=\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\.This proves the gradient condition \([10](https://arxiv.org/html/2608.23765#S3.E10)\)\.

We now derive the specific form ofQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)in equation \([11](https://arxiv.org/html/2608.23765#S3.E11)\)\. First, note that[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)implies thatW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is linear and self\-adjoint with respect to the Frobenius inner product\. Indeed, for all𝐀,𝐁∈ℝd1×d2\\mathbf\{A\},\\mathbf\{B\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}, orthogonality of𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}and𝐕𝐗\\mathbf\{V\}\_\{\\mathbf\{X\}\}gives

⟨W𝐗,ε​\(𝐀\),𝐁⟩F\\displaystyle\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{A\}\),\\mathbf\{B\}\\rangle\_\{F\}=⟨𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐀𝐕𝐗\),𝐔𝐗⊤​𝐁𝐕𝐗⟩F\\displaystyle=\\langle\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{A\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\),\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{B\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\\rangle\_\{F\}=⟨𝐔𝐗⊤​𝐀𝐕𝐗,𝐇𝝈,ε∘\(𝐔𝐗⊤​𝐁𝐕𝐗\)⟩F=⟨𝐀,W𝐗,ε​\(𝐁\)⟩F,\\displaystyle=\\langle\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{A\}\\mathbf\{V\}\_\{\\mathbf\{X\}\},\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\\circ\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\mathbf\{B\}\\mathbf\{V\}\_\{\\mathbf\{X\}\}\)\\rangle\_\{F\}=\\langle\\mathbf\{A\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{B\}\)\\rangle\_\{F\},where the middle identity holds for every real core matrix𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\(symmetry of𝐇𝝈,ε\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}is not required\), since Hadamard multiplication is entrywise and thus self\-adjoint for the Frobenius product\. Using the definition[eq\.8](https://arxiv.org/html/2608.23765#S3.E8), the gradient condition \([10](https://arxiv.org/html/2608.23765#S3.E10)\), and this self\-adjointness, we obtain

Qε​\(𝐙∣𝐗\)\\displaystyle Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)=𝒥ε​\(𝐗\)\+⟨W𝐗,ε​\(𝐗\),𝐙−𝐗⟩F\+12​⟨𝐙−𝐗,W𝐗,ε​\(𝐙−𝐗\)⟩F\\displaystyle=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{Z\}\-\\mathbf\{X\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\}\-\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\-\\mathbf\{X\}\)\\rangle\_\{F\}=𝒥ε​\(𝐗\)\+⟨W𝐗,ε​\(𝐗\),𝐙⟩F−⟨W𝐗,ε​\(𝐗\),𝐗⟩F\\displaystyle=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{Z\}\\rangle\_\{F\}\-\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}\+12​⟨𝐙,W𝐗,ε​\(𝐙\)⟩F−⟨𝐙,W𝐗,ε​\(𝐗\)⟩F\+12​⟨𝐗,W𝐗,ε​\(𝐗\)⟩F\\displaystyle\\quad\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\-\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\langle\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}=𝒥ε​\(𝐗\)\+12​⟨𝐙,W𝐗,ε​\(𝐙\)⟩F−12​⟨𝐗,W𝐗,ε​\(𝐗\)⟩F\.\\displaystyle=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\frac\{1\}\{2\}\\langle\\mathbf\{Z\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}\.This proves \([11](https://arxiv.org/html/2608.23765#S3.E11)\)\. Finally, the statementQε​\(𝐗∣𝐗\)=𝒥ε​\(𝐗\)Q\_\{\\varepsilon\}\(\\mathbf\{X\}\\mid\\mathbf\{X\}\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)follows immediately as12​⟨𝐗,W𝐗,ε​\(𝐗\)⟩F−12​⟨𝐗,W𝐗,ε​\(𝐗\)⟩F=0\\frac\{1\}\{2\}\\langle\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle\\mathbf\{X\},W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\)\\rangle\_\{F\}=0\.

### B\.2Lipschitz Gradients of Smoothed Nuclear Norm

In[Section3\.1](https://arxiv.org/html/2608.23765#S3.SS1), it was claimed that theε\\varepsilon\-smoothed nuclear norm𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)is differentiable with a Lipschitz\-continuous gradient\. We now provide a proof of this claim\.

In particular, we observe that the gradient𝐗↦∇𝒥ε​\(𝐗\)\\mathbf\{X\}\\mapsto\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)of[eq\.95](https://arxiv.org/html/2608.23765#A2.E95)is a non\-Hermitian Loewner operator\([Löwner, 1934](https://arxiv.org/html/2608.23765#bib.bib90);[Sun and Sun, 2008](https://arxiv.org/html/2608.23765#bib.bib91);[Ding et al\., 2018](https://arxiv.org/html/2608.23765#bib.bib92)\)\. The framework of\([Andersson et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib53)\)provides an exact quantification of the Lipschitz properties of such operators if the function that is applied to each singular value is the same, which is the case here withgε:σ↦σmax⁡\(σ,ε\)g\_\{\\varepsilon\}:\\sigma\\mapsto\\frac\{\\sigma\}\{\\max\(\\sigma,\\varepsilon\)\}\.gεg\_\{\\varepsilon\}is Lipschitz continuous with Lipschitz constantL=1εL=\\frac\{1\}\{\\varepsilon\}, so by\([Andersson et al\., 2016](https://arxiv.org/html/2608.23765#bib.bib53), Theorem 1\.1\), the gradient𝐗↦∇𝒥ε​\(𝐗\)\\mathbf\{X\}\\mapsto\\nabla\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)is Lipschitz continuous with Lipschitz constantL=1εL=\\frac\{1\}\{\\varepsilon\}with respect to the Frobenius norm, which proves the claim\.

### B\.3Prior Art of Majorization Proofs for Low\-Rank IRLS Algorithms

For quadratic model functions defined by one\-sided weight operators[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[5](https://arxiv.org/html/2608.23765#S3.E5)and associated IRLS methods\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Mohan and Fazel, 2012](https://arxiv.org/html/2608.23765#bib.bib48)\),[eq\.14](https://arxiv.org/html/2608.23765#S3.E14)follows from the*global majorization property*as established in previous works\([Fornasier et al\., 2011](https://arxiv.org/html/2608.23765#bib.bib54);[Kümmerle, 2019](https://arxiv.org/html/2608.23765#bib.bib97), Section 2\.3\.2\), which we restate below for completeness\.

###### Proposition 5\(Global Majorization of One\-Sided Quadratic Models\)\.

Letε\>0\\varepsilon\>0, let𝒥ε:ℝd1×d2→ℝ\\mathcal\{J\}\_\{\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rbe theε\\varepsilon\-smoothed nuclear norm[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)andQε\(⋅∣𝐗\):ℝd1×d2→ℝQ\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\):\\R^\{d\_\{1\}\\times d\_\{2\}\}\\to\\Rbe the quadratic model function of[eq\.8](https://arxiv.org/html/2608.23765#S3.E8)defined by the left\-sided or right\-sided weight operator[eq\.7](https://arxiv.org/html/2608.23765#S3.E7)using core matrix[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)or[eq\.5](https://arxiv.org/html/2608.23765#S3.E5), respectively\. Then,Qε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)majorizes𝒥ε\\mathcal\{J\}\_\{\\varepsilon\}globally:

Qε​\(𝐙∣𝐗\)≥𝒥ε​\(𝐙\)for each​𝐙,𝐗∈ℝd1×d2\.Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\\qquad\\text\{for each \}\\mathbf\{Z\},\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\.\(96\)

It is easy to see that[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)implies[eq\.14](https://arxiv.org/html/2608.23765#S3.E14)by choosing𝐙=𝐗\(k\)\\mathbf\{Z\}=\\mathbf\{X\}^\{\(k\)\}and𝐗=𝐗\(k−1\)\\mathbf\{X\}=\\mathbf\{X\}^\{\(k\-1\)\}\. For one\-sided weight operators as used in the algorithms of[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54)and[Mohan and Fazel \(2012\)](https://arxiv.org/html/2608.23765#bib.bib48),[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)can be shown, for example, using variational arguments via the definition of a suitable auxiliary functional, which is in line with the literature on IRLS methods for separable objectives such asℓ1\\ell\_\{1\}\-type norms\([Geman and Reynolds, 1992](https://arxiv.org/html/2608.23765#bib.bib5);[Charbonnier et al\., 1997](https://arxiv.org/html/2608.23765#bib.bib6);[Daubechies et al\., 2010](https://arxiv.org/html/2608.23765#bib.bib38);[Ochs et al\., 2015](https://arxiv.org/html/2608.23765#bib.bib57)\)\.

In the remainder of this section, we provide two different proofs for[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)in the case of one\-sided weights in[SectionB\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1)and[SectionB\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2), respectively, and a third proof in[SectionB\.3\.3](https://arxiv.org/html/2608.23765#A2.SS3.SSS3)via a specialization of the proof strategy of[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)\. Finally, we discuss why these proof strategies do not extend to the case of harmonic\-mean weights in[SectionB\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4)\.

#### B\.3\.1Proof of[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)Using Concavity Arguments

Apart from low\-rank matrix optimization problems, the vast majority of the theory of iteratively reweighted least squares algorithms is specialized to coordinatewise separable objectives such asℓ1\\ell\_\{1\}\-norms or other functions that are sums of coordinatewise vector functions\. For such cases, a variety of works have established frameworks for how to derive suitable quadratic auxiliary objectivesQε\(⋅∣𝐗\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{X\}\)that provably majorize a given \(smoothed\) surrogate function\. We refer to[Geman and Reynolds \(1992\)](https://arxiv.org/html/2608.23765#bib.bib5)and[Charbonnier et al\. \(1997\)](https://arxiv.org/html/2608.23765#bib.bib6)for classical works in the image processing literature \(in which this methodology is commonly referred to as*half\-quadratic minimization*\) and the work of[Ochs et al\. \(2015\)](https://arxiv.org/html/2608.23765#bib.bib57)for an exposition tailored to computer vision\.

It turns out that the proof strategies tailored to separable objectives can be adapted to show the majorization property[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for one\-sided weight operators with core matrices[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)or[eq\.5](https://arxiv.org/html/2608.23765#S3.E5)relatively straightforwardly\. The key idea is to use the concavity of the term𝒥ε​\(𝐙\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)in a matrix variable after a suitable change of variables, and has been detailed by[Kümmerle \(2019, Section 2\.3\.2\)](https://arxiv.org/html/2608.23765#bib.bib97)for rank surrogate optimization\. We present the proof below\.

First Proof of[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for One\-Sided WeightsWe show the statement without loss of generality for the case of left\-sided weight operators\. We recall that the smoothed nuclear norm surrogate[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)is given by𝒥ε​\(𝐙\):=∑i=1djε​\(σi​\(𝐙\)\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\):=\\sum\_\{i=1\}^\{d\}j\_\{\\varepsilon\}\(\\sigma\_\{i\}\(\\mathbf\{Z\}\)\)with real\-valued penalization functionsjε:ℝ→ℝj\_\{\\varepsilon\}:\\R\\to\\Rsuch that

jε​\(σ\):=\{\|σ\|,if​\|σ\|\>ε,σ22​ε\+ε2,if​\|σ\|≤ε\.j\_\{\\varepsilon\}\(\\sigma\):=\\begin\{cases\}\|\\sigma\|,&\\text\{ if \}\|\\sigma\|\>\\varepsilon,\\\\ \\frac\{\\sigma^\{2\}\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\},&\\text\{ if \}\|\\sigma\|\\leq\\varepsilon\.\\end\{cases\}Accordingly, we define the functiongε:ℝ≥0→ℝg\_\{\\varepsilon\}:\\R\_\{\\geq 0\}\\to\\Rby

gε​\(λ\):=jε​\(λ\)=\{λ,if​λ\>ε2,λ2​ε\+ε2,if​0≤λ≤ε2\.\\displaystyle g\_\{\\varepsilon\}\(\\lambda\):=j\_\{\\varepsilon\}\(\\sqrt\{\\lambda\}\)=\\begin\{cases\}\\sqrt\{\\lambda\},&\\text\{ if \}\\lambda\>\\varepsilon^\{2\},\\\\ \\frac\{\\lambda\}\{2\\varepsilon\}\+\\frac\{\\varepsilon\}\{2\},&\\text\{ if \}0\\leq\\lambda\\leq\\varepsilon^\{2\}\.\\end\{cases\}For a PSD matrix𝐌∈ℝd1×d1\\mathbf\{M\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}, letGε​\(𝐌\):=tr⁡\[gε​\(𝐌\)\]G\_\{\\varepsilon\}\(\\mathbf\{M\}\):=\\mathrm\{tr\}\[g\_\{\\varepsilon\}\(\\mathbf\{M\}\)\], wheregε​\(𝐌\)g\_\{\\varepsilon\}\(\\mathbf\{M\}\)is defined via the standard functional calculus for symmetric matrices, i\.e\., if𝐌=𝐔​diag⁡\(λ1,…,λd1\)​𝐔⊤\\mathbf\{M\}=\\mathbf\{U\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\mathbf\{U\}^\{\\top\}is the eigendecomposition of𝐌\\mathbf\{M\}, thengε​\(𝐌\)=𝐔​diag⁡\(gε​\(λ1\),…,gε​\(λd1\)\)​𝐔⊤g\_\{\\varepsilon\}\(\\mathbf\{M\}\)=\\mathbf\{U\}\\diag\(g\_\{\\varepsilon\}\(\\lambda\_\{1\}\),\\ldots,g\_\{\\varepsilon\}\(\\lambda\_\{d\_\{1\}\}\)\)\\mathbf\{U\}^\{\\top\}\. Next, we notice that for any𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\mathbb\{R\}^\{d\_\{1\}\\times d\_\{2\}\}we have

𝒥ε​\(𝐙\)=∑i=1djε​\(σi​\(𝐙\)\)=∑i=1dgε​\(σi2​\(𝐙\)\)=Gε​\(𝐙𝐙⊤\)−max⁡\(0,d1−d2\)​ε2\.\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)=\\sum\_\{i=1\}^\{d\}j\_\{\\varepsilon\}\(\\sigma\_\{i\}\(\\mathbf\{Z\}\)\)=\\sum\_\{i=1\}^\{d\}g\_\{\\varepsilon\}\(\\sigma\_\{i\}^\{2\}\(\\mathbf\{Z\}\)\)=G\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}\)\-\\max\(0,d\_\{1\}\-d\_\{2\}\)\\frac\{\\varepsilon\}\{2\}\.\(97\)Sincegε′​\(λ\)=12​max⁡\(λ,ε\)g\_\{\\varepsilon\}^\{\\prime\}\(\\lambda\)=\\frac\{1\}\{2\\max\(\\sqrt\{\\lambda\},\\varepsilon\)\}is decreasing on the interval\(0,∞\)\(0,\\infty\), the functiongεg\_\{\\varepsilon\}is concave on this interval, and so is the induced spectral trace functionGεG\_\{\\varepsilon\}\(see, e\.g\.,[Carlen, 2010](https://arxiv.org/html/2608.23765#bib.bib96), Theorem 2\.10\)\. We also have that∇Gε​\(𝐗𝐗⊤\)=𝐔𝐗​diag⁡\(gε′​\(σi2​\(𝐗\)\)\)i=1d1​𝐔𝐗⊤\\nabla G\_\{\\varepsilon\}\(\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\)=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(g\_\{\\varepsilon\}^\{\\prime\}\(\\sigma^\{2\}\_\{i\}\(\\mathbf\{X\}\)\)\)\_\{i=1\}^\{d\_\{1\}\}\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}by[Lewis and Sendov \(2005, Proposition 6\.2\)](https://arxiv.org/html/2608.23765#bib.bib84), where we recall that

𝐗𝐗⊤=𝐔𝐗​diag⁡\(σ12​\(𝐗\),…,σd12​\(𝐗\)\)​𝐔𝐗⊤\\displaystyle\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\sigma\_\{1\}^\{2\}\(\\mathbf\{X\}\),\\ldots,\\sigma\_\{d\_\{1\}\}^\{2\}\(\\mathbf\{X\}\)\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}is an eigendecomposition of𝐗𝐗⊤\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}with square matrix𝐔𝐗∈ℝd1×d1\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}and the convention thatσi​\(𝐗\)=0\\sigma\_\{i\}\(\\mathbf\{X\}\)=0fori\>d=min⁡\(d1,d2\)i\>d=\\min\(d\_\{1\},d\_\{2\}\)\. Thus, we obtain that

∇Gε​\(𝐗𝐗⊤\)\\displaystyle\\nabla G\_\{\\varepsilon\}\(\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\)=12​𝐔𝐗​diag⁡\(1max⁡\(σi​\(𝐗\),ε\)\)i=1d1​𝐔𝐗⊤\\displaystyle=\\frac\{1\}\{2\}\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\\left\(\\frac\{1\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\}\\right\)\_\{i=1\}^\{d\_\{1\}\}\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}=12​\(𝐔𝐗​diag⁡\(max⁡\(σi​\(𝐗\),ε\)\)i=1d1​𝐔𝐗⊤\)−1=12​𝐋𝐔−1,\\displaystyle=\\frac\{1\}\{2\}\\left\(\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\\left\(\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\\right\)\_\{i=1\}^\{d\_\{1\}\}\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\right\)^\{\-1\}=\\frac\{1\}\{2\}\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\},where𝐋𝐔=𝐔𝐗​diag​\(max⁡\(σ1​\(𝐗\),ε\),…,max⁡\(σd1​\(𝐗\),ε\)\)​𝐔𝐗⊤∈ℝd1×d1\\mathbf\{L\}\_\{\\mathbf\{U\}\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\max\(\\sigma\_\{1\}\(\\mathbf\{X\}\),\\varepsilon\),\\ldots,\\max\(\\sigma\_\{d\_\{1\}\}\(\\mathbf\{X\}\),\\varepsilon\)\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}, cf\. also[Section3\.1](https://arxiv.org/html/2608.23765#S3.SS1)\. Due to the concavity of the entrywise functionsgεg\_\{\\varepsilon\}and\([Lewis and Sendov, 2005](https://arxiv.org/html/2608.23765#bib.bib84), Proposition 6\.1\), we know thatGε:ℝd1×d1→ℝG\_\{\\varepsilon\}:\\R^\{d\_\{1\}\\times d\_\{1\}\}\\to\\Ris also a concave function\. Consequently, we can upper boundGε​\(𝐙𝐙⊤\)G\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}\)by its linearization in𝐗𝐗⊤\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}such that

𝒥ε​\(𝐙\)\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)=Gε​\(𝐙𝐙⊤\)−max⁡\(0,d1−d2\)​ε2\\displaystyle=G\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}\)\-\\max\(0,d\_\{1\}\-d\_\{2\}\)\\frac\{\\varepsilon\}\{2\}≤Gε​\(𝐗𝐗⊤\)\+⟨∇Gε​\(𝐗𝐗⊤\),𝐙𝐙⊤−𝐗𝐗⊤⟩−max⁡\(0,d1−d2\)​ε2\\displaystyle\\leq G\_\{\\varepsilon\}\(\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\)\+\\langle\\nabla G\_\{\\varepsilon\}\(\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\),\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}\-\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\\rangle\-\\max\(0,d\_\{1\}\-d\_\{2\}\)\\frac\{\\varepsilon\}\{2\}=Gε​\(𝐗𝐗⊤\)\+12​⟨𝐋𝐔−1​𝐙,𝐙⟩F−12​⟨𝐋𝐔−1​𝐗,𝐗⟩F−max⁡\(0,d1−d2\)​ε2\\displaystyle=G\_\{\\varepsilon\}\(\\mathbf\{X\}\\mathbf\{X\}^\{\\top\}\)\+\\frac\{1\}\{2\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\},\\mathbf\{Z\}\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{X\},\\mathbf\{X\}\\rangle\_\{F\}\-\\max\(0,d\_\{1\}\-d\_\{2\}\)\\frac\{\\varepsilon\}\{2\}=\(a\)​𝒥ε​\(𝐗\)\+12​⟨W𝐗,ε​\(𝐙\),𝐙⟩F−12​⟨W𝐗,ε​\(𝐗\),𝐗⟩F\\displaystyle\\overset\{\(a\)\}\{=\}\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}=\(b\)​Qε​\(𝐙∣𝐗\),\\displaystyle\\overset\{\(b\)\}\{=\}Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\),where we used[eq\.97](https://arxiv.org/html/2608.23765#A2.E97)for𝐙=𝐗\\mathbf\{Z\}=\\mathbf\{X\}in equality \(a\) as well as that𝐖𝐗,ε​\(𝐙\)=𝐋𝐔−1​𝐙\\mathbf\{W\}\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}for all𝐙∈ℝd1×d2\\mathbf\{Z\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}\. In equality \(b\), we used the consequence[eq\.11](https://arxiv.org/html/2608.23765#S3.E11)of the gradient condition[eq\.10](https://arxiv.org/html/2608.23765#S3.E10)\. This concludes the proof\.

Extending this proof strategy to weight operatorsW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)with harmonic\-mean core matrices[eq\.6](https://arxiv.org/html/2608.23765#S3.E6)has remained elusive to the authors; a simple change of variables does not suffice as the weight operator cannot be represented as simply a left\- \(or right\-\)matrix multiplication in this case\.

#### B\.3\.2Proof of[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)Using Variational Calculus

The one\-sided majorization property can also be derived from the auxiliary variational formulation underlying the IRLS\-M algorithm of[Fornasier et al\. \(2011\)](https://arxiv.org/html/2608.23765#bib.bib54)\. We spell out the argument for the left\-sided weights; the right\-sided case follows by applying the same argument to the transposed matrices\.

Second Proof of[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for One\-Sided WeightsLet𝐗∈ℝd1×d2\\mathbf\{X\}\\in\\R^\{d\_\{1\}\\times d\_\{2\}\}be fixed and assume that the left\-sided weight core matrix[eq\.4](https://arxiv.org/html/2608.23765#S3.E4)is used\. For a symmetric positive definite matrix𝐌∈ℝd1×d1\\mathbf\{M\}\\in\\R^\{d\_\{1\}\\times d\_\{1\}\}, define

ΦεL​\(𝐙,𝐌\):=12​⟨𝐌𝐙,𝐙⟩F\+12​tr⁡\(𝐌−1\),\\Phi\_\{\\varepsilon\}^\{L\}\(\\mathbf\{Z\},\\mathbf\{M\}\):=\\frac\{1\}\{2\}\\langle\\mathbf\{M\}\\mathbf\{Z\},\\mathbf\{Z\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\trace\(\\mathbf\{M\}^\{\-1\}\),\(98\)on the restricted domain for the second matrix variable𝐌\\mathbf\{M\}such that0≺𝐌⪯ε−1​Idd10\\prec\\mathbf\{M\}\\preceq\\varepsilon^\{\-1\}\\Id\_\{d\_\{1\}\}\. This is the left\-sided auxiliary functional used by[Fornasier et al\. \(2011, Section 5\)](https://arxiv.org/html/2608.23765#bib.bib54)\. If𝐙=𝐔𝐙​dg⁡\(𝝈⁡\(𝐙\)\)​𝐕𝐙⊤\\mathbf\{Z\}=\\mathbf\{U\}\_\{\\mathbf\{Z\}\}\\dg\(\\bm\{\\sigma\}\(\\mathbf\{Z\}\)\)\\mathbf\{V\}\_\{\\mathbf\{Z\}\}^\{\\top\}is a full singular value decomposition and we use the conventionσi​\(𝐙\)=0\\sigma\_\{i\}\(\\mathbf\{Z\}\)=0fori\>d=min⁡\(d1,d2\)i\>d=\\min\(d\_\{1\},d\_\{2\}\), then minimizing the preceding functional over𝐌\\mathbf\{M\}gives

𝐌𝐙,εL=𝐔𝐙​diag⁡\(1max⁡\(σi​\(𝐙\),ε\)\)i=1d1​𝐔𝐙⊤\.\\mathbf\{M\}\_\{\\mathbf\{Z\},\\varepsilon\}^\{L\}=\\mathbf\{U\}\_\{\\mathbf\{Z\}\}\\diag\\left\(\\frac\{1\}\{\\max\(\\sigma\_\{i\}\(\\mathbf\{Z\}\),\\varepsilon\)\}\\right\)\_\{i=1\}^\{d\_\{1\}\}\\mathbf\{U\}\_\{\\mathbf\{Z\}\}^\{\\top\}\.\(99\)To see this, rotate the auxiliary variable into the left singular\-vector basis and write𝐍:=𝐔𝐙⊤​𝐌𝐔𝐙\\mathbf\{N\}:=\\mathbf\{U\}\_\{\\mathbf\{Z\}\}^\{\\top\}\\mathbf\{M\}\\mathbf\{U\}\_\{\\mathbf\{Z\}\}\. Then0≺𝐍⪯ε−1​Idd10\\prec\\mathbf\{N\}\\preceq\\varepsilon^\{\-1\}\\Id\_\{d\_\{1\}\}and, withai:=σi2​\(𝐙\)a\_\{i\}:=\\sigma\_\{i\}^\{2\}\(\\mathbf\{Z\}\)fori=1,…,d1i=1,\\ldots,d\_\{1\},

ΦεL​\(𝐙,𝐌\)=12​∑i=1d1ai​𝐍i​i\+12​tr⁡\(𝐍−1\)\.\\Phi\_\{\\varepsilon\}^\{L\}\(\\mathbf\{Z\},\\mathbf\{M\}\)=\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\_\{1\}\}a\_\{i\}\\mathbf\{N\}\_\{ii\}\+\\frac\{1\}\{2\}\\trace\(\\mathbf\{N\}^\{\-1\}\)\.For every positive definite𝐍\\mathbf\{N\}and every coordinate vector𝐞i\\mathbf\{e\}\_\{i\}, the Cauchy–Schwarz inequality gives

1=\(\(𝐍−1/2𝐞i\)⊤\(𝐍1/2𝐞i\)\)2≤\(𝐍−1\)i​i𝐍i​i,1=\\left\(\(\\mathbf\{N\}^\{\-1/2\}\\mathbf\{e\}\_\{i\}\)^\{\\top\}\(\\mathbf\{N\}^\{1/2\}\\mathbf\{e\}\_\{i\}\)\\right\)^\{2\}\\leq\(\\mathbf\{N\}^\{\-1\}\)\_\{ii\}\\mathbf\{N\}\_\{ii\},and hence\(𝐍−1\)i​i≥1/𝐍i​i\(\\mathbf\{N\}^\{\-1\}\)\_\{ii\}\\geq 1/\\mathbf\{N\}\_\{ii\}\. Since0<𝐍i​i≤ε−10<\\mathbf\{N\}\_\{ii\}\\leq\\varepsilon^\{\-1\}, it follows that

ΦεL​\(𝐙,𝐌\)\\displaystyle\\Phi\_\{\\varepsilon\}^\{L\}\(\\mathbf\{Z\},\\mathbf\{M\}\)≥12​∑i=1d1\(ai​𝐍i​i\+1𝐍i​i\)\\displaystyle\\geq\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\_\{1\}\}\\left\(a\_\{i\}\\mathbf\{N\}\_\{ii\}\+\\frac\{1\}\{\\mathbf\{N\}\_\{ii\}\}\\right\)≥12​∑i=1d1min0<m≤ε−1⁡\(ai​m\+1m\)\.\\displaystyle\\geq\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\_\{1\}\}\\min\_\{0<m\\leq\\varepsilon^\{\-1\}\}\\left\(a\_\{i\}m\+\\frac\{1\}\{m\}\\right\)\.The scalar minimizer ismi⋆=1/max⁡\(σi​\(𝐙\),ε\)m\_\{i\}^\{\\star\}=1/\\max\(\\sigma\_\{i\}\(\\mathbf\{Z\}\),\\varepsilon\): ifσi​\(𝐙\)≥ε\\sigma\_\{i\}\(\\mathbf\{Z\}\)\\geq\\varepsilon, this is the unconstrained critical pointm=1/σi​\(𝐙\)m=1/\\sigma\_\{i\}\(\\mathbf\{Z\}\), whereas ifσi​\(𝐙\)<ε\\sigma\_\{i\}\(\\mathbf\{Z\}\)<\\varepsilonthe minimum over\(0,ε−1\]\(0,\\varepsilon^\{\-1\}\]is attained at the boundarym=ε−1m=\\varepsilon^\{\-1\}\. Choosing𝐍=diag⁡\(m1⋆,…,md1⋆\)\\mathbf\{N\}=\\diag\(m\_\{1\}^\{\\star\},\\ldots,m\_\{d\_\{1\}\}^\{\\star\}\)makes the preceding lower bounds equalities\. Transforming back from𝐍\\mathbf\{N\}to𝐌\\mathbf\{M\}proves[eq\.99](https://arxiv.org/html/2608.23765#A2.E99)\. At this minimizer, theii\-th scalar contribution isjε​\(σi​\(𝐙\)\)j\_\{\\varepsilon\}\(\\sigma\_\{i\}\(\\mathbf\{Z\}\)\)fori≤di\\leq dandε/2\\varepsilon/2fori\>di\>d\. Consequently,

min0≺𝐌⪯ε−1​Idd1⁡ΦεL​\(𝐙,𝐌\)=𝒥ε​\(𝐙\)\+d1−d2​ε\.\\min\_\{0\\prec\\mathbf\{M\}\\preceq\\varepsilon^\{\-1\}\\Id\_\{d\_\{1\}\}\}\\Phi\_\{\\varepsilon\}^\{L\}\(\\mathbf\{Z\},\\mathbf\{M\}\)=\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\+\\frac\{d\_\{1\}\-d\}\{2\}\\varepsilon\.\(100\)The additional constant appears only whend1\>d2d\_\{1\}\>d\_\{2\}, because the remainingd1−dd\_\{1\}\-dzero eigenvalue directions of𝐙𝐙⊤\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}each contributeε/2\\varepsilon/2\.

We now freeze the auxiliary variable at the minimizer associated with the base point𝐗\\mathbf\{X\}\. Writing𝐋𝐔:=𝐔𝐗​diag⁡\(max⁡\(σi​\(𝐗\),ε\)\)i=1d1​𝐔𝐗⊤\\mathbf\{L\}\_\{\\mathbf\{U\}\}:=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\\left\(\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\\right\)\_\{i=1\}^\{d\_\{1\}\}\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\}, the minimizer \([99](https://arxiv.org/html/2608.23765#A2.E99)\) for𝐙=𝐗\\mathbf\{Z\}=\\mathbf\{X\}is𝐌𝐗,εL=𝐋𝐔−1\\mathbf\{M\}\_\{\\mathbf\{X\},\\varepsilon\}^\{L\}=\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\. It turns out that this coincides exactly \(see also[Section3\.1](https://arxiv.org/html/2608.23765#S3.SS1)\) with the action of the left\-sided weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)[eqs\.4](https://arxiv.org/html/2608.23765#S3.E4)and[7](https://arxiv.org/html/2608.23765#S3.E7)such thatW𝐗,ε​\(𝐙\)=𝐋𝐔−1​𝐙=𝐌𝐗,εL​𝐙W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}=\\mathbf\{M\}\_\{\\mathbf\{X\},\\varepsilon\}^\{L\}\\mathbf\{Z\}\. Using \([100](https://arxiv.org/html/2608.23765#A2.E100)\) first at𝐙\\mathbf\{Z\}and then at𝐗\\mathbf\{X\}, we obtain

𝒥ε​\(𝐙\)\+d1−d2​ε\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\+\\frac\{d\_\{1\}\-d\}\{2\}\\varepsilon≤ΦεL​\(𝐙,𝐌𝐗,εL\)\\displaystyle\\leq\\Phi\_\{\\varepsilon\}^\{L\}\(\\mathbf\{Z\},\\mathbf\{M\}\_\{\\mathbf\{X\},\\varepsilon\}^\{L\}\)=12​⟨W𝐗,ε​\(𝐙\),𝐙⟩F\+12​tr⁡\(𝐋𝐔\)\\displaystyle=\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\+\\frac\{1\}\{2\}\\trace\(\\mathbf\{L\}\_\{\\mathbf\{U\}\}\)=12​⟨W𝐗,ε​\(𝐙\),𝐙⟩F\+𝒥ε​\(𝐗\)\+d1−d2​ε−12​⟨W𝐗,ε​\(𝐗\),𝐗⟩F\.\\displaystyle=\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\+\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\frac\{d\_\{1\}\-d\}\{2\}\\varepsilon\-\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}\.After cancelling the dimension\-dependent constant,[eq\.11](https://arxiv.org/html/2608.23765#S3.E11)yields

𝒥ε​\(𝐙\)\\displaystyle\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)≤𝒥ε​\(𝐗\)\+12​⟨W𝐗,ε​\(𝐙\),𝐙⟩F−12​⟨W𝐗,ε​\(𝐗\),𝐗⟩F=Qε​\(𝐙∣𝐗\),\\displaystyle\\leq\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{X\}\)\+\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\-\\frac\{1\}\{2\}\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}=Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\),which proves[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for left\-sided weights\. The right\-sided case follows analogously by replacing𝐙𝐙⊤\\mathbf\{Z\}\\mathbf\{Z\}^\{\\top\}by𝐙⊤​𝐙\\mathbf\{Z\}^\{\\top\}\\mathbf\{Z\}and𝐋𝐔−1​𝐙\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}by𝐙𝐋𝐕−1\\mathbf\{Z\}\\mathbf\{L\}\_\{\\mathbf\{V\}\}^\{\-1\}\.

#### B\.3\.3Proof of[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)Based on[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)

For completeness, we provide an application of the proof strategy of[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)to the quadratic model majorization result of[Proposition5](https://arxiv.org/html/2608.23765#Thmproposition5)for one\-sided weight operators, providing an alternative for the existing proofs presented in the preceding sections\. Compared to the harmonic\-mean arguments of[SectionA\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1), the argument is shortened significantly\.

Third Proof of[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for One\-Sided WeightsWe show the argument for left\-sided weights; the proof for right\-sided weights follows by transposition\. The computation of⟨W𝐗,ε​\(𝐗\),𝐗⟩F\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{X\}\),\\mathbf\{X\}\\rangle\_\{F\}and the resulting representation ofQε​\(𝐙∣𝐗\)Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)are identical to the beginning of the proof of[Theorem1](https://arxiv.org/html/2608.23765#Thmtheorem1)in[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1), because all admissible weight cores have diagonal entries\(𝐇𝝈,ε\)i​i=1/max⁡\(σi​\(𝐗\),ε\)\(\\mathbf\{H\}\_\{\\bm\{\\sigma\},\\varepsilon\}\)\_\{ii\}=1/\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)\. Thus, the only ingredient specific to the one\-sided case is the lower bound for⟨W𝐗,ε​\(𝐙\),𝐙⟩F\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}\. We recall from[Section3\.1](https://arxiv.org/html/2608.23765#S3.SS1)that the action of the left\-sided weight operatorW𝐗,ε​\(⋅\)W\_\{\\mathbf\{X\},\\varepsilon\}\(\\cdot\)is given by

W𝐗,ε​\(𝐙\)=𝐋𝐔−1​𝐙W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\)=\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\}\\mathbf\{Z\}with

𝐋𝐔=𝐔𝐗​diag⁡\(λ1,…,λd1\)​𝐔𝐗⊤,\\mathbf\{L\}\_\{\\mathbf\{U\}\}=\\mathbf\{U\}\_\{\\mathbf\{X\}\}\\diag\(\\lambda\_\{1\},\\ldots,\\lambda\_\{d\_\{1\}\}\)\\mathbf\{U\}\_\{\\mathbf\{X\}\}^\{\\top\},where𝐔𝐗\\mathbf\{U\}\_\{\\mathbf\{X\}\}is the matrix of left singular vectors of𝐗\\mathbf\{X\}andλi=max⁡\(σi​\(𝐗\),ε\)\\lambda\_\{i\}=\\max\(\\sigma\_\{i\}\(\\mathbf\{X\}\),\\varepsilon\)for1≤i≤d11\\leq i\\leq d\_\{1\}\. Let𝐙=∑k=1dσk​\(𝐙\)​𝐮k​𝐯k⊤\\mathbf\{Z\}=\\sum\_\{k=1\}^\{d\}\\sigma\_\{k\}\(\\mathbf\{Z\}\)\\mathbf\{u\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}be an SVD of𝐙\\mathbf\{Z\}\. Then

⟨W𝐗,ε​\(𝐙\),𝐙⟩F=∑k=1dσk2​\(𝐙\)​⟨𝐋𝐔−1,𝐮k​𝐮k⊤⟩F≥∑k=1dσk2​\(𝐙\)⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\\displaystyle\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}=\\sum\_\{k=1\}^\{d\}\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\\geq\\sum\_\{k=1\}^\{d\}\\frac\{\\sigma\_\{k\}^\{2\}\(\\mathbf\{Z\}\)\}\{\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\}where the inequality follows by the Cauchy\-Schwarz estimate1≤∥𝐋𝐔−1/2𝐮k∥2∥𝐋𝐔1/2𝐮k∥21\\leq\\\|\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{\-1/2\}\\mathbf\{u\}\_\{k\}\\\|\_\{2\}\\\|\\mathbf\{L\}\_\{\\mathbf\{U\}\}^\{1/2\}\\mathbf\{u\}\_\{k\}\\\|\_\{2\}taken to the power two\. This is precisely the one\-sided analogue of Lemma[2](https://arxiv.org/html/2608.23765#Thmlemma2)\. The remaining spectral and scalar estimates are the same as in the previous section, with the averaged quantity⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F/2\+⟨𝐋𝐕,𝐯k​𝐯k⊤⟩F/2\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2\+\\langle\\mathbf\{L\}\_\{\\mathbf\{V\}\},\\mathbf\{v\}\_\{k\}\\mathbf\{v\}\_\{k\}^\{\\top\}\\rangle\_\{F\}/2replaced by⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\. Indeed, Wielandt’s minimax principle gives

∑k=1dλk≥∑k=1d⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F,\\sum\_\{k=1\}^\{d\}\\lambda\_\{k\}\\geq\\sum\_\{k=1\}^\{d\}\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\},and the scalar case distinction from the harmonic proof applies withαk=⟨𝐋𝐔,𝐮k​𝐮k⊤⟩F\\alpha\_\{k\}=\\langle\\mathbf\{L\}\_\{\\mathbf\{U\}\},\\mathbf\{u\}\_\{k\}\\mathbf\{u\}\_\{k\}^\{\\top\}\\rangle\_\{F\}\. This establishes the desired majorizationQε​\(𝐙∣𝐗\)≥𝒥ε​\(𝐙\)Q\_\{\\varepsilon\}\(\\mathbf\{Z\}\\mid\\mathbf\{X\}\)\\geq\\mathcal\{J\}\_\{\\varepsilon\}\(\\mathbf\{Z\}\)\.

#### B\.3\.4Challenges for Harmonic\-Mean Majorization Proofs

It is natural to ask whether the variational proof from[SectionB\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2)can be adapted to harmonic\-mean weights\. A proof attempt in this direction was made by[Kümmerle and Sigl \(2018, Definition 13 and Lemma 14\)](https://arxiv.org/html/2608.23765#bib.bib36)for a smoothed Schatten\-ppsurrogate, which is a generalization of the smoothed nuclear norm𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)of[eq\.3](https://arxiv.org/html/2608.23765#S1.E3)to include nonconvex Schatten\-ppquasi\-norms corresponding to0<p≤10<p\\leq 1\(the Schatten\-11norm coincides with the nuclear norm\)\. We briefly outline this proof strategy and point out why it does not directly provide a complete proof of the harmonic\-mean majorization property\.

The idea\(by[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), pp\. 26–27\)is to introduce, for fixed𝐗\\mathbf\{X\}and smoothing parameterε\\varepsilon, an auxiliary matrix variable𝐌\\mathbf\{M\}and the weight operator matrix𝐖~​\(𝐌\)\\widetilde\{\\mathbf\{W\}\}\(\\mathbf\{M\}\)defined as

𝐖~​\(𝐌\)=2​\[Idd2⊗\(𝐌𝐌⊤\)1/2\]​\[\(𝐌𝐌⊤\)1/2⊕\(𝐌⊤​𝐌\)1/2\]−1​\[\(𝐌⊤​𝐌\)1/2⊗Idd1\],\\widetilde\{\\mathbf\{W\}\}\(\\mathbf\{M\}\)=2\\big\[\\Id\_\{d\_\{2\}\}\\otimes\(\\mathbf\{M\}\\mathbf\{M\}^\{\\top\}\)^\{1/2\}\\big\]\\big\[\(\\mathbf\{M\}\\mathbf\{M\}^\{\\top\}\)^\{1/2\}\\oplus\(\\mathbf\{M\}^\{\\top\}\\mathbf\{M\}\)^\{1/2\}\\big\]^\{\-1\}\\big\[\(\\mathbf\{M\}^\{\\top\}\\mathbf\{M\}\)^\{1/2\}\\otimes\\Id\_\{d\_\{1\}\}\\big\],where⊕\\oplusdenotes the Kronecker sum and⊗\\otimesthe Kronecker product\. Here,𝐖~​\(𝐌\)\\widetilde\{\\mathbf\{W\}\}\(\\mathbf\{M\}\)is of size\(d1​d2\)×\(d1​d2\)\(d\_\{1\}d\_\{2\}\)\\times\(d\_\{1\}d\_\{2\}\)\. In the square non\-singular case, this is rewritten as the harmonic mean2​\(𝐖~1−1\+𝐖~2−1\)−12\(\\widetilde\{\\mathbf\{W\}\}\_\{1\}^\{\-1\}\+\\widetilde\{\\mathbf\{W\}\}\_\{2\}^\{\-1\}\)^\{\-1\}, where𝐖~1:ℝd1​d2→ℝd1​d2\\widetilde\{\\mathbf\{W\}\}\_\{1\}:\\R^\{d\_\{1\}d\_\{2\}\}\\to\\R^\{d\_\{1\}d\_\{2\}\}is the matrix representation of the left\-sided matrix multiplication operatorIdd2⊗\(𝐌𝐌⊤\)1/2\\Id\_\{d\_\{2\}\}\\otimes\(\\mathbf\{M\}\\mathbf\{M\}^\{\\top\}\)^\{1/2\}and𝐖~2:ℝd1​d2→ℝd1​d2\\widetilde\{\\mathbf\{W\}\}\_\{2\}:\\R^\{d\_\{1\}d\_\{2\}\}\\to\\R^\{d\_\{1\}d\_\{2\}\}of the right\-sided matrix multiplication operator\(𝐌⊤​𝐌\)1/2⊗Idd1\(\\mathbf\{M\}^\{\\top\}\\mathbf\{M\}\)^\{1/2\}\\otimes\\Id\_\{d\_\{1\}\}, respectively\. With this notation, an analogue of[eq\.98](https://arxiv.org/html/2608.23765#A2.E98)can be defined as

Φε​\(𝐙,𝐌\):=12​⟨𝐙vec,𝐖~​\(𝐌\)​𝐙vec⟩\+ε22​∑i=1dσi​\(𝐌\)\+12​∑i=1dσi​\(𝐌\)−1,\\Phi\_\{\\varepsilon\}\(\\mathbf\{Z\},\\mathbf\{M\}\):=\\frac\{1\}\{2\}\\left\\langle\\mathbf\{Z\}\_\{\\operatorname\{vec\}\},\\widetilde\{\\mathbf\{W\}\}\(\\mathbf\{M\}\)\\mathbf\{Z\}\_\{\\operatorname\{vec\}\}\\right\\rangle\+\\frac\{\\varepsilon^\{2\}\}\{2\}\\sum\_\{i=1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{M\}\)\+\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\}\\sigma\_\{i\}\(\\mathbf\{M\}\)^\{\-1\},\(101\)which is essentially Definition 13 of[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)forp=1p=1\. The claimed minimizer ofΦε​\(𝐙,⋅\)\\Phi\_\{\\varepsilon\}\(\\mathbf\{Z\},\\cdot\)with respect to𝐌\\mathbf\{M\}\(stated by[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), Lemma 14\)is aligned with the singular vectors of𝐙\\mathbf\{Z\}, namely

𝐌opt=𝐔𝐙diag\(\(σi\(𝐙\)2\+ε2\)−1/2\)i=1d𝐕𝐙⊤\.\\mathbf\{M\}\_\{\\operatorname\{opt\}\}=\\mathbf\{U\}\_\{\\mathbf\{Z\}\}\\diag\\left\(\(\\sigma\_\{i\}\(\\mathbf\{Z\}\)^\{2\}\+\\varepsilon^\{2\}\)^\{\-1/2\}\\right\)\_\{i=1\}^\{d\}\\mathbf\{V\}\_\{\\mathbf\{Z\}\}^\{\\top\}\.\(102\)If this variational characterization were available, freezing the auxiliary variable at the minimizer𝐌opt\\mathbf\{M\}\_\{\\operatorname\{opt\}\}associated with the base point𝐙\\mathbf\{Z\}would provide an alternating\-minimization explanation for the harmonic\-mean weight update and prove a majorization statement akin to[eq\.96](https://arxiv.org/html/2608.23765#A2.E96)for a quadratic model functionQε\(⋅∣𝐙\)Q\_\{\\varepsilon\}\(\\cdot\\mid\\mathbf\{Z\}\)using a harmonic\-mean weight operator\(with the minor technical difference that[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), use a slightly different Schatten\-ppsmoothing than𝒥ε​\(⋅\)\\mathcal\{J\}\_\{\\varepsilon\}\(\\cdot\)from this paper\)\.

However, it turns out that[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)do not provide a complete proof of the optimality of[eq\.102](https://arxiv.org/html/2608.23765#A2.E102)with respect toΦε​\(𝐙,⋅\)\\Phi\_\{\\varepsilon\}\(\\mathbf\{Z\},\\cdot\)of[eq\.101](https://arxiv.org/html/2608.23765#A2.E101)\. The main issue is that the harmonic\-mean weight depends simultaneously on the left and right singular spaces through an inverse Kronecker\-sum, or equivalently through a Sylvester operator\. Consequently, the auxiliary minimization in𝐌\\mathbf\{M\}does not decouple into independent scalar minimizations after one change of basis, in contrast to the one\-sided functionalΦεL\\Phi\_\{\\varepsilon\}^\{L\}in[SectionB\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2)\.

Specifically, there are at least two substantive gaps in the critical\-point calculation of[Kümmerle and Sigl \(2018, Lemma 14\)](https://arxiv.org/html/2608.23765#bib.bib36): First, the differentiation of matrix square roots and inverse square roots is treated as if the scalar chain rule applied directly to\(𝐌⊤𝐌\)−1/2\(\\mathbf\{M\}^\{\\top\}\\mathbf\{M\}\)^\{\-1/2\}and\(𝐌𝐌⊤\)−1/2\(\\mathbf\{M\}\\mathbf\{M\}^\{\\top\}\)^\{\-1/2\}\(see[Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), p\. 42, eq\. \(58\)\)\. This is not valid for noncommuting matrix perturbations; the Fréchet derivative of a matrix power involves divided differences or an equivalent Sylvester\-type operator\([Daletskii and Krein, 1965](https://arxiv.org/html/2608.23765#bib.bib4)\)and\([Noferini, 2017](https://arxiv.org/html/2608.23765#bib.bib94), Theorem 3\.8\)\. Thus, the stationarity equation\([Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), Eq\. \(60\)\)used later in the proof of\([Kümmerle and Sigl, 2018](https://arxiv.org/html/2608.23765#bib.bib36), Section B\.2\)is not justified by the cited matrix calculus rules\.

The second substantive gap is that, after deriving the stationarity equation, the proof of[Kümmerle and Sigl \(2018, Section B\.2\)](https://arxiv.org/html/2608.23765#bib.bib36)uses informal linear\-algebra implications to conclude that the singular vectors of the auxiliary variable𝐌\\mathbf\{M\}must align with those of𝐙\\mathbf\{Z\}\. For example, the commutation relation𝚺4​𝐁𝐁⊤=𝐁𝐁⊤​𝚺4\\mathbf\{\\Sigma\}^\{4\}\\mathbf\{B\}\\mathbf\{B\}^\{\\top\}=\\mathbf\{B\}\\mathbf\{B\}^\{\\top\}\\mathbf\{\\Sigma\}^\{4\}obtained by[Kümmerle and Sigl \(2018, Eq\. \(80\)\)](https://arxiv.org/html/2608.23765#bib.bib36)does not force𝐁𝐁⊤\\mathbf\{B\}\\mathbf\{B\}^\{\\top\}to be diagonal unless additional nondegeneracy assumptions on the diagonal entries of𝚺\\mathbf\{\\Sigma\}are imposed\. Even diagonality of𝐁𝐁⊤\\mathbf\{B\}\\mathbf\{B\}^\{\\top\}would not by itself force𝐁\\mathbf\{B\}to be diagonal, contrary to the third paragraph after eq\. \(80\) of[Kümmerle and Sigl \(2018\)](https://arxiv.org/html/2608.23765#bib.bib36)\. Hence, the key alignment claim needed to identify the supposed global minimizer𝐌opt\\mathbf\{M\}\_\{\\operatorname\{opt\}\}is not established\.

Thus, while the variational strategy correctly identifies the formal harmonic\-mean weight update, it leaves open the essential inequality needed for majorization\. The work done in[SectionA\.1](https://arxiv.org/html/2608.23765#A1.SS1)provides a direct lower bound for the weighted inner product⟨W𝐗,ε​\(𝐙\),𝐙⟩F\\langle W\_\{\\mathbf\{X\},\\varepsilon\}\(\\mathbf\{Z\}\),\\mathbf\{Z\}\\rangle\_\{F\}, instead of relying exclusively on such a variational envelope\.

We believe that the statement of[Kümmerle and Sigl \(2018, Lemma 14\)](https://arxiv.org/html/2608.23765#bib.bib36)is correct forp=1p=1, but likely incorrect for0<p<10<p<1\. As the focus of this paper is the nuclear norm objective corresponding top=1p=1, we leave a clarification of majorization properties of IRLS\-type quadratic models for nonconvex rank surrogates to future work\.

## References

- Ahmedet al\.\(2014\)A\. Ahmed, B\. Recht, and J\. RombergBlind deconvolution using convex programming\.IEEE Trans\. Inf\. Theory60\(3\),pp\. 1711–1732\.Cited by:[§5\.1](https://arxiv.org/html/2608.23765#S5.SS1.p1.1)\.
- Amselet al\.\(2026\)N\. Amsel, D\. Persson, C\. Musco, and R\. M\. GowerThe Polar Express: optimal matrix sign methods and their application to the Muon algorithm\.InInternational Conference on Learning Representations \(ICLR\),pp\. 138323–138360\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Anderssonet al\.\(2016\)F\. Andersson, M\. Carlsson, and K\. PerfektOperator\-Lipschitz estimates for the singular value functional calculus\.Proceedings of the American Mathematical Society144\(5\),pp\. 1867–1875\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§B\.2](https://arxiv.org/html/2608.23765#A2.SS2.p2.1)\.
- Andoet al\.\(1987\)T\. Ando, R\. A\. Horn, and C\. R\. JohnsonThe singular values of a Hadamard product: a basic inequality\.Linear and Multilinear Algebra21\(4\),pp\. 345–365\.Cited by:[§A\.3\.2](https://arxiv.org/html/2608.23765#A1.SS3.SSS2.p3.1),[§A\.3\.2](https://arxiv.org/html/2608.23765#A1.SS3.SSS2.p4.1)\.
- Aravkinet al\.\(2019\)A\. Aravkin, J\. V\. Burke, and D\. HeIRLS for Sparse Recovery Revisited: Examples of Failure and a Remedy\.arXiv preprint arXiv:1910\.07095\.Cited by:[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1)\.
- Bach \(2019\)F\. BachThe “η\\eta\-trick” or the effectiveness of reweighted least\-squares\.Note:Blog article, available at[https://francisbach\.com/the\-η\-trick\-or\-the\-effectiveness\-of\-reweighted\-least\-squares/](https://francisbach.com/the-%CE%B7-trick-or-the-effectiveness-of-reweighted-least-squares/)Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- Beck and Sabach \(2015\)A\. Beck and S\. SabachWeiszfeld’s method: old and new results\.Journal of Optimization Theory and Applications164\(1\),pp\. 1–40\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- Bernstein and Newhouse \(2025\)J\. Bernstein and L\. NewhouseModular duality in deep learning\.InProceedings of the 42nd International Conference on Machine Learning \(ICML\),Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Bhatia \(1997\)R\. BhatiaMatrix analysis\.Graduate Texts in Mathematics, Vol\.169,Springer\-Verlag,New York\.Cited by:[§A\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1.Px1.p11.1),[§A\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1.Px1.p2.1),[§A\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1.Px1.p3.1),[§A\.1\.1](https://arxiv.org/html/2608.23765#A1.SS1.SSS1.Px2.p5.1),[§A\.1](https://arxiv.org/html/2608.23765#A1.SS1.p5.5)\.
- Bhatia \(2007\)R\. BhatiaPositive definite matrices\.Princeton, NJ: Princeton University Press\(English\)\.Cited by:[§A\.3\.2](https://arxiv.org/html/2608.23765#A1.SS3.SSS2.p6.1)\.
- Bullen \(2003\)P\.S\. BullenHandbook of means and their inequalities\.Mathematics and Its Applications, Vol\.560,Springer Science & Business Media\.Cited by:[§A\.2\.1](https://arxiv.org/html/2608.23765#A1.SS2.SSS1.p2.4),[§A\.2\.3](https://arxiv.org/html/2608.23765#A1.SS2.SSS3.p2.1),[§A\.4\.1](https://arxiv.org/html/2608.23765#A1.SS4.SSS1.p5.1),[§3\.4](https://arxiv.org/html/2608.23765#S3.SS4.p4.1),[Definition 3](https://arxiv.org/html/2608.23765#Thmdefinition3)\.
- Burer and Monteiro \(2005\)S\. Burer and R\. D\. C\. MonteiroLocal minima and convergence in low\-rank semidefinite programming\.Math\. Program\.103\(3\(A\)\),pp\. 427–444\(English\)\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Burke and Ferris \(1993\)J\. V\. Burke and M\. C\. FerrisWeak sharp minima in mathematical programming\.SIAM Journal on Control and Optimization31\(5\),pp\. 1340–1359\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2)\.
- Caiet al\.\(2010\)J\. Cai, E\. J\. Candès, and Z\. ShenA Singular Value Thresholding Algorithm For Matrix Completion\.SIAM J\. Optim\.20\(4\),pp\. 1956–1982\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- Caiet al\.\(2025\)J\. Cai, T\. Wu, and R\. XiaFast non\-convex matrix sensing with optimal sample complexity\.InProceedings of the Forty\-first Conference on Uncertainty in Artificial Intelligence,Vol\.286,pp\. 497–507\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Cai and Zhang \(2015\)T\. T\. Cai and A\. ZhangROP: matrix recovery via rank\-one projections\.Ann\. Stat\.43\(1\),pp\. 102–138\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.2),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p2.1),[footnote 1](https://arxiv.org/html/2608.23765#footnote1)\.
- Cai and Li \(2017\)Y\. Cai and S\. LiConvergence and stability of iteratively reweighted least squares for low\-rank matrix recovery\.Inverse Problems and Imaging11\(4\),pp\. 643–661\.Cited by:[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.7.1.1.1),[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§1](https://arxiv.org/html/2608.23765#S1.p5.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p1.1)\.
- Candèset al\.\(2013\)E\. J\. Candès, Y\. Eldar, T\. Strohmer, and V\. VoroninskiPhase Retrieval via Matrix Completion\.SIAM J\. Imag\. Sci\.6\(1\),pp\. 199–225\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Candès and Recht \(2009\)E\. J\. Candès and B\. RechtExact matrix completion via convex optimization\.Found\. Comput\. Math\.9\(6\),pp\. 717–772\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Carlen \(2010\)E\. CarlenTrace inequalities and quantum entropy: an introductory course\.Entropy and the quantum529,pp\. 73–140\.Cited by:[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p3.4)\.
- Carlsonet al\.\(2015\)D\. Carlson, V\. Cevher, and L\. CarinStochastic spectral descent for restricted Boltzmann machines\.InProceedings of the 18th International Conference on Artificial Intelligence and Statistics \(AISTATS\),Vol\.38,pp\. 111–119\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Charbonnieret al\.\(1997\)P\. Charbonnier, L\. Blanc\-Feraud, G\. Aubert, and M\. BarlaudDeterministic Edge\-Preserving Regularization in Computed Imaging\.IEEE Trans\. Image Process\.6\(2\),pp\. 298–311\.Cited by:[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p1.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1)\.
- Chartrand and Yin \(2008\)R\. Chartrand and W\. T\. YinIteratively reweighted algorithms for compressive sensing\.InIEEE International Conference on Acoustics, Speech and Signal Processing \(ICASSP\),pp\. 3869–3872\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1)\.
- Chenet al\.\(2025\)L\. Chen, J\. Li, and Q\. LiuMuon optimizes under spectral norm constraints\.arXiv preprint arXiv:2506\.15054\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Chen \(2012\)X\. ChenSmoothing methods for nonsmooth, nonconvex minimization\.Math\. Program\.134\(1\),pp\. 71–99\.External Links:ISBN 1436\-4646Cited by:[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p2.1)\.
- Chen and Chi \(2018\)Y\. Chen and Y\. ChiHarnessing structures in big data via guaranteed low\-rank matrix estimation: recent theory and fast algorithms via convex and nonconvex optimization\.IEEE Signal Processing Magazine35\(4\),pp\. 14–31\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p5.1)\.
- Chenet al\.\(2019\)Y\. Chen, Y\. Chi, J\. Fan, and C\. MaGradient descent with random initialization: fast global convergence for nonconvex phase retrieval\.Math\. Program\.176\(1\),pp\. 5–37\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Chenet al\.\(2021\)Y\. Chen, Y\. Chi, J\. Fan, and C\. MaSpectral methods for data science: a statistical perspective\.Found\. Trends Mach\. Learn\.14\(5\),pp\. 1–246\(English\)\.Cited by:[§A\.4\.1](https://arxiv.org/html/2608.23765#A1.SS4.SSS1.p9.10)\.
- Chiet al\.\(2019\)Y\. Chi, Y\. M\. Lu, and Y\. ChenNonconvex optimization meets low\-rank matrix factorization: an overview\.IEEE Transactions on Signal Processing67\(20\),pp\. 5239–5269\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Cohenet al\.\(2009\)A\. Cohen, W\. Dahmen, and R\. DeVoreCompressed sensing and bestkk\-term approximation\.Journal of the American Mathematical Society22\(1\),pp\. 211–231\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2)\.
- Colbrook \(2022\)M\. J\. ColbrookWARPd: a linearly convergent first\-order primal\-dual algorithm for inverse problems with approximate sharpness conditions\.SIAM Journal on Imaging Sciences15\(3\),pp\. 1539–1575\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- Daletskii and Krein \(1965\)Yu\. L\. Daletskii and S\. G\. KreinIntegration and differentiation of functions of hermitian operators and applications to the theory of perturbations\.AMS Translations \(2\)47\(1\-30\),pp\. 10–1090\.Cited by:[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p4.1)\.
- Daubechieset al\.\(2010\)I\. Daubechies, R\. DeVore, M\. Fornasier, and C\. S\. GüntürkIteratively reweighted least squares minimization for sparse recovery\.Comm\. Pure Appl\. Math\.63,pp\. 1–38\.Cited by:[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1),[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.3.1.1.1),[§1](https://arxiv.org/html/2608.23765#S1.p3.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p2.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1)\.
- Davenport and Romberg \(2016\)M\. A\. Davenport and J\. RombergAn overview of low\-rank matrix recovery from incomplete observations\.IEEE Journal of Selected Topics in Signal Processing10\(4\),pp\. 608–622\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- DeepSeek\-AI \(2026\)DeepSeek\-AIDeepSeek\-V4: towards highly efficient million\-token context intelligence\.arXiv preprint arXiv:2606\.19348\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Diamond and Boyd \(2016\)S\. Diamond and S\. BoydCVXPY: A Python\-embedded modeling language for convex optimization\.J\. Mach\. Learn\. Res\.17\(83\),pp\. 1–5\.Cited by:[§5\.5](https://arxiv.org/html/2608.23765#S5.SS5.p3.1)\.
- Dinget al\.\(2018\)C\. Ding, D\. Sun, J\. Sun, and K\. TohSpectral operators of matrices\.Math\. Program\.168\(1\),pp\. 509–531\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§B\.2](https://arxiv.org/html/2608.23765#A2.SS2.p2.1)\.
- Ding and Wang \(2026\)L\. Ding and A\. L\. WangSharpness and conditioning of nonsmooth convex formulations in statistical signal recovery\.SIAM J\. Optim\.36\(3\),pp\. 1269–1296\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- Fazel \(2002\)M\. FazelMatrix rank minimization with applications\.Ph\.D\. Thesis, Electrical Engineering Department, Stanford University,Electrical Engineering Department\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Fornasieret al\.\(2011\)M\. Fornasier, H\. Rauhut, and R\. WardLow\-rank matrix recovery via iteratively reweighted least squares minimization\.SIAM J\. Optim\.21\(4\),pp\. 1614–1640\.Cited by:[§B\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2.p1.1),[§B\.3\.2](https://arxiv.org/html/2608.23765#A2.SS3.SSS2.p2.2),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p1.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1),[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.6.1.1.1),[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§1](https://arxiv.org/html/2608.23765#S1.p5.1),[§1](https://arxiv.org/html/2608.23765#S1.p7.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p8.1),[§3\.2](https://arxiv.org/html/2608.23765#S3.SS2.p1.3),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1),[§3\.4](https://arxiv.org/html/2608.23765#S3.SS4.p1.2),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p1.1),[§6](https://arxiv.org/html/2608.23765#S6.p1.1),[Lemma 10](https://arxiv.org/html/2608.23765#Thmlemma10)\.
- Foucart and Rauhut \(2013\)S\. Foucart and H\. RauhutA mathematical introduction to compressive sensing\.Applied and Numerical Harmonic Analysis,Springer New York\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p2.1),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2)\.
- Fuchset al\.\(2022\)T\. Fuchs, D\. Gross, P\. Jung, F\. Krahmer, R\. Kueng, and D\. StögerProof methods for robust low\-rank matrix recovery\.InCompressed Sensing in Information Processing,pp\. 37–75\(English\)\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- Geman and Reynolds \(1992\)D\. Geman and G\. ReynoldsConstrained Restoration and the Recovery of Discontinuities\.IEEE Trans\. Pattern Anal\. Mach\. Intell\.14\(03\),pp\. 367–383\.Cited by:[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p1.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1)\.
- Ghoshet al\.\(2024\)I\. Ghosh, A\. Tasissa, and C\. KümmerleSample\-Efficient Geometry Reconstruction from Euclidean Distances Using Non\-Convex Optimization\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.37,pp\. 77226–77268\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p2.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p8.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p2.1),[§5\.2](https://arxiv.org/html/2608.23765#S5.SS2.p1.2)\.
- Goffin \(1977\)J\. GoffinOn convergence rates of subgradient optimization methods\.Math\. Program\.13\(1\),pp\. 329–347\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2)\.
- Gorodnitsky and Rao \(1997\)I\. F\. Gorodnitsky and B\. D\. RaoSparse signal reconstruction from limited data using focuss: a re\-weighted minimum norm algorithm\.IEEE Trans\. Signal Process\.45\(3\),pp\. 600–616\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1)\.
- Grosset al\.\(2010\)D\. Gross, Y\. Liu, S\. T\. Flammia, S\. Becker, and J\. EisertQuantum state tomography via compressed sensing\.Physical Review Letters105\(15\),pp\. 150401\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Gross \(2011\)D\. GrossRecovering low\-rank matrices from few coefficients in any basis\.IEEE Trans\. Inf\. Theory57\(3\),pp\. 1548–1566\(English\)\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1)\.
- Hawkins and Ben\-Israel \(1973\)J\.B\. Hawkins and A\. Ben\-IsraelOn generalized matrix functions\.Linear and Multilinear Algebra1\(2\),pp\. 163–171\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2)\.
- Holland and Welsch \(1977\)P\. W\. Holland and R\. E\. WelschRobust regression using iteratively reweighted least\-squares\.Communications in Statistics \- Theory and Methods6,pp\. 813–827\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- Horn \(1990\)R\. A\. HornThe Hadamard product\.InProc\. Symp\. Appl\. Math\.,Vol\.40,pp\. 87–169\.Cited by:[§A\.3\.2](https://arxiv.org/html/2608.23765#A1.SS3.SSS2.p4.1)\.
- Huber \(1964\)P\. J\. HuberRobust estimation of a location parameter\.The Annals of Mathematical Statistics35\(1\),pp\. 73–101\.Cited by:[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p1.1)\.
- Jainet al\.\(2010\)P\. Jain, R\. Meka, and I\. DhillonGuaranteed rank minimization via singular value projection\.Advances in Neural Information Processing Systems \(NIPS\)23\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Jordanet al\.\(2024\)K\. Jordan, Y\. Jin, V\. Boza, J\. You, F\. Cesista, L\. Newhouse, and J\. BernsteinMuon: an optimizer for hidden layers in neural networks\.Note:[https://kellerjordan\.github\.io/posts/muon/](https://kellerjordan.github.io/posts/muon/)Blog postCited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Junget al\.\(2018\)P\. Jung, F\. Krahmer, and D\. StögerBlind demixing and deconvolution at near\-optimal rate\.IEEE Trans\. Inf\. Theory64\(2\),pp\. 704–727\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Kabanavaet al\.\(2016\)M\. Kabanava, R\. Kueng, H\. Rauhut, and U\. TerstiegeStable low\-rank matrix recovery via null space properties\.Inf\. Inference5\(4\),pp\. 405–441\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p2.1)\.
- Kimi Team \(2025\)Kimi TeamKimi K2: open agentic intelligence\.arXiv preprint arXiv:2507\.20534\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Korenet al\.\(2009\)Y\. Koren, R\. Bell, and C\. VolinskyMatrix Factorization Techniques for Recommender Systems\.Computer42\(8\),pp\. 30–37\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Kraemer \(2025\)S\. KraemerAffine rank minimization via asymptotic log\-det iteratively reweighted least squares\.J\. Mach\. Learn\. Res\.26\(92\),pp\. 1–44\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p2.1),[§5\.3](https://arxiv.org/html/2608.23765#S5.SS3.p1.1),[§6](https://arxiv.org/html/2608.23765#S6.p4.1)\.
- Kuenget al\.\(2017\)R\. Kueng, H\. Rauhut, and U\. TerstiegeLow rank matrix recovery from rank one measurements\.Appl\. Comput\. Harmon\. Anal\.42\(1\),pp\. 88–116\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p2.1)\.
- Kümmerleet al\.\(2021\)C\. Kümmerle, C\. Mayrink Verdun, and D\. StögerIteratively Reweighted Least Squares for Basis Pursuit with Global Linear Convergence Rate\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.34,pp\. 2873–2886\.Cited by:[§A\.3\.1](https://arxiv.org/html/2608.23765#A1.SS3.SSS1.p3.1),[§A\.3\.1](https://arxiv.org/html/2608.23765#A1.SS3.SSS1.p5.5),[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.4.1.1.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2),[Remark 2](https://arxiv.org/html/2608.23765#Thmremark2.p1.1.1)\.
- Kümmerle and Mayrink Verdun \(2021\)C\. Kümmerle and C\. Mayrink VerdunA Scalable Second Order Method for Ill\-Conditioned Matrix Completion from Few Samples\.InProceedings of the 38th International Conference on Machine Learning \(ICML\),Vol\.139,pp\. 5872–5883\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p2.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p8.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p2.1),[§5\.2](https://arxiv.org/html/2608.23765#S5.SS2.p1.2),[§6](https://arxiv.org/html/2608.23765#S6.p4.1),[Definition 2](https://arxiv.org/html/2608.23765#Thmdefinition2)\.
- Kümmerle and Sigl \(2018\)C\. Kümmerle and J\. SiglHarmonic mean iteratively reweighted least squares for low\-rank matrix recovery\.J\. Mach\. Learn\. Res\.19\(1\),pp\. 1815–1863\.Cited by:[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p1.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p2.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p2.3),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p2.4),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p3.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p4.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p5.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p7.1),[Appendix B](https://arxiv.org/html/2608.23765#A2.p1.1),[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.8.1.1.1),[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p2.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p8.1),[§3\.2](https://arxiv.org/html/2608.23765#S3.SS2.p1.3),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p2.1),[§3\.4](https://arxiv.org/html/2608.23765#S3.SS4.p1.2),[§3\.4](https://arxiv.org/html/2608.23765#S3.SS4.p9.1),[Definition 2](https://arxiv.org/html/2608.23765#Thmdefinition2)\.
- Kümmerle and Stöger \(2024\)C\. Kümmerle and D\. StögerLinear convergence of iteratively reweighted least squares for nuclear norm minimization\.In2024 IEEE 13th Sensor Array and Multichannel Signal Processing Workshop \(SAM\),pp\. 1–5\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p10.1)\.
- Kümmerle \(2019\)C\. KümmerleUnderstanding and enhancing data recovery algorithms: from noise\-blind sparse recovery to reweighted methods for low\-rank matrix optimization\.Ph\.D\. Thesis,Technische Universität München,Munich, Germany\.External Links:[Link](https://mediatum.ub.tum.de/doc/1521436/1521436.pdf)Cited by:[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p2.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p1.1)\.
- Laiet al\.\(2013\)M\. Lai, Y\. Xu, and W\. YinImproved iteratively reweighted least squares for unconstrained smoothedℓq\\ell\_\{q\}minimization\.SIAM J\. Numer\. Anal\.51,pp\. 927–957\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p1.1)\.
- Lange \(2016\)K\. LangeMM Optimization Algorithms\.Society for Industrial and Applied Mathematics,Philadelphia, PA\.External Links:https://epubs\.siam\.org/doi/pdf/10\.1137/1\.9781611974409Cited by:[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p2.1),[Remark 1](https://arxiv.org/html/2608.23765#Thmremark1.p1.1.1)\.
- Langeet al\.\(2000\)K\. Lange, D\. R\. Hunter, and I\. YangOptimization transfer using surrogate objective functions\.Journal of Computational and Graphical Statistics9\(1\),pp\. 1–20\.Cited by:[Remark 1](https://arxiv.org/html/2608.23765#Thmremark1.p1.1.1)\.
- Lermanet al\.\(2025\)G\. Lerman, K\. Li, T\. Maunu, and T\. ZhangGlobal convergence of iteratively reweighted least squares for robust subspace recovery\.arXiv preprint arXiv:2506\.20533\.Cited by:[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1)\.
- Lewis and Sendov \(2005\)A\. S\. Lewis and H\. S\. SendovNonsmooth Analysis of Singular Values\. Part I: Theory\.Set\-Valued Analysis13\(3\),pp\. 213–241\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p7.3),[§B\.1](https://arxiv.org/html/2608.23765#A2.SS1.p2.1),[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p3.4),[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p3.6)\.
- Liet al\.\(2019\)X\. Li, S\. Ling, T\. Strohmer, and K\. WeiRapid, robust, and reliable blind deconvolution via nonconvex optimization\.Applied and Computational Harmonic Analysis47\(3\),pp\. 893–934\.Cited by:[§5\.1](https://arxiv.org/html/2608.23765#S5.SS1.p1.1)\.
- Ling and Strohmer \(2017\)S\. Ling and T\. StrohmerBlind deconvolution meets blind demixing: algorithms and performance bounds\.IEEE Trans\. Inf\. Theory63\(7\),pp\. 4497–4520\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Liuet al\.\(2025\)J\. Liu, J\. Su, X\. Yao, Z\. Jiang, G\. Lai, Y\. Du, Y\. Qin, W\. Xu, E\. Lu, J\. Yan,et al\.Muon is scalable for LLM training\.arXiv preprint arXiv:2502\.16982\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Löwner \(1934\)K\. LöwnerÜber monotone Matrixfunktionen\.Mathematische Zeitschrift38\(1\),pp\. 177–216\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§B\.2](https://arxiv.org/html/2608.23765#A2.SS2.p2.1)\.
- Luoet al\.\(2023\)Y\. Luo, W\. Huang, X\. Li, and A\. ZhangRecursive importance sketching for rank constrained least squares: algorithms and high\-order convergence\.Operations Research\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Maet al\.\(2020\)C\. Ma, K\. Wang, Y\. Chi, and Y\. ChenImplicit regularization in nonconvex statistical estimation: gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution\.Found\. Comput\. Math\.20\(3\),pp\. 451–632\(English\)\.Cited by:[§5\.1](https://arxiv.org/html/2608.23765#S5.SS1.p1.1)\.
- Mohan and Fazel \(2012\)K\. Mohan and M\. FazelIterative reweighted algorithms for matrix rank minimization\.J\. Mach\. Learn\. Res\.13\(1\),pp\. 3441–3473\.Cited by:[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p1.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1),[Table 1](https://arxiv.org/html/2608.23765#S1.T1.5.6.1.1.1),[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§1](https://arxiv.org/html/2608.23765#S1.p5.1),[§1](https://arxiv.org/html/2608.23765#S1.p7.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p3.1),[§3\.4](https://arxiv.org/html/2608.23765#S3.SS4.p1.2),[§6](https://arxiv.org/html/2608.23765#S6.p1.1)\.
- Nesterov and Nemirovskii \(1994\)Y\. Nesterov and A\. NemirovskiiInterior\-point polynomial algorithms in convex programming\.SIAM Studies in Applied and Numerical Mathematics, Vol\.13,Society for Industrial and Applied Mathematics,Philadelphia\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.2)\.
- Nikolova and Ng \(2005\)M\. Nikolova and M\. K\. NgAnalysis of half\-quadratic minimization methods for signal and image recovery\.SIAM J\. Sci\. Comput\.27\(3\),pp\. 937–966\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- Noferini \(2017\)V\. NoferiniA Formula for the Fréchet Derivative of a Generalized Matrix Function\.SIAM J\. Matrix Anal\. Appl\.38\(2\),pp\. 434–457\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p3.1),[§B\.3\.4](https://arxiv.org/html/2608.23765#A2.SS3.SSS4.p4.1)\.
- Ochset al\.\(2015\)P\. Ochs, A\. Dosovitskiy, T\. Brox, and T\. PockOn iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision\.SIAM J\. Imaging Sci\.8\(1\),pp\. 331–372\.Cited by:[§B\.3\.1](https://arxiv.org/html/2608.23765#A2.SS3.SSS1.p1.1),[§B\.3](https://arxiv.org/html/2608.23765#A2.SS3.p2.1),[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- O’Donoghueet al\.\(2016\)B\. O’Donoghue, E\. Chu, N\. Parikh, and S\. BoydConic optimization via operator splitting and homogeneous self\-dual embedding\.Journal of Optimization Theory and Applications169\(3\),pp\. 1042–1068\.External Links:[Link](http://stanford.edu/~boyd/papers/scs.html)Cited by:[§5\.5](https://arxiv.org/html/2608.23765#S5.SS5.p3.1)\.
- Penget al\.\(2022\)L\. Peng, C\. Kümmerle, and R\. VidalGlobal Linear and Local Superlinear Convergence of IRLS for Non\-Smooth Robust Regression\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.35,pp\. 28972–28987\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p1.1)\.
- Penget al\.\(2023\)L\. Peng, C\. Kümmerle, and R\. VidalOn the Convergence of IRLS and Its Variants in Outlier\-Robust Estimation\.InIEEE Conference on Computer Vision and Pattern Recognition \(CVPR\),pp\. 17808–17818\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1),[§3\.3](https://arxiv.org/html/2608.23765#S3.SS3.p3.1)\.
- Pethicket al\.\(2025\)T\. Pethick, W\. Xie, K\. Antonakopoulos, Z\. Zhu, A\. Silveti\-Falls, and V\. CevherTraining deep learning models with norm\-constrained LMOs\.InProceedings of the 42nd International Conference on Machine Learning \(ICML\),Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p1.1)\.
- Radhakrishnanet al\.\(2024\)A\. Radhakrishnan, D\. Beaglehole, P\. Pandit, and M\. BelkinMechanism for feature learning in neural networks and backpropagation\-free machine learning models\.Science383\(6690\),pp\. 1461–1467\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p3.1),[§6](https://arxiv.org/html/2608.23765#S6.p4.1)\.
- Radhakrishnanet al\.\(2025\)A\. Radhakrishnan, M\. Belkin, and D\. DrusvyatskiyLinear recursive feature machines provably recover low\-rank matrices\.Proceedings of the National Academy of Sciences122\(13\),pp\. e2411325122\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px2.p3.1),[§5\.3](https://arxiv.org/html/2608.23765#S5.SS3.p1.1),[§6](https://arxiv.org/html/2608.23765#S6.p4.1),[footnote 2](https://arxiv.org/html/2608.23765#footnote2)\.
- Rao and Kreutz\-Delgado \(1999\)B\. D\. Rao and K\. Kreutz\-DelgadoAn affine scaling methodology for best basis selection\.IEEE Trans\. Signal Process\.47\(1\),pp\. 187–200\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1)\.
- Rechtet al\.\(2011\)B\. Recht, W\. Xu, and B\. HassibiNull space conditions and thresholds for rank minimization\.Math\. Program\.127\(1\),pp\. 175–202\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p1.1)\.
- Rechtet al\.\(2010\)B\. Recht, M\. Fazel, and P\. A\. ParriloGuaranteed minimum\-rank solutions of linear matrix equations via nuclear norm minimization\.SIAM Rev\.52\(3\),pp\. 471–501\(English\)\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p2.1),[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p2.1)\.
- Romanov and Gavish \(2018\)E\. Romanov and M\. GavishNear\-optimal matrix recovery from random linear measurements\.Proceedings of the National Academy of Sciences115\(28\),pp\. 7200–7205\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.2)\.
- Rouletet al\.\(2020\)V\. Roulet, N\. Boumal, and A\. d’AspremontComputational complexity versus statistical performance on sparse recovery problems\.Information and Inference: A Journal of the IMA9\(1\),pp\. 1–32\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p3.2)\.
- Soltanolkotabiet al\.\(2025\)M\. Soltanolkotabi, D\. Stöger, and C\. XieImplicit balancing and regularization: generalization and convergence guarantees for overparameterized asymmetric matrix sensing\.IEEE Trans\. Inf\. Theory71\(4\),pp\. 2991–3037\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Stöger and Soltanolkotabi \(2021\)D\. Stöger and M\. SoltanolkotabiSmall random initialization is akin to spectral learning: optimization and generalization guarantees for overparameterized low\-rank matrix reconstruction\.Advances in Neural Information Processing Systems \(NeurIPS\)34,pp\. 23831–23843\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Sun and Sun \(2008\)D\. Sun and J\. SunLöwner’s Operator and Spectral Functions in Euclidean Jordan Algebras\.Math\. Oper\. Res\.33\(2\),pp\. 421–445\.Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p2.2),[§B\.2](https://arxiv.org/html/2608.23765#A2.SS2.p2.1)\.
- Sunet al\.\(2017\)Y\. Sun, P\. Babu, and D\. P\. PalomarMajorization\-Minimization algorithms in signal processing, communications, and machine learning\.IEEE Transactions on Signal Processing65\(3\),pp\. 794–816\.Cited by:[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p2.1)\.
- Tariqet al\.\(2024\)S\. Tariq, A\. Farooq, J\. U\. Rehman, T\. Q\. Duong, and H\. ShinEfficient quantum state estimation with low\-rank matrix completion\.EPJ Quantum Technology11\(1\),pp\. 50\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p2.1)\.
- Tonget al\.\(2021\)T\. Tong, C\. Ma, and Y\. ChiAccelerating ill\-conditioned low\-rank matrix estimation via scaled gradient descent\.J\. Mach\. Learn\. Res\.22\(150\),pp\. 1–63\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Tuet al\.\(2016\)S\. Tu, R\. Boczar, M\. Simchowitz, M\. Soltanolkotabi, and B\. RechtLow\-rank solutions of linear matrix equations via procrustes flow\.InInternational Conference on Machine Learning \(ICML\),pp\. 964–973\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p5.1),[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.
- Vandereycken \(2013\)B\. VandereyckenLow\-Rank Matrix Completion by Riemannian Optimization\.SIAM J\. Optim\.23\(2\),pp\. 1214–1236\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1),[§5\.4](https://arxiv.org/html/2608.23765#S5.SS4.p2.1)\.
- Wedin \(1972\)P\.\-Å\. WedinPerturbation bounds in connection with singular value decomposition\.BIT12\(1\),pp\. 99–111\.Cited by:[§A\.4\.1](https://arxiv.org/html/2608.23765#A1.SS4.SSS1.p9.10)\.
- Weiszfeld \(1937\)E\. WeiszfeldSur le point pour lequel la somme des distances de n points donnés est minimum\.Tohoku Mathematical Journal, First Series43,pp\. 355–386\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p3.1)\.
- Wipf and Nagarajan \(2010\)D\. Wipf and S\. NagarajanIterative reweightedℓ1\\ell\_\{1\}andℓ2\\ell\_\{2\}methods for finding sparse solutions\.IEEE Journal of Selected Topics in Signal Processing4\(2\),pp\. 317–329\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px3.p1.1),[§3\.1](https://arxiv.org/html/2608.23765#S3.SS1.p2.1)\.
- Yang \(2009\)Z\. YangA study on nonsymmetric matrix\-valued functions\.Master’s thesis, Department of Mathematics, National University of Singapore\.Note:Master’s thesis, Department of Mathematics, National University of Singapore, available at[https://www\.polyu\.edu\.hk/ama/profile/dfsun/files/Main\_YZ\.pdf](https://www.polyu.edu.hk/ama/profile/dfsun/files/Main_YZ.pdf)Cited by:[§A\.2\.2](https://arxiv.org/html/2608.23765#A1.SS2.SSS2.p3.1)\.
- Yi and Xu \(2020\)J\. Yi and W\. XuNecessary and sufficient null space condition for nuclear norm minimization in low\-rank matrix recovery\.IEEE Trans\. Inf\. Theory66\(10\),pp\. 6597–6604\.Cited by:[§4\.1](https://arxiv.org/html/2608.23765#S4.SS1.p1.1)\.
- Zhuet al\.\(2025\)L\. Zhu, D\. Davis, D\. Drusvyatskiy, and M\. FazelIteratively reweighted kernel machines efficiently learn sparse functions\.arXiv preprint arXiv:2505\.08277\.Cited by:[§1](https://arxiv.org/html/2608.23765#S1.p4.2),[§6](https://arxiv.org/html/2608.23765#S6.p4.1)\.
- Zilber and Nadler \(2022\)P\. Zilber and B\. NadlerGNMR: a provable one\-line algorithm for low rank matrix recovery\.SIAM J\. Math\. Data Sci\.4\(2\),pp\. 909–934\(English\)\.Cited by:[§2](https://arxiv.org/html/2608.23765#S2.SS0.SSS0.Px1.p1.1)\.