Multi-Source Wasserstein Distributionally Robust Graph Learning

# Multi-Source Wasserstein Distributionally Robust Graph Learning
Source: [https://arxiv.org/html/2608.19914](https://arxiv.org/html/2608.19914)
Yifan XiaJinshan ZhongXiaojing ShenThanks:The work was supported by the Sichuan Provincial Natural Science Foundation Project \(Innovative Research Group\) 2026NSFSCZY0056\.\(Corresponding author: Xiaojing Shen\.\) Chuansen Peng, Jinshan Zhong and Xiaojing Shen are with School of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China\. \(e\-mail: pengchuansen@stu\.scu\.edu\.cn; 2023322010061@stu\.scu\.edu\.cn; shenxj@scu\.edu\.cn\)\. Yifan Xia is with School of Statistics and Data Science, Southwestern University of Finance and Economics, Chengdu, Sichuan, 611130, China\. \(e\-mail: xiayifan@swufe\.edu\.cn\)\.

###### Abstract

Network topology inference from graph signals is central to graph signal processing, with applications in brain connectivity, sensor networks, and social networks\. In practice, target\-domain samples are scarce, while heterogeneous source\-domain data are abundant\. Fusing these sources into a reliable nominal distribution is challenging: Euclidean averaging works when sources are homogeneous but degrades sharply as inter\-source divergence grows, collapsing distinct geometries into an inflated, biased consensus\. We exploit the Wasserstein metric’s distribution\-preserving fusion to counter this heterogeneity while preserving each source’s intrinsic geometry\. We propose MS\-WDRO, a multi\-source Wasserstein distributionally robust graph learning framework that fuses heterogeneous sources via their weighted Wasserstein barycenter, a geometrically principled nominal distribution, then builds an ambiguity ball around it to hedge residual uncertainty\. Minimizing worst\-case risk over this ball yields a tractable regularized Laplacian estimator, solved efficiently via a provably convergent ADMM scheme\. We establish non\-asymptotic guarantees: a finite\-sample concentration bound for the empirical barycenter, a pooling bias lower bound proving naive aggregation is suboptimal, and an out\-of\-sample excess risk bound decaying at a parametric rate with only logarithmic dependence on source count\. To calibrate the four coupled hyperparameters governing robustness, sparsity, and source fusion, we unroll the solver into a differentiable architecture trained end\-to\-end, achieving data\-adaptive calibration beyond cross\-validation while retaining interpretability\. Experiments on synthetic benchmarks and the multi\-site ABIDE I neuroimaging dataset show MS\-WDRO consistently outperforms seven baselines in graph recovery accuracy, sample efficiency, and downstream diagnostic utility, with the largest gains in the sample\-scarce regime motivating this work\.

###### Index Terms:

Graph signal processing, network topology inference, distributionally robust optimization, Wasserstein barycenter, algorithm unrolling\. heterogeneous multi\-source learning

## IIntroduction

Network\-structured data pervade modern science and engineering: gene regulatory networks govern cellular function, functional brain connectomes encode cognitive processes, power grids exchange energy across interconnected nodes, and social platforms generate opinion dynamics over relational graphs\[[1](https://arxiv.org/html/2608.19914#bib.bib1),[2](https://arxiv.org/html/2608.19914#bib.bib2),[3](https://arxiv.org/html/2608.19914#bib.bib3),[4](https://arxiv.org/html/2608.19914#bib.bib4),[5](https://arxiv.org/html/2608.19914#bib.bib5),[6](https://arxiv.org/html/2608.19914#bib.bib6)\]\. Graph signal processing \(GSP\) has emerged as a principled mathematical framework for analyzing signals defined on such irregular domains, extending classical tools from Fourier analysis, filtering, and sampling theory to data indexed by graph vertices\[[7](https://arxiv.org/html/2608.19914#bib.bib7),[8](https://arxiv.org/html/2608.19914#bib.bib8),[9](https://arxiv.org/html/2608.19914#bib.bib9)\]\. A central and often foundational challenge in this paradigm is*network topology inference*, the problem of recovering the underlying graph structure from observed nodal signals\. The inferred topology is not merely an end in itself; it serves as the substrate for graph\-based filtering, signal interpolation, anomaly detection, and disease biomarker discovery in neuroimaging studies of conditions such as autism spectrum disorder\. In all of these settings, the quality of downstream analysis depends critically on the fidelity of the inferred graph, placing a premium on topology estimators that are both statistically principled and computationally tractable\.

The vast majority of graph learning methods are built on a generative signal model that captures the relationship between the observed signals and the latent graph\. Under the widely adopted*smooth graph signal model*, nodal observations are assumed to vary slowly across connected vertices, and the combinatorial graph Laplacian naturally emerges as the precision matrix of the corresponding degenerate Gaussian distribution\. Building on this model, Dong et al\.\[[10](https://arxiv.org/html/2608.19914#bib.bib10)\]formulated graph learning as a constrained maximum\-likelihood estimation problem and derived efficient block coordinate descent algorithms\. Kalofolias\[[11](https://arxiv.org/html/2608.19914#bib.bib11)\]recast a closely related objective as a smooth convex program amenable to proximal gradient solvers, while Egilmez et al\.\[[12](https://arxiv.org/html/2608.19914#bib.bib12)\]incorporated structural constraints on the Laplacian and developed specialized ADMM procedures with convergence guarantees\. Several subsequent works extended these foundations: Kumar et al\.\[[13](https://arxiv.org/html/2608.19914#bib.bib13)\]unified a broad class of structured graph learning problems through spectral constraints; Sardellitti et al\.\[[14](https://arxiv.org/html/2608.19914#bib.bib14)\]leveraged transform learning to infer graph topology without specifying a parametric signal model; and accelerated first\-order methods have been developed to handle large\-scale instances\[[15](https://arxiv.org/html/2608.19914#bib.bib15)\]\. Beyond the smoothness paradigm, complementary lines of work infer topology from stationary graph signals whose power spectrum is aligned with the graph Fourier basis\[[16](https://arxiv.org/html/2608.19914#bib.bib16)\], from diffusion\-process observations\[[17](https://arxiv.org/html/2608.19914#bib.bib17)\], from spectral templates derived from signal second\-order statistics\[[18](https://arxiv.org/html/2608.19914#bib.bib18)\], and from joint network topology inference across multiple related graphs\[[19](https://arxiv.org/html/2608.19914#bib.bib19)\]\. In the probabilistic graphical model literature, sparse precision matrix estimation\[[20](https://arxiv.org/html/2608.19914#bib.bib20)\]provides an alternative route to graph structure recovery when Gaussian assumptions hold\. Despite their considerable diversity in signal model and algorithmic strategy, these approaches share a common requirement: they rely on a sufficiently large and representative collection of signal observations drawn from the*same*target distribution\. Their statistical performance degrades severely when samples are scarce or when the observations do not faithfully reflect the distribution of interest\.

In practice, graph signal observations are finite in number, contaminated by noise, and subject to distributional shift arising from sensor drift, protocol variation, or environmental change\. Under these conditions, the empirical risk minimization \(ERM\) principle may lead to overfitting: the learned graph Laplacian is calibrated to the finite observed samples but generalizes poorly to signals drawn from the true underlying distribution\[[21](https://arxiv.org/html/2608.19914#bib.bib21)\]\. Distributionally robust optimization \(DRO\) offers a principled remedy by replacing the nominal empirical distribution with a worst\-case distribution drawn from an ambiguity set, thereby immunizing the estimator against deviations between the training and true distributions\[[22](https://arxiv.org/html/2608.19914#bib.bib22),[23](https://arxiv.org/html/2608.19914#bib.bib23)\]\. Among the many choices of ambiguity set, those defined through the Wasserstein metric have attracted particular attention owing to their geometric interpretability, their tight connection to adversarial perturbations, and their ability to yield tractable convex reformulations of otherwise intractable minimax problems\[[24](https://arxiv.org/html/2608.19914#bib.bib24),[25](https://arxiv.org/html/2608.19914#bib.bib25),[26](https://arxiv.org/html/2608.19914#bib.bib26)\]\. A Wasserstein ball of prescribed radius centered at the empirical distribution certifies that the learned estimator performs well for any distribution that lies within a controlled optimal transport distance from the training data, a guarantee that purely moment\-based ambiguity sets\[[22](https://arxiv.org/html/2608.19914#bib.bib22),[23](https://arxiv.org/html/2608.19914#bib.bib23)\]cannot provide in the absence of shape assumptions\. The foundations of Wasserstein DRO, duality theory, tractable reformulations, and finite\-sample performance guarantees, have been developed in considerable depth\[[24](https://arxiv.org/html/2608.19914#bib.bib24)\], and the framework has been applied to a broad array of statistical learning problems including logistic regression\[[27](https://arxiv.org/html/2608.19914#bib.bib27)\]and adversarially robust training\[[28](https://arxiv.org/html/2608.19914#bib.bib28)\]\. Most recently, Wasserstein DRO has been specialized to graph Laplacian estimation from smooth signals\[[29](https://arxiv.org/html/2608.19914#bib.bib29)\], demonstrating that distributional robustness yields measurable improvements in graph recovery accuracy when target\-domain samples are limited\. These developments concentrate on the single\-source setting, in which all observations are assumed to arise from one underlying distribution\. Motivated by this body of work, the present paper extends Wasserstein distributional robustness from single source to the multi\-source regime, in which signals collected across several related yet heterogeneous domains must be reconciled into a single robust estimator\. This extension raises challenges of its own: heterogeneous sources cannot be pooled directly without incurring a mixing bias, and the ambiguity set itself must be constructed jointly from multiple empirical distributions rather than centered on a single one, calling for a principled mechanism to aggregate heterogeneous source information before robustness can be meaningfully enforced\.

Many high\-impact applications of graph learning feature a target domain from which signal samples are difficult or impossible to collect in sufficient quantity, while multiple related but heterogeneous source domains provide abundant data\. In functional neuroimaging, a rare clinical cohort at a small acquisition site, such as the CMU site in the ABIDE I consortium\[[30](https://arxiv.org/html/2608.19914#bib.bib30)\], constitutes the target domain, while data from larger studies conducted at different scanning sites under distinct imaging protocols serve as source domains, inducing site\-specific distributional shift that is well documented in the brain connectivity literature\[[31](https://arxiv.org/html/2608.19914#bib.bib31)\]\. Similar multi\-source configurations arise in distributed sensor networks, where sensors with different hardware characteristics monitor a shared physical phenomenon; in clinical federated learning, where data heterogeneity across hospitals precludes simple aggregation; and in social network analysis, where platform\-specific behavioral norms differentiate the signal statistics across source and target domains\. The transfer learning literature has long recognized that naive aggregation of heterogeneous source data by pooling all observations into a single empirical distribution introduces a*mixing bias*\[[32](https://arxiv.org/html/2608.19914#bib.bib32),[33](https://arxiv.org/html/2608.19914#bib.bib33)\]whose magnitude grows with inter\-source divergence, and that constructing a shared model from such a biased nominal distribution leads to systematically degraded target\-domain performance\[[34](https://arxiv.org/html/2608.19914#bib.bib34)\]\. A principled approach to multi\-source distribution aggregation is provided by*optimal transport*theory\[[35](https://arxiv.org/html/2608.19914#bib.bib35)\]: the*Wasserstein barycenter*\[[36](https://arxiv.org/html/2608.19914#bib.bib36)\]defines the Fréchet mean of a family of distributions in the Wasserstein metric space and produces a consensus representation that preserves the intrinsic geometric structure of each source, in sharp contrast to the covariance inflation inherent to the mixture distribution\. Efficient algorithms for Wasserstein barycenter computation\[[37](https://arxiv.org/html/2608.19914#bib.bib37),[38](https://arxiv.org/html/2608.19914#bib.bib38)\]and the Gelbrich lower bound relating the barycenter to source second\-order moments\[[39](https://arxiv.org/html/2608.19914#bib.bib39)\]have rendered this construction computationally tractable at the scale of practical graph learning problems\. Optimal transport has also proved fruitful for domain adaptation by geometrically aligning source and target distributions\[[40](https://arxiv.org/html/2608.19914#bib.bib40)\]; the present work leverages its complementary role of fusing source distributions into a structurally sound nominal distribution for robust graph estimation\.

Despite clear practical motivation, the intersection of multi\-source learning, Wasserstein distributional robustness, and smooth\-signal graph topology inference remains largely unexplored\. A further challenge is the joint calibration of multiple interacting hyperparameters inherent in any WDRO\-based graph learning framework: the ambiguity set radius determines the degree of distributional robustness; the sparsity regularization coefficient controls graph density; the augmented Lagrangian penalty governs algorithmic convergence; and the barycentric fusion weights modulate the relative contribution of each source domain\. These parameters interact nonlinearly and cannot be calibrated in isolation\. Conventional cross\-validation over the resulting joint parameter space is computationally prohibitive in the multi\-source regime, and concentration\-inequality\-based radius selection, while theoretically motivated, is inherently conservative and ignores the mutual coupling among the parameters\. A principled, end\-to\-end trainable framework that simultaneously addresses multi\-source fusion, distributional robustness, and automatic hyperparameter calibration is therefore strongly needed\.

Algorithm unrolling, the technique of mapping the iterations of an optimization algorithm onto the layers of a neural network while treating algorithmic parameters as learnable variables\[[41](https://arxiv.org/html/2608.19914#bib.bib41)\], offers a compelling solution to the hyperparameter calibration challenge\. By embedding the iterative solver into a differentiable architecture and training end\-to\-end on a supervised corpus of graph\-signal pairs, all parameters can be jointly calibrated through backpropagation, capturing their mutual interactions in a manner that any sequential or grid\-search strategy fundamentally misses\. The resulting unrolled network retains full algorithmic interpretability, each layer executes a precisely specified proximal, eigendecomposition, or dual\-ascent substep, while acquiring data\-adaptive calibration capability unavailable to the fixed\-parameter solver\. This paradigm has been successfully applied to compressive sensing magnetic resonance imaging via deep ADMM\-Net\[[42](https://arxiv.org/html/2608.19914#bib.bib42)\], to sparse graph recovery via GLAD\[[43](https://arxiv.org/html/2608.19914#bib.bib43)\], and to graph signal denoising\[[44](https://arxiv.org/html/2608.19914#bib.bib44)\]\. The survey by Monga et al\.\[[41](https://arxiv.org/html/2608.19914#bib.bib41)\]provides a comprehensive account of the theoretical and practical principles underlying the paradigm, documenting its advantages in interpretability, sample efficiency, and computational scalability relative to generic deep architectures\. A critical prerequisite enabling differentiable unrolling in our setting is the ADMM solver structure, whose per\-iteration updates admit closed\-form analytical expressions that are differentiable with respect to the hyperparameters and therefore compatible with gradient\-based end\-to\-end training\.

This paper proposes MS\-WDRO \(Multi\-SourceWassers\-teinDistributionallyRobustOptimization for graph learning\), a framework that jointly addresses multi\-source data fusion, distributional robustness, and automatic hyperparameter learning for network topology inference from smooth graph signals\. The main contributions are as follows\.

*\(i\) Multi\-source WDRO framework\.*We formulate a novel graph Laplacian estimation framework that fuses heterogeneous source distributions by computing their weighted Wasserstein barycenter as the nominal distribution of a Wasserstein ambiguity set\. Minimizing the worst\-case expected log\-likelihood loss over the resulting barycentric ambiguity ball yields a principled minimax problem that simultaneously exploits multi\-source information and guards against residual distributional uncertainty\. The framework encompasses single\-source WDRO as a special case and formally subsumes naive pooling while provably outperforming it\.

*\(ii\) Tractable reformulation and efficient algorithm\.*We derive a closed\-form tractable reformulation of the minimax problem via Wasserstein strong duality, reducing it to a regularized Laplacian estimation problem with an explicit Frobenius\-norm robustness penalty and sparsity term\. A two\-block ADMM solver with closed\-form per\-iteration updates is developed, and its global convergence, objective convergence, primal consensus, and dual convergence, governed by a provably monotone primal\-dual Lyapunov potential, is rigorously established in Theorem[III\.2](https://arxiv.org/html/2608.19914#S3.Thmtheorem2)\. Explicit non\-asymptotic guarantees are given, including an ergodicO⁡\(1/K\)O\(1/K\)primal\-dual gap\-function rate \(Corollary[III\.2](https://arxiv.org/html/2608.19914#S3.Thmcorollary2)\)\.

*\(iii\) Rigorous statistical theory\.*We establish three complementary non\-asymptotic guarantees\. First, a finite\-sample concentration bound for the empirical Wasserstein barycenter quantifies how accurately the empirical barycenter approximates its population counterpart as source sample sizes grow\. Second, a formal pooling bias lower bound, proved via the Lieb concavity theorem for quantum fidelity\[[45](https://arxiv.org/html/2608.19914#bib.bib45)\]and the Haar measure twirling identity\[[46](https://arxiv.org/html/2608.19914#bib.bib46)\], confirms that, under a common\-covariance, heterogeneous\-mean Gaussian source model, naive data pooling incurs a strictly positive, irreducible Wasserstein estimation bias that scales with the between\-source mean dispersionℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}, establishing an asymptotic separation between the barycentric and pooling approaches in this regime\. Third, an out\-of\-sample excess risk bound derived through Rademacher complexity analysis shows that the excess risk of the MS\-WDRO estimator decays at the parametric rate and depends on the number of sources only logarithmically, with a concentration correction computable entirely from training data\.

*\(iv\) Algorithm unrolling for joint hyperparameter learning\.*We embed the MS\-WDRO solver into a differentiable multi\-layer architecture by interpreting each iteration as a computational layer parameterized by four learnable quantities: the ambiguity set radius, the sparsity regularization coefficient, the augmented Lagrangian penalty, and the barycentric fusion weights\. End\-to\-end supervised training jointly calibrates all parameters through backpropagation, enabling an implicit annealing schedule across layers, progressively shrinking robustness margins, relaxing sparsity, tightening the dual step size, and adapting source weights, that is difficult to realize efficiently by conventional cross\-validation\. The total number of trainable parameters grows only linearly in network depth and source count, making the architecture highly parameter\-efficient\.

*\(v\) Comprehensive experimental validation\.*We conduct experiments on synthetic graph benchmarks with controlled inter\-source heterogeneity and on the ABIDE I multi\-site functional neuroimaging dataset, comparing against seven competitive baselines spanning classical optimization\-based methods, deep graph learning algorithms, and the state\-of\-the\-art single\-source WDRO approach\. MS\-WDRO consistently and significantly outperforms all baselines in graph recovery accuracy and sample efficiency across the full range of target sample sizes, achieves superior downstream diagnostic classification on the neuroimaging dataset, and empirically validates the theoretical out\-of\-sample bound at the predicted convergence rate\.

The remainder of the paper is organized as follows\. Section[II](https://arxiv.org/html/2608.19914#S2)defines the graph signal model, reviews the Wasserstein distance and barycenter, and states the baseline Laplacian estimation problem\. Section[III](https://arxiv.org/html/2608.19914#S3)presents the Wasserstein barycentric ambiguity set, derives the tractable minimax reformulation, and develops the ADMM solver\. Section[IV](https://arxiv.org/html/2608.19914#S4)provides the complete theoretical analysis, including barycenter concentration, the pooling bias lower bound, data\-driven radius selection, and the out\-of\-sample excess risk guarantee\. Section[V](https://arxiv.org/html/2608.19914#S5)introduces the algorithm unrolling architecture and its end\-to\-end training procedure\. Numerical experiments are reported in Section[VI](https://arxiv.org/html/2608.19914#S6), and Section[VII](https://arxiv.org/html/2608.19914#S7)concludes with a discussion of limitations and future directions\.

## IISignal Models and Problem Formulation

This section lays the groundwork for the multi\-source distributionally robust framework developed in the sequel\. We proceed in three steps\. We first specify the generative model that links observed graph signals to the underlying topology, showing that smoothness on the graph induces a degenerate Gaussian likelihood whose precision matrix coincides with the graph Laplacian \(Section[II\-A](https://arxiv.org/html/2608.19914#S2.SS1)\)\. Building on this likelihood, we formulate the canonical single\-distribution graph learning problem as a regularized maximum\-likelihood estimator \(Section[II\-B](https://arxiv.org/html/2608.19914#S2.SS2)\), which serves as the point of departure for the robust formulation introduced later\. Because that baseline estimator implicitly treats the empirical distribution of a single data source as exact, it provides no principled mechanism for combining several heterogeneous sources or for hedging against sampling error, both of which are central concerns of this paper\. We therefore close the section by introducing the optimal\-transport tools required to overcome this limitation, namely the Wasserstein distance, the Wasserstein barycenter, and its second\-moment surrogate, the Gelbrich distance \(Section[II\-C](https://arxiv.org/html/2608.19914#S2.SS3)\); these constructs furnish the geometric and computational vocabulary on which the multi\-source Wasserstein distributionally robust formulation of Section[III](https://arxiv.org/html/2608.19914#S3)is built\.

### II\-AGraph Signal Model

Let a weighted, undirected graph be represented by the triple𝒢=\(𝒱,ℰ,𝐋\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\},\\mathbf\{L\}\), where𝒱\\mathcal\{V\}is a finite vertex set of cardinalityNN,ℰ⊆𝒱×𝒱\\mathcal\{E\}\\subseteq\\mathcal\{V\}\\times\\mathcal\{V\}is the edge set, and𝐋∈ℝN×N\\mathbf\{L\}\\in\\mathbb\{R\}^\{N\\times N\}is the combinatorial graph Laplacian, whose off\-diagonal entryLi​jL\_\{ij\}encodes \(the negative of\) the edge weight between nodesiiandjj\. Under the standing assumption of no self\-loops,𝐋\\mathbf\{L\}satisfies𝐋𝟏=𝟎\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}\(Here,𝟏\\mathbf\{1\}is the all\-ones vector\) and is a symmetric positive semi\-definite matrix\[[19](https://arxiv.org/html/2608.19914#bib.bib19)\]\. The set of admissible Laplacians is accordingly given by

ℒ=\{𝐋∣𝐋⪰0,𝐋𝟏=𝟎,Li​j=Lj​i≤0,i≠j\}\.\\displaystyle\\mathcal\{L\}=\\bigl\\\{\\mathbf\{L\}\\mid\\mathbf\{L\}\\succeq 0,\\ \\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\},\\ L\_\{ij\}=L\_\{ji\}\\leq 0,\\ i\\neq j\\bigr\\\}\.\(1\)A graph signal is a vector𝐱=\[x1,…,xN\]T∈ℝN\\mathbf\{x\}=\[x\_\{1\},\\ldots,x\_\{N\}\]^\{T\}\\in\\mathbb\{R\}^\{N\}assigning a scalar valuexix\_\{i\}to each nodeii\. To relate such signals to the graph topology, we adopt the standard graph filtering formalism: writing the eigendecomposition of the Laplacian as𝐋=𝐔​𝚲​𝐔T\\mathbf\{L\}=\\mathbf\{U\}\\bm\{\\Lambda\}\\mathbf\{U\}^\{T\}, with𝚲=diag⁡\(λ1,…,λN\)\\bm\{\\Lambda\}=\\mathrm\{diag\}\(\\lambda\_\{1\},\\ldots,\\lambda\_\{N\}\)collecting the non\-decreasing graph frequencies0≤λ1≤⋯≤λN0\\leq\\lambda\_\{1\}\\leq\\cdots\\leq\\lambda\_\{N\}, the graph Fourier transform of𝐱\\mathbf\{x\}is defined as𝐱^=𝐔T​𝐱\\hat\{\\mathbf\{x\}\}=\\mathbf\{U\}^\{T\}\\mathbf\{x\}, with inverse transform𝐱=𝐔​𝐱^\\mathbf\{x\}=\\mathbf\{U\}\\hat\{\\mathbf\{x\}\}\. Given the eigendecomposition𝐋=𝐔​𝚲​𝐔T\\mathbf\{L\}=\\mathbf\{U\}\\bm\{\\Lambda\}\\mathbf\{U\}^\{T\}, a graph filter with spectral responseh⁡\(⋅\)h\(\\cdot\), defined ash⁡\(𝐋\)=𝐔​h​\(𝚲\)​𝐔Th\(\\mathbf\{L\}\)=\\mathbf\{U\}h\(\\bm\{\\Lambda\}\)\\mathbf\{U\}^\{T\}, generates signals from a white input𝐱0\\mathbf\{x\}\_\{0\}via

𝐱=𝝁\+h⁡\(𝐋\)​𝐱0,\\displaystyle\\mathbf\{x\}=\\bm\{\\mu\}\+h\(\\mathbf\{L\}\)\\mathbf\{x\}\_\{0\},\(2\)where𝝁\\bm\{\\mu\}denotes the signal mean\. Taking𝐱0\\mathbf\{x\}\_\{0\}to be standard multivariate Gaussian noise renders𝐱\\mathbf\{x\}Gaussian with covarianceh2​\(𝐋\)≜h⁡\(𝐋\)​h​\(𝐋\)Th^\{2\}\(\\mathbf\{L\}\)\\triangleq h\(\\mathbf\{L\}\)h\(\\mathbf\{L\}\)^\{T\}:

𝐱∼𝒩⁡\(𝝁,h2​\(𝐋\)\)\.\\displaystyle\\mathbf\{x\}\\sim\\mathcal\{N\}\\bigl\(\\bm\{\\mu\},\\,h^\{2\}\(\\mathbf\{L\}\)\\bigr\)\.\(3\)The choice of filterh⁡\(⋅\)h\(\\cdot\)determines the qualitative behavior of the resulting signal family\. Following the widely adopted smoothness prior for graph signals\[[10](https://arxiv.org/html/2608.19914#bib.bib10),[11](https://arxiv.org/html/2608.19914#bib.bib11)\], we select the low\-pass filterh⁡\(𝐋\)=𝐋†h\(\\mathbf\{L\}\)=\\sqrt\{\\mathbf\{L\}^\{\\dagger\}\}, where\(⋅\)†\(\\cdot\)^\{\\dagger\}denotes the Moore–Penrose pseudoinverse; this choice attenuates high graph\-frequency components and thereby favors signals that vary slowly across connected vertices, consistent with the smoothness assumption underlying most graph learning methods reviewed in the Introduction\. Substituting this filter into \([3](https://arxiv.org/html/2608.19914#S2.E3)\) shows that a smooth graph signal follows a degenerate Gaussian law whose precision matrix is precisely the graph Laplacian:

f⁡\(𝐱\)=𝒩⁡\(𝝁,𝐋†\)\.\\displaystyle f\(\\mathbf\{x\}\)=\\mathcal\{N\}\\bigl\(\\bm\{\\mu\},\\,\\mathbf\{L\}^\{\\dagger\}\\bigr\)\.\(4\)Equation \([4](https://arxiv.org/html/2608.19914#S2.E4)\) is the key structural fact exploited throughout the paper: it reduces the topological problem of estimating𝐋\\mathbf\{L\}to the statistical problem of estimating the precision matrix of a Gaussian graphical model, subject to the combinatorial constraints in \([1](https://arxiv.org/html/2608.19914#S2.E1)\)\.

### II\-BBaseline Graph Learning Problem

We first formulate the canonical estimation problem implied by the signal model \([4](https://arxiv.org/html/2608.19914#S2.E4)\) under the classical assumption that all observed signals are drawn i\.i\.d\. from a single, fixed distribution\. Let𝐗=\[𝐱1,…,𝐱n\]∈ℝN×n\\mathbf\{X\}=\[\\mathbf\{x\}\_\{1\},\\ldots,\\mathbf\{x\}\_\{n\}\]\\in\\mathbb\{R\}^\{N\\times n\}collectnnsuch i\.i\.d\. samples\. The empirical log\-likelihood of a candidate Laplacian𝐋\\mathbf\{L\}given𝐗\\mathbf\{X\}follows directly from \([4](https://arxiv.org/html/2608.19914#S2.E4)\):

ℱn​\(𝐋∣𝐗\)=1n​∑i=1nlog⁡f⁡\(𝐱i\)=log⁡\|𝐋\|\+−tr⁡\(𝚺^​𝐋\),\\displaystyle\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\log f\(\\mathbf\{x\}\_\{i\}\)=\\log\|\\mathbf\{L\}\|\_\{\+\}\-\\mathrm\{tr\}\(\\hat\{\\bm\{\\Sigma\}\}\\mathbf\{L\}\),\(5\)where\|𝐋\|\+\|\\mathbf\{L\}\|\_\{\+\}denotes the pseudo\-determinant of𝐋\\mathbf\{L\}\(the product of its nonzero eigenvalues, required since𝐋\\mathbf\{L\}is singular by construction\) and𝚺^=1n​∑i=1n\(𝐱i−𝝁\)​\(𝐱i−𝝁\)T\\hat\{\\bm\{\\Sigma\}\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\(\\mathbf\{x\}\_\{i\}\-\\bm\{\\mu\}\)\(\\mathbf\{x\}\_\{i\}\-\\bm\{\\mu\}\)^\{T\}is the sample covariance matrix of the observed signals\. Maximizingℱn​\(𝐋∣𝐗\)\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)alone is generally ill\-posed and prone to overfitting whennnis small relative toNN; we therefore augment it with a regularization term𝒫⁡\(𝐋\)\\mathcal\{P\}\(\\mathbf\{L\}\), weighted byρ\>0\\rho\>0, that promotes structural properties such as sparsity, e\.g\.,𝒫⁡\(𝐋\)=‖𝐋‖F2\\mathcal\{P\}\(\\mathbf\{L\}\)=\\\|\\mathbf\{L\}\\\|\_\{F\}^\{2\}or𝒫⁡\(𝐋\)=−log⁡\(diag⁡\(𝐋\)\)\\mathcal\{P\}\(\\mathbf\{L\}\)=\-\\log\(\\mathrm\{diag\}\(\\mathbf\{L\}\)\)\. The resulting baseline graph learning problem is the regularized maximum\-likelihood estimator

max𝐋ℱn​\(𝐋∣𝐗\)−ρ​𝒫​\(𝐋\)s\.t\.​𝐋∈ℒ\.\\displaystyle\\max\_\{\\mathbf\{L\}\}\\quad\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)\-\\rho\\,\\mathcal\{P\}\(\\mathbf\{L\}\)\\quad\\text\{s\.t\.\}\\;\\mathbf\{L\}\\in\\mathcal\{L\}\.\(6\)Problem \([6](https://arxiv.org/html/2608.19914#S2.E6)\) is the workhorse formulation underlying classical smooth\-signal graph learning\. It implicitly identifies the empirical distribution of𝐗\\mathbf\{X\}with the true data\-generating distribution and offers no safeguard when this identification fails, whether becausennis too small for𝚺^\\hat\{\\bm\{\\Sigma\}\}to be a reliable estimate of the population covariance, or because𝐗\\mathbf\{X\}is itself pooled, in whole or in part, from sources whose distributions differ from that of the intended target\. Both failure modes are the norm rather than the exception in the multi\-source, small\-target\-sample regime motivating this paper, which calls for replacing the single fixed nominal distribution in \([6](https://arxiv.org/html/2608.19914#S2.E6)\) with an ambiguity set that is robust to distributional uncertainty while remaining informed by all available source data\. Constructing such a set requires a notion of distance between probability distributions together with a principled way of aggregating several of them; we introduce both next\.

### II\-CWasserstein Distance, Barycenter, and Gelbrich Bound

We adopt the Wasserstein distance as the governing metric for distributional uncertainty, owing to its ability to compare distributions with disjoint or singular support, as is the case for the degenerate Gaussian law in \([4](https://arxiv.org/html/2608.19914#S2.E4)\), and to its well\-known tractable convex reformulations in distributionally robust optimization\[[24](https://arxiv.org/html/2608.19914#bib.bib24),[47](https://arxiv.org/html/2608.19914#bib.bib47)\]\.

###### Definition II\.1\(Wasserstein Distance\)

Forp∈\[1,∞\)p\\in\[1,\\infty\), the order\-ppWasserstein distance between probability distributionsℙ1\\mathbb\{P\}\_\{1\}andℙ2\\mathbb\{P\}\_\{2\}onℝN\\mathbb\{R\}^\{N\}is

Wp​\(ℙ1,ℙ2\)\\displaystyle W\_\{p\}\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)=\(infπ∈𝒰⁡\(ℙ1,ℙ2\)∫ℝN×ℝNC​\(𝐚1,𝐚2\)p​π​\(d​𝐚1,d​𝐚2\)\)1p,\\displaystyle=\\left\(\\inf\_\{\\pi\\in\\mathcal\{U\}\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)\}\\int\_\{\\mathbb\{R\}^\{N\}\\times\\mathbb\{R\}^\{N\}\}C\(\\mathbf\{a\}\_\{1\},\\mathbf\{a\}\_\{2\}\)^\{p\}\\,\\pi\(d\\mathbf\{a\}\_\{1\},d\\mathbf\{a\}\_\{2\}\)\\right\)^\{\\frac\{1\}\{p\}\},\(7\)where𝒰⁡\(ℙ1,ℙ2\)\\mathcal\{U\}\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)denotes the set of couplings ofℙ1\\mathbb\{P\}\_\{1\}andℙ2\\mathbb\{P\}\_\{2\}, i\.e\., joint distributions onℝN×ℝN\\mathbb\{R\}^\{N\}\\times\\mathbb\{R\}^\{N\}with marginalsℙ1\\mathbb\{P\}\_\{1\}andℙ2\\mathbb\{P\}\_\{2\}, andC⁡\(⋅,⋅\)C\(\\cdot,\\cdot\)is a transportation cost function\.

Throughout, we takeC⁡\(𝐚1,𝐚2\)=‖𝐚1−𝐚2‖pC\(\\mathbf\{a\}\_\{1\},\\mathbf\{a\}\_\{2\}\)=\\\|\\mathbf\{a\}\_\{1\}\-\\mathbf\{a\}\_\{2\}\\\|\_\{p\}, so thatWp​\(ℙ1,ℙ2\)W\_\{p\}\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)quantifies the minimal cost of transporting the mass ofℙ1\\mathbb\{P\}\_\{1\}ontoℙ2\\mathbb\{P\}\_\{2\}under anℓp\\ell\_\{p\}transportation cost\.

When the data of interest arise from several distributions rather than one, it is natural to seek a single representative distribution that best summarizes them jointly in the Wasserstein metric\. This role is played by the Wasserstein barycenter, the Fréchet mean of a distribution over the Wasserstein space\.

###### Definition II\.2\(Wasserstein Barycenter\)

Forp∈\[1,∞\)p\\in\[1,\\infty\), thepp\-Wasserstein barycenter of a distributionℙ\\mathbb\{P\}over probability measures is

bp​\(ℙ\):=arg​minν⁡𝔼ρ∼ℙ​\[Wpp​\(ν,ρ\)\],\\displaystyle b\_\{p\}\(\\mathbb\{P\}\):=\\operatorname\*\{arg\\,min\}\_\{\\nu\}\\,\\mathbb\{E\}\_\{\\rho\\sim\\mathbb\{P\}\}\\bigl\[W\_\{p\}^\{p\}\(\\nu,\\rho\)\\bigr\],\(8\)whereρ\\rhois a random measure distributed according toℙ\\mathbb\{P\}\. In particular, takingℙ=∑m=1Mλm​δρm\\mathbb\{P\}=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\delta\_\{\\rho\_\{m\}\}for weights𝛌=\(λm\)m∈\[M\]∈△M\\bm\{\\lambda\}=\(\\lambda\_\{m\}\)\_\{m\\in\[M\]\}\\in\\triangle^\{M\}on the probability simplex recovers the*empirical*pp\-Wasserstein barycenter ofMMmeasuresρ1,…,ρm\\rho\_\{1\},\\ldots,\\rho\_\{m\},

b^𝝀,p\(ρ1,…,ρM\):=arg​minν∑m=1MλmWpp\(ν,ρm\)\.\\displaystyle\\hat\{b\}\_\{\\bm\{\\lambda\},p\}\(\\rho\_\{1\},\\ldots,\\rho\_\{M\}\):=\\operatorname\*\{arg\\,min\}\_\{\\nu\}\\,\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\,W\_\{p\}^\{p\}\(\\nu,\\rho\_\{m\}\)\.\(9\)

Equation \([9](https://arxiv.org/html/2608.19914#S2.E9)\) will later serve to fuse theMMheterogeneous source distributions into a single nominal distribution for the ambiguity set of Section[III](https://arxiv.org/html/2608.19914#S3), with𝝀\\bm\{\\lambda\}controlling each source’s relative contribution\.

EvaluatingWpW\_\{p\}or its barycenter exactly is, in general, NP\-hard outside a few special cases\[[48](https://arxiv.org/html/2608.19914#bib.bib48)\]\. Since the signal model of Section[II\-A](https://arxiv.org/html/2608.19914#S2.SS1)identifies each source distribution with a \(degenerate\) Gaussian law, and hence with its first two moments, we can instead work with a moment\-based surrogate that is tractable for arbitrary covariance matrices and coincides with the22\-Wasserstein distance whenever both distributions are Gaussian: the Gelbrich distance\.

###### Definition II\.3\(Gelbrich Distance\)

For distributionsℙ1,ℙ2\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}with meansμ1,μ2\\mu\_\{1\},\\mu\_\{2\}and covariance matricesΣ1,Σ2∈𝕊\+N\\Sigma\_\{1\},\\Sigma\_\{2\}\\in\\mathbb\{S\}\_\{\+\}^\{N\}, the Gelbrich distance between them is

G⁡\(ℙ1,ℙ2\):=‖μ1−μ2‖22\+B2​\(Σ1,Σ2\),\\displaystyle G\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\):=\\sqrt\{\\\|\\mu\_\{1\}\-\\mu\_\{2\}\\\|\_\{2\}^\{2\}\+B^\{2\}\(\\Sigma\_\{1\},\\Sigma\_\{2\}\)\},\(10\)where

B2​\(Σ1,Σ2\)\\displaystyle B^\{2\}\(\\Sigma\_\{1\},\\Sigma\_\{2\}\):=tr⁡\(Σ1\)\+tr⁡\(Σ2\)\\displaystyle:=\\mathrm\{tr\}\(\\Sigma\_\{1\}\)\+\\mathrm\{tr\}\(\\Sigma\_\{2\}\)−2​tr​\(\(Σ11/2​Σ2​Σ11/2\)1/2\)\\displaystyle\\quad\-2\\,\\mathrm\{tr\}\\Bigl\(\\bigl\(\\Sigma\_\{1\}^\{1/2\}\\Sigma\_\{2\}\\Sigma\_\{1\}^\{1/2\}\\bigr\)^\{1/2\}\\Bigr\)\(11\)is the squared Bures–Wasserstein distance betweenΣ1\\Sigma\_\{1\}andΣ2\\Sigma\_\{2\}\.

BecauseG⁡\(ℙ1,ℙ2\)G\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)depends onℙ1,ℙ2\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}only through their first two moments, it provides a closed\-form, provably tight lower bound onW2​\(ℙ1,ℙ2\)W\_\{2\}\(\\mathbb\{P\}\_\{1\},\\mathbb\{P\}\_\{2\}\)that is exact in the Gaussian case relevant here\[[39](https://arxiv.org/html/2608.19914#bib.bib39)\], and it is this tractability that makes the Wasserstein barycentric ambiguity set of Section[III](https://arxiv.org/html/2608.19914#S3)computationally viable at the scale of the graph learning problem formulated in \([6](https://arxiv.org/html/2608.19914#S2.E6)\)\.

## IIIWasserstein Distributionally Robust Graph Learning

This section develops the proposed distributionally robust graph learning framework in four steps\. We begin by revisiting the baseline estimator of Section[II\-B](https://arxiv.org/html/2608.19914#S2.SS2)through the lens of empirical risk minimization and robustifying it against a single source’s sampling uncertainty via a Wasserstein ambiguity set \(Section[III\-A](https://arxiv.org/html/2608.19914#S3.SS1)\)\. We then extend this single\-source construction to the heterogeneous multi\-source setting motivating this paper, replacing the naively pooled empirical distribution with a Wasserstein barycentric ambiguity set that fuses theMMsource distributions while explicitly accounting for their residual sampling and heterogeneity error \(Section[III\-B](https://arxiv.org/html/2608.19914#S3.SS2)\)\. The resulting min–max problem is then reduced to a single\-level, tractable convex program via Wasserstein strong duality \(Section[III\-C](https://arxiv.org/html/2608.19914#S3.SS3)\), which in turn admits an efficient ADMM solver with closed\-form per\-block updates \(Section[III\-D](https://arxiv.org/html/2608.19914#S3.SS4)\)\.

### III\-ABasic Formulation

The baseline problem \([6](https://arxiv.org/html/2608.19914#S2.E6)\) can be equivalently cast as an empirical risk minimization \(ERM\) problem,

min𝐋∈ℒ𝔼𝐱∼ℙn​\[−ℱn​\(𝐋∣𝐗\)\]\+ρ​𝒫​\(𝐋\),\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\quad\\mathbb\{E\}\_\{\\mathbf\{x\}\\sim\\mathbb\{P\}\_\{n\}\}\[\-\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)\]\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\),\(12\)whereℙn\\mathbb\{P\}\_\{n\}is the empirical distribution of the observed signals𝐱1,…,𝐱n\\mathbf\{x\}\_\{1\},\\ldots,\\mathbf\{x\}\_\{n\},

ℙn=1n​∑i=1nδ𝐱i,\\displaystyle\\mathbb\{P\}\_\{n\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\delta\_\{\\mathbf\{x\}\_\{i\}\},\(13\)withδ𝐱i\\delta\_\{\\mathbf\{x\}\_\{i\}\}the Dirac measure at𝐱i\\mathbf\{x\}\_\{i\}\. Formulation \([12](https://arxiv.org/html/2608.19914#S3.E12)\) makes explicit that the baseline estimator optimizes the risk exclusively with respect toℙn\\mathbb\{P\}\_\{n\}, which inherits two well\-known deficiencies from the finiteness of the sample:ℙn\\mathbb\{P\}\_\{n\}may deviate substantially from the true data\-generating distribution when observations are scarce or noisy, and an estimator tuned toℙn\\mathbb\{P\}\_\{n\}alone is prone to overfit the observed signals at the expense of generalization to unseen ones\. Both deficiencies point to the same remedy, the learned graph should remain accurate not only forℙn\\mathbb\{P\}\_\{n\}, but for every distribution plausibly consistent with the observed data\.

We formalize this requirement by replacing the single pointℙn\\mathbb\{P\}\_\{n\}with a Wasserstein ball around it,

ℳ=\{ℙ∈𝒩N:Wp​\(ℙ,ℙn\)≤ϵ\},\\displaystyle\\mathcal\{M\}=\\\{\\mathbb\{P\}\\in\\mathcal\{N\}^\{N\}:W\_\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{n\}\)\\leq\\epsilon\\\},\(14\)where𝒩N\\mathcal\{N\}^\{N\}denote the the family of all normal distributions onℝN\\mathbb\{R\}^\{N\},ϵ≥0\\epsilon\\geq 0is the ambiguity set radius andp≥1p\\geq 1is left unspecified for now, as Section[III\-C](https://arxiv.org/html/2608.19914#S3.SS3)shows that the subsequent reformulation is independent of this choice\. Learning a graph that performs well uniformly overℳ\\mathcal\{M\}amounts to minimizing the worst\-case expected risk,

min𝐋∈ℒ⁡R⁡\(𝐋,𝐱\)\+ρ​𝒫​\(𝐋\)\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}R\(\\mathbf\{L\};\\mathbf\{x\}\)\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)=min𝐋∈ℒ⁡maxℙ∈ℳ​𝔼𝐱∼ℙ​\[−ℱn​\(𝐋∣𝐗\)\]\+ρ​𝒫​\(𝐋\),\\displaystyle=\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\max\_\{\\mathbb\{P\}\\in\\mathcal\{M\}\}\\mathbb\{E\}\_\{\\mathbf\{x\}\\sim\\mathbb\{P\}\}\[\-\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)\]\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\),\(15\)whereR⁡\(𝐋,𝐱\)=maxℙ∈ℳ⁡𝔼𝐱∼ℙ​\[−ℱn​\(𝐋∣𝐗\)\]R\(\\mathbf\{L\};\\mathbf\{x\}\)=\\max\_\{\\mathbb\{P\}\\in\\mathcal\{M\}\}\\mathbb\{E\}\_\{\\mathbf\{x\}\\sim\\mathbb\{P\}\}\[\-\\mathcal\{F\}\_\{n\}\(\\mathbf\{L\}\\mid\\mathbf\{X\}\)\]is the worst\-case expected risk\. The rationale behind \([15](https://arxiv.org/html/2608.19914#S3.E15)\) is that ifℳ\\mathcal\{M\}contains the true distributionℙ⋆\\mathbb\{P\}^\{\\star\}, then driving down the worst\-case risk overℳ\\mathcal\{M\}simultaneously controls the risk atℙ⋆\\mathbb\{P\}^\{\\star\}: the expected risk of the learned graph𝐋^\\hat\{\\mathbf\{L\}\}underℙ⋆\\mathbb\{P\}^\{\\star\}is bounded byR⁡\(𝐋^,𝐱\)R\(\\hat\{\\mathbf\{L\}\};\\mathbf\{x\}\), so that𝐋^\\hat\{\\mathbf\{L\}\}remains reliable even though it was estimated from a finite, possibly unrepresentative sample\. The radiusϵ\\epsilongoverns a robustness–conservatism trade\-off intrinsic to this construction: a largerϵ\\epsilonraises the likelihood thatℳ\\mathcal\{M\}coversℙ⋆\\mathbb\{P\}^\{\\star\}, but also admits increasingly implausible distributions into the worst case, inflatingR⁡\(𝐋,𝐱\)R\(\\mathbf\{L\};\\mathbf\{x\}\)and yielding an overly conservative estimator;ϵ\\epsilonmust therefore be chosen with care, a question we return to in Section[IV](https://arxiv.org/html/2608.19914#S4)\.

### III\-BWasserstein Barycentric Distributionally Robust Optimization

The single\-source construction of Section[III\-A](https://arxiv.org/html/2608.19914#S3.SS1)guards against sampling error but offers no mechanism for incorporating data from related source domains\. We now extend it to the heterogeneous multi\-source setting that motivates this paper\. Suppose each sourcem∈\[M\]m\\in\[M\]contributes a local sample𝒟m=\{𝐳m,1,…,𝐳m,nm\}\\mathcal\{D\}\_\{m\}=\\\{\\mathbf\{z\}\_\{m,1\},\\ldots,\\mathbf\{z\}\_\{m,n\_\{m\}\}\\\}of sizenmn\_\{m\}, withn=∑m=1Mnmn=\\sum\_\{m=1\}^\{M\}n\_\{m\}, and empirical distributionℙ^m=1nm​∑i=1nmδ𝐳m,i\\hat\{\\mathbb\{P\}\}\_\{m\}=\\frac\{1\}\{n\_\{m\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\delta\_\{\\mathbf\{z\}\_\{m,i\}\}\. The most direct extension of \([12](https://arxiv.org/html/2608.19914#S3.E12)\) to this setting is the weighted ERM problem

min⁡∑m=1M𝐋∈ℒ⁡λmnm​∑i=1nmℓ⁡\(𝐋,𝐳m,i\)\.\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\sum\_\{m=1\}^\{M\}\\frac\{\\lambda\_\{m\}\}\{n\_\{m\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\ell\(\\mathbf\{L\},\\mathbf\{z\}\_\{m,i\}\)\.\(16\)
Observe thatℙ𝝀\\mathbb\{P\}\_\{\\bm\{\\lambda\}\}, viewed as a mixture, is precisely the*Euclidean*barycenter ofℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}in the space of probability measures\. This observation suggests a natural generalization: replacing the Euclidean barycenter with the geometrically richer*Wasserstein*barycenterb^𝝀,p​\(ℙ1,…,ℙM\)\\hat\{b\}\_\{\\bm\{\\lambda\},p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}\)of Definition[II\.2](https://arxiv.org/html/2608.19914#S2.Thmdefinition2), so that \([16](https://arxiv.org/html/2608.19914#S3.E16)\) becomes

min𝐋∈ℒ⁡𝔼𝐳∼b^𝝀,p​\(ℙ1,…,ℙM\)​\[ℓ⁡\(𝐋,𝐳\)\],\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\mathbb\{E\}\_\{\\mathbf\{z\}\\sim\\hat\{b\}\_\{\\bm\{\\lambda\},p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}\)\}\[\\ell\(\\mathbf\{L\},\\mathbf\{z\}\)\],\(17\)with the corresponding empirical surrogate, computed from the source samples𝒟1,…,𝒟M\\mathcal\{D\}\_\{1\},\\ldots,\\mathcal\{D\}\_\{M\}via their empirical distributionsℙ^1,…,ℙ^M\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\}, given by

min𝐋∈ℒ⁡𝔼𝐳∼b^𝝀,p​\(ℙ^1,…,ℙ^M\)​\[ℓ⁡\(𝐋,𝐳\)\]\.\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\mathbb\{E\}\_\{\\mathbf\{z\}\\sim\\hat\{b\}\_\{\\bm\{\\lambda\},p\}\(\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\}\)\}\[\\ell\(\\mathbf\{L\},\\mathbf\{z\}\)\]\.\(18\)Unlike the mixtureℙ^𝝀\\hat\{\\mathbb\{P\}\}\_\{\\bm\{\\lambda\}\}, the Wasserstein barycenter of \([18](https://arxiv.org/html/2608.19914#S3.E18)\) respects the geometry of each source distribution rather than simply averaging their probability mass, and consequently provides a more faithful consensus representation of heterogeneous sources \(a claim made precise in Section[IV\-B](https://arxiv.org/html/2608.19914#S4.SS2)\)\. This advantage notwithstanding, the barycenter of the*empirical*distributionsℙ^1,…,ℙ^M\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\}remains only a finite\-sample estimate of the barycenter of the true source distributionsℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}, and residual discrepancy between the two is unavoidable due to sampling error within each source and heterogeneity across sources\. Exactly as in Section[III\-A](https://arxiv.org/html/2608.19914#S3.SS1), we hedge against this residual uncertainty by enclosing the empirical barycenter in a Wasserstein ball rather than treating it as exact\.

###### Definition III\.1

For𝛌∈△M\\bm\{\\lambda\}\\in\\triangle^\{M\}, thepp\-Wasserstein barycentric ambiguity set of radiusϵ≥0\\epsilon\\geq 0, centered at the Wasserstein barycenter ofℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}\(assumed to exist\)𝐏¯𝛌,p=b^𝛌,p​\(ℙ1,…,ℙM\)\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}=\\hat\{b\}\_\{\\bm\{\\lambda\},p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}\), is

𝒲ϵ,p​\(ℙ1,…,ℙM,𝝀\):=\{ℙ:Wp​\(ℙ,𝐏¯𝝀,p\)≤ϵ\}\.\\displaystyle\\mathcal\{W\}\_\{\\epsilon,p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\};\\bm\{\\lambda\}\):=\\\{\\mathbb\{P\}:W\_\{p\}\(\\mathbb\{P\},\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}\)\\leq\\epsilon\\\}\.\(19\)

###### Theorem III\.1

For𝛌∈△M\\bm\{\\lambda\}\\in\\triangle^\{M\}, suppose a𝛌\\bm\{\\lambda\}\-weightedpp\-Wasserstein barycenter ofℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}exists\. Then, for anyϵ≥0\\epsilon\\geq 0,

ℳ~ϵ,p​\(ℙ1,…,ℙM,𝝀\)⊆𝒲2p⋅ϵ,p​\(ℙ1,…,ℙM,𝝀\)\.\\displaystyle\\tilde\{\\mathcal\{M\}\}\_\{\\epsilon,p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\};\\bm\{\\lambda\}\)\\subseteq\\mathcal\{W\}\_\{2^\{p\}\\cdot\\epsilon,p\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\};\\bm\{\\lambda\}\)\.\(20\)

Proof\. Let𝐏¯𝝀,p\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}denote the Wasserstein barycenter ofℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}for𝝀∈△M\\bm\{\\lambda\}\\in\\triangle^\{M\}\. By the triangle inequality and the inequality\(a\+b\)p≤2p−1​\(ap\+bp\)\(a\+b\)^\{p\}\\leq 2^\{p\-1\}\(a^\{p\}\+b^\{p\}\)fora,b≥0a,b\\geq 0andp∈\[1,∞\)p\\in\[1,\\infty\), for anyℙ\\mathbb\{P\}andm∈\[M\]m\\in\[M\],

Wpp​\(ℙ,𝐏¯𝝀,p\)\\displaystyle W\_\{p\}^\{p\}\(\\mathbb\{P\},\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}\)≤\(Wp​\(ℙ,ℙm\)\+Wp​\(𝐏¯𝝀,p,ℙm\)\)p\\displaystyle\\leq\\bigl\(W\_\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)\+W\_\{p\}\(\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\},\\mathbb\{P\}\_\{m\}\)\\bigr\)^\{p\}≤2p−1​\(Wpp​\(ℙ,ℙm\)\+Wpp​\(𝐏¯𝝀,p,ℙm\)\)\.\\displaystyle\\leq 2^\{p\-1\}\\bigl\(W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)\+W\_\{p\}^\{p\}\(\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\},\\mathbb\{P\}\_\{m\}\)\\bigr\)\.Taking the𝝀\\bm\{\\lambda\}\-weighted average overm∈\[M\]m\\in\[M\]and using the definition of the Wasserstein barycenter to bound∑m=1Mλm​Wpp​\(𝐏¯𝝀,p,ℙm\)≤∑m=1MWpp​\(ℙ,ℙm\)\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{p\}^\{p\}\(\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\},\\mathbb\{P\}\_\{m\}\)\\leq\\sum\_\{m=1\}^\{M\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)yields

Wpp​\(ℙ,𝐏¯𝝀,p\)=∑m=1Mλm​Wpp​\(ℙ,𝐏¯𝝀,p\)\\displaystyle W\_\{p\}^\{p\}\(\\mathbb\{P\},\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}\)=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}\)≤2p−1​∑m=1Mλm​Wpp​\(ℙ,ℙk\)\+2p−1​∑m=1MWpp​\(ℙ,ℙm\)\\displaystyle\\leq 2^\{p\-1\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{k\}\)\+2^\{p\-1\}\\sum\_\{m=1\}^\{M\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)=2p​∑m=1Mλm​Wpp​\(ℙ,ℙm\)\.\\displaystyle=2^\{p\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)\.Consequently, for anyϵ≥0\\epsilon\\geq 0,

∑m=1Mλm​Wpp​\(ℙ,ℙm\)≤ϵ⟹Wpp​\(ℙ,𝐏¯𝝀,p\)≤2p⋅ϵ,\\displaystyle\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{p\}^\{p\}\(\\mathbb\{P\},\\mathbb\{P\}\_\{m\}\)\\leq\\epsilon\\;\\Longrightarrow\\;W\_\{p\}^\{p\}\(\\mathbb\{P\},\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}\)\\leq 2^\{p\}\\cdot\\epsilon,which is the desired inclusion\.□\\square

With the barycentric ambiguity set in hand, the multi\-source counterpart of the min–max problem \([15](https://arxiv.org/html/2608.19914#S3.E15)\) reads

inf𝐋∈ℒ\{−log⁡\|𝐋\|\+\+supℙ∈𝒲ϵ,p𝔼𝐳∼ℙ​\[tr⁡\(Σ​𝐋\)\]\+ρ​𝒫​\(𝐋\)\}\.\\displaystyle\\inf\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\Bigl\\\{\-\\log\|\\mathbf\{L\}\|\_\{\+\}\+\\sup\_\{\\mathbb\{P\}\\in\\mathcal\{W\}\_\{\\epsilon,p\}\}\\mathbb\{E\}\_\{\\mathbf\{z\}\\sim\\mathbb\{P\}\}\[\\mathrm\{tr\}\(\\Sigma\\mathbf\{L\}\)\]\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)\\Bigr\\\}\.\(21\)Evaluating \([21](https://arxiv.org/html/2608.19914#S3.E21)\) in practice requires an explicit handle on the center of the ambiguity set,𝐏¯𝝀,p\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},p\}, which we now characterize under the Gaussian source model implied by the signal model of Section[II\-A](https://arxiv.org/html/2608.19914#S2.SS1)\.

###### Assumption III\.1

ℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}areMMpossibly degenerate Gaussian distributionsℙm=𝒩⁡\(μm,Σm\)\\mathbb\{P\}\_\{m\}=\\mathcal\{N\}\(\\mu\_\{m\},\\Sigma\_\{m\}\)withμm∈ℝN\\mu\_\{m\}\\in\\mathbb\{R\}^\{N\}andΣm∈𝕊\+N\\Sigma\_\{m\}\\in\\mathbb\{S\}\_\{\+\}^\{N\}form∈\[M\]m\\in\[M\]\.

###### Proposition III\.1

Under Assumption[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1), the22\-Wasserstein barycenter ofℙ1,…,ℙM\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}, restricted to the Gaussian manifold, is𝐏¯𝛌,2=b^𝛌,2​\(ℙ1,…,ℙM\)=𝒩⁡\(μ¯𝛌,Σ¯𝛌\)\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\}=\\hat\{b\}\_\{\\bm\{\\lambda\},2\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\}\)=\\mathcal\{N\}\(\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\},\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\), with

μ¯𝝀\\displaystyle\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}=∑m=1Mλm​μm,\\displaystyle=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\mu\_\{m\},Σ¯𝝀\\displaystyle\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}=arg​minΣ∈𝕊\+N∑m=1MλmB2\(Σ,Σm\),\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\Sigma\\in\\mathbb\{S\}\_\{\+\}^\{N\}\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}B^\{2\}\(\\Sigma,\\Sigma\_\{m\}\),\(22\)whereB⁡\(⋅,⋅\)B\(\\cdot,\\cdot\)is the Bures–Wasserstein distance of Definition[II\.3](https://arxiv.org/html/2608.19914#S2.Thmdefinition3)\(see \([11](https://arxiv.org/html/2608.19914#S2.E11)\)\)\. Moreover, if all source covariances share a common null space,⋂m=1Mnull⁡\(Σm\)=:𝒦≠\{𝟎\}\\bigcap\_\{m=1\}^\{M\}\\mathrm\{null\}\(\\Sigma\_\{m\}\)=:\\mathcal\{K\}\\neq\\\{\\mathbf\{0\}\\\}\(in particular𝒦⊇span​\{𝟏\}\\mathcal\{K\}\\supseteq\\mathrm\{span\}\\\{\\mathbf\{1\}\\\}under Remark[III\.3](https://arxiv.org/html/2608.19914#S3.Thmremark3)\), thenΣ¯𝛌\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}inherits this null space,𝒦⊆null⁡\(Σ¯𝛌\)\\mathcal\{K\}\\subseteq\\mathrm\{null\}\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\), soΣ¯𝛌\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}is itself rank\-deficient and, in particular, is never positive definite whenever𝒦≠\{𝟎\}\\mathcal\{K\}\\neq\\\{\\mathbf\{0\}\\\}\.

Proof\.Existence and uniqueness of the minimizer in \([22](https://arxiv.org/html/2608.19914#S3.E22)\) on the closed cone𝕊\+N\\mathbb\{S\}\_\{\+\}^\{N\}follow from\[[38](https://arxiv.org/html/2608.19914#bib.bib38), Thm\. 1\]\. Let𝐯∈𝒦\\mathbf\{v\}\\in\\mathcal\{K\}; sinceΣm​𝐯=𝟎\\Sigma\_\{m\}\\mathbf\{v\}=\\mathbf\{0\}for everymm, we haveΣm1/2​𝐯=𝟎\\Sigma\_\{m\}^\{1/2\}\\mathbf\{v\}=\\mathbf\{0\}, and hence𝐯T​Σ​𝐯=0\\mathbf\{v\}^\{T\}\\Sigma\\mathbf\{v\}=0for the fixed pointΣ=Σ¯𝝀\\Sigma=\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}of

Σ=∑m=1Mλm​Σ1/2​Σm​Σ1/2\\displaystyle\\Sigma=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\sqrt\{\\Sigma^\{1/2\}\\Sigma\_\{m\}\\Sigma^\{1/2\}\}\(23\)\(each summand on the right annihilates𝐯\\mathbf\{v\}on both sides by the same argument, so𝐯\\mathbf\{v\}remains in the null space at every iterate, and hence at the fixed point\), which, sinceΣ¯𝝀⪰0\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\succeq 0, forcesΣ¯𝝀​𝐯=𝟎\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{v\}=\\mathbf\{0\}\.□\\hfill\\square

Because Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)showsΣ¯𝝀​𝟏=𝟎\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{1\}=\\mathbf\{0\}, the fixed\-point map \([23](https://arxiv.org/html/2608.19914#S3.E23)\) is ill\-posed if applied verbatim on𝕊\+⁣\+N\\mathbb\{S\}\_\{\+\+\}^\{N\}, since it requiresΣ−1/2\\Sigma^\{\-1/2\}at a singular matrix\. We instead solve \([23](https://arxiv.org/html/2608.19914#S3.E23)\) in the\(N−1\)\(N\-1\)\-dimensional subspace𝟏⟂\\mathbf\{1\}^\{\\perp\}, where positive definiteness is generically restored\. Let𝐔⟂∈ℝN×\(N−1\)\\mathbf\{U\}\_\{\\perp\}\\in\\mathbb\{R\}^\{N\\times\(N\-1\)\}be an orthonormal basis of𝟏⟂\\mathbf\{1\}^\{\\perp\}and define the reduced source covariancesΣm⟂:=𝐔⟂T​Σm​𝐔⟂∈𝕊\+N−1\\Sigma\_\{m\}^\{\\perp\}:=\\mathbf\{U\}\_\{\\perp\}^\{T\}\\Sigma\_\{m\}\\mathbf\{U\}\_\{\\perp\}\\in\\mathbb\{S\}\_\{\+\}^\{N\-1\}, which are generically positive definite whenever each source graph is connected \(so thatnull⁡\(𝐋m\)=span​\{𝟏\}\\mathrm\{null\}\(\\mathbf\{L\}\_\{m\}\)=\\mathrm\{span\}\\\{\\mathbf\{1\}\\\}exactly\)\. The reduced barycenterΣ¯𝝀⟂∈𝕊\+⁣\+N−1\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\\in\\mathbb\{S\}\_\{\+\+\}^\{N\-1\}is then computed by the well\-posed fixed\-point iteration

\(Σ¯𝝀⟂\)k\+1=ℋ⁡\(\(Σ¯𝝀⟂\)k,𝝀,\{Σm⟂\}m=1M\)\\displaystyle\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\+1\}=\\mathcal\{H\}\\bigl\(\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\},\\bm\{\\lambda\},\\\{\\Sigma\_\{m\}^\{\\perp\}\\\}\_\{m=1\}^\{M\}\\bigr\)=\(\(Σ¯𝝀⟂\)k\)−1/2\(∑m=1Mλm\(\(Σ¯𝝀⟂\)k\)1/2​Σm⟂​\(\(Σ¯𝝀⟂\)k\)1/2\)2\\displaystyle=\\bigl\(\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\}\\bigr\)^\{\-1/2\}\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\sqrt\{\(\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\}\)^\{1/2\}\\Sigma\_\{m\}^\{\\perp\}\(\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\}\)^\{1/2\}\}\\Bigr\)^\{2\}⋅\(\(Σ¯𝝀⟂\)k\)−1/2,\\displaystyle\\quad\\cdot\\bigl\(\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\)^\{k\}\\bigr\)^\{\-1/2\},\(24\)which converges at a linear rate to the unique positive definite fixed pointΣ¯𝝀⟂\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}in𝕊\+⁣\+N−1\\mathbb\{S\}\_\{\+\+\}^\{N\-1\}\[[38](https://arxiv.org/html/2608.19914#bib.bib38),[49](https://arxiv.org/html/2608.19914#bib.bib49)\]\. The ambient barycenter covariance is then recovered exactly, and with the correct rank and null space, by the lifting

Σ¯𝝀=𝐔⟂​Σ¯𝝀⟂​𝐔⟂T∈𝕊\+N,Σ¯𝝀​𝟏=𝟎\.\\displaystyle\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}=\\mathbf\{U\}\_\{\\perp\}\\,\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}^\{\\perp\}\\,\\mathbf\{U\}\_\{\\perp\}^\{T\}\\in\\mathbb\{S\}\_\{\+\}^\{N\},\\qquad\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{1\}=\\mathbf\{0\}\.\(25\)
###### Lemma III\.1

For every𝐋∈ℒ\\mathbf\{L\}\\in\\mathcal\{L\}and everyΣ∈𝕊N\\Sigma\\in\\mathbb\{S\}^\{N\},

tr⁡\(Σ​𝐋\)=tr⁡\(𝐏​Σ​𝐏​𝐋\),𝐏:=𝐈−1N​𝟏𝟏T\.\\displaystyle\\mathrm\{tr\}\(\\Sigma\\mathbf\{L\}\)=\\mathrm\{tr\}\(\\mathbf\{P\}\\Sigma\\mathbf\{P\}\\,\\mathbf\{L\}\),\\qquad\\mathbf\{P\}:=\\mathbf\{I\}\-\\tfrac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\.\(26\)

Proof\.Since𝐋𝟏=𝟎\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}and𝐋=𝐋T\\mathbf\{L\}=\\mathbf\{L\}^\{T\}, we have𝐋𝟏𝟏T=𝟎\\mathbf\{L\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}=\\mathbf\{0\}and𝟏𝟏T​𝐋=\(𝐋𝟏𝟏T\)T=𝟎\\mathbf\{1\}\\mathbf\{1\}^\{T\}\\mathbf\{L\}=\(\\mathbf\{L\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\)^\{T\}=\\mathbf\{0\}, so𝐏𝐋=𝐋𝐏=𝐋\\mathbf\{P\}\\mathbf\{L\}=\\mathbf\{L\}\\mathbf\{P\}=\\mathbf\{L\}, i\.e\.𝐋=𝐏𝐋𝐏\\mathbf\{L\}=\\mathbf\{P\}\\mathbf\{L\}\\mathbf\{P\}\. Then, by cyclicity of the trace,tr⁡\(Σ​𝐋\)=tr⁡\(Σ​𝐏𝐋𝐏\)=tr⁡\(𝐏​Σ​𝐏​𝐋\)\\mathrm\{tr\}\(\\Sigma\\mathbf\{L\}\)=\\mathrm\{tr\}\(\\Sigma\\,\\mathbf\{P\}\\mathbf\{L\}\\mathbf\{P\}\)=\\mathrm\{tr\}\(\\mathbf\{P\}\\Sigma\\mathbf\{P\}\\,\\mathbf\{L\}\)\.□\\hfill\\square

###### Assumption III\.2

There exists a constantR<∞R<\\inftysuch that‖𝐳‖2≤R\\\|\\mathbf\{z\}\\\|\_\{2\}\\leq Ralmost surely under every source distributionℙm\\mathbb\{P\}\_\{m\},m∈\[M\]m\\in\[M\], and under every distributionℙ\\mathbb\{P\}contained in the vector\-space ambiguity set𝒲ϵ,2​\(ℙ1,…,ℙM,𝛌\)\\mathcal\{W\}\_\{\\epsilon,2\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\};\\bm\{\\lambda\}\)of \([19](https://arxiv.org/html/2608.19914#S3.E19)\)\.

###### Lemma III\.2

LetΦ:ℝN→𝕊N×N\\Phi:\\mathbb\{R\}^\{N\}\\to\\mathbb\{S\}^\{N\\times N\},Φ⁡\(𝐳\):=𝐳𝐳T\\Phi\(\\mathbf\{z\}\):=\\mathbf\{z\}\\mathbf\{z\}^\{T\}\. Under Assumption[III\.2](https://arxiv.org/html/2608.19914#S3.Thmassumption2),Φ\\Phiis2​R2R\-Lipschitz from\(\{∥𝐳∥2≤R\},∥⋅∥2\)\(\\\{\\\|\\mathbf\{z\}\\\|\_\{2\}\\leq R\\\},\\\|\\cdot\\\|\_\{2\}\)to\(𝕊N×N,∥⋅∥F\)\(\\mathbb\{S\}^\{N\\times N\},\\\|\\cdot\\\|\_\{F\}\), and consequently, for any two distributionsℙ,ℙ′\\mathbb\{P\},\\mathbb\{P\}^\{\\prime\}supported on\{∥𝐳∥2≤R\}\\\{\\\|\\mathbf\{z\}\\\|\_\{2\}\\leq R\\\},

W2\(Φ\#ℙ,Φ\#ℙ′;∥⋅∥F\)≤2RW2\(ℙ,ℙ′;∥⋅∥2\)\.\\displaystyle W\_\{2\}\\bigl\(\\Phi\_\{\\\#\}\\mathbb\{P\},\\Phi\_\{\\\#\}\\mathbb\{P\}^\{\\prime\};\\\|\\cdot\\\|\_\{F\}\\bigr\)\\leq 2R\\,W\_\{2\}\(\\mathbb\{P\},\\mathbb\{P\}^\{\\prime\};\\\|\\cdot\\\|\_\{2\}\)\.\(27\)

Proof\.For‖𝐳1‖2,‖𝐳2‖2≤R\\\|\\mathbf\{z\}\_\{1\}\\\|\_\{2\},\\\|\\mathbf\{z\}\_\{2\}\\\|\_\{2\}\\leq R,

‖Φ⁡\(𝐳1\)−Φ⁡\(𝐳2\)‖F=‖𝐳1​\(𝐳1−𝐳2\)T\+\(𝐳1−𝐳2\)​𝐳2T‖F\\displaystyle\\\|\\Phi\(\\mathbf\{z\}\_\{1\}\)\-\\Phi\(\\mathbf\{z\}\_\{2\}\)\\\|\_\{F\}=\\bigl\\\|\\mathbf\{z\}\_\{1\}\(\\mathbf\{z\}\_\{1\}\-\\mathbf\{z\}\_\{2\}\)^\{T\}\+\(\\mathbf\{z\}\_\{1\}\-\\mathbf\{z\}\_\{2\}\)\\mathbf\{z\}\_\{2\}^\{T\}\\bigr\\\|\_\{F\}≤\(‖𝐳1‖2\+‖𝐳2‖2\)​‖𝐳1−𝐳2‖2≤2​R​‖𝐳1−𝐳2‖2,\\displaystyle\\leq\\bigl\(\\\|\\mathbf\{z\}\_\{1\}\\\|\_\{2\}\+\\\|\\mathbf\{z\}\_\{2\}\\\|\_\{2\}\\bigr\)\\\|\\mathbf\{z\}\_\{1\}\-\\mathbf\{z\}\_\{2\}\\\|\_\{2\}\\leq 2R\\\|\\mathbf\{z\}\_\{1\}\-\\mathbf\{z\}\_\{2\}\\\|\_\{2\},soΦ\\Phiis2​R2R\-Lipschitz on\{∥𝐳∥2≤R\}\\\{\\\|\\mathbf\{z\}\\\|\_\{2\}\\leq R\\\}\. Letπ⋆\\pi^\{\\star\}be an optimal coupling attainingW2​\(ℙ,ℙ′\)W\_\{2\}\(\\mathbb\{P\},\\mathbb\{P\}^\{\\prime\}\)\. Then\(Φ,Φ\)\#​π⋆\(\\Phi,\\Phi\)\_\{\\\#\}\\pi^\{\\star\}is a feasible coupling ofΦ\#​ℙ\\Phi\_\{\\\#\}\\mathbb\{P\}andΦ\#​ℙ′\\Phi\_\{\\\#\}\\mathbb\{P\}^\{\\prime\}, and

W22​\(Φ\#​ℙ,Φ\#​ℙ′\)≤∫‖Φ⁡\(𝐳1\)−Φ⁡\(𝐳2\)‖F2​d​π⋆​\(𝐳1,𝐳2\)\\displaystyle W\_\{2\}^\{2\}\(\\Phi\_\{\\\#\}\\mathbb\{P\},\\Phi\_\{\\\#\}\\mathbb\{P\}^\{\\prime\}\)\\leq\\int\\\|\\Phi\(\\mathbf\{z\}\_\{1\}\)\-\\Phi\(\\mathbf\{z\}\_\{2\}\)\\\|\_\{F\}^\{2\}\\,d\\pi^\{\\star\}\(\\mathbf\{z\}\_\{1\},\\mathbf\{z\}\_\{2\}\)≤4​R2​∫‖𝐳1−𝐳2‖22​d​π⋆​\(𝐳1,𝐳2\)=4​R2​W22​\(ℙ,ℙ′\),\\displaystyle\\leq 4R^\{2\}\\int\\\|\\mathbf\{z\}\_\{1\}\-\\mathbf\{z\}\_\{2\}\\\|\_\{2\}^\{2\}\\,d\\pi^\{\\star\}\(\\mathbf\{z\}\_\{1\},\\mathbf\{z\}\_\{2\}\)=4R^\{2\}W\_\{2\}^\{2\}\(\\mathbb\{P\},\\mathbb\{P\}^\{\\prime\}\),which gives \([27](https://arxiv.org/html/2608.19914#S3.E27)\)\.□\\hfill\\square

###### Corollary III\.1

Under Assumption[III\.2](https://arxiv.org/html/2608.19914#S3.Thmassumption2), the image of the vector\-space ambiguity set𝒲ϵ,2\\mathcal\{W\}\_\{\\epsilon,2\}of \([19](https://arxiv.org/html/2608.19914#S3.E19)\) under the outer\-product map satisfies

Φ\#​𝒲ϵ,2​\(ℙ1,…,ℙM,𝝀\)\\displaystyle\\Phi\_\{\\\#\}\\mathcal\{W\}\_\{\\epsilon,2\}\(\\mathbb\{P\}\_\{1\},\\ldots,\\mathbb\{P\}\_\{M\};\\bm\{\\lambda\}\)⊆\{ℚon𝕊N×N:W2\(ℚ,Φ\#𝐏¯𝝀,2;∥⋅∥F\)≤2Rϵ\}\.\\displaystyle\\subseteq\\bigl\\\{\\mathbb\{Q\}\\ \\text\{on\}\\ \\mathbb\{S\}^\{N\\times N\}:W\_\{2\}\(\\mathbb\{Q\},\\Phi\_\{\\\#\}\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\};\\\|\\cdot\\\|\_\{F\}\)\\leq 2R\\epsilon\\bigr\\\}\.\(28\)

Proof\.Immediate from Lemma[III\.2](https://arxiv.org/html/2608.19914#S3.Thmlemma2)applied withℙ′=𝐏¯𝝀,2\\mathbb\{P\}^\{\\prime\}=\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\}to everyℙ∈𝒲ϵ,2\\mathbb\{P\}\\in\\mathcal\{W\}\_\{\\epsilon,2\}\.□\\hfill\\square

Corollary[III\.1](https://arxiv.org/html/2608.19914#S3.Thmcorollary1)is the precise sense in which the covariance\-space representation of Section[III\-C](https://arxiv.org/html/2608.19914#S3.SS3)is compatible with the vector\-space ambiguity set of Section[III\-B](https://arxiv.org/html/2608.19914#S3.SS2): the pushforward mapΦ\\Phiis not an isometry, so the radius is*not*preserved verbatim, but it is provably inflated by the fixed, computable factor2​R2R\. We use this containment, rather than an unproven identity of the two spaces, to derive a tractable and provably valid upper bound on the true worst\-case risk in Section[III\-C](https://arxiv.org/html/2608.19914#S3.SS3)\.

### III\-CTractable Reformulation

Problem \([21](https://arxiv.org/html/2608.19914#S3.E21)\) appears intractable at first sight, since its inner supremum lacks a closed form\. Tractability is restored by lifting the problem into the space of rank\-one covariance matrices via the outer\-product mapΦ⁡\(𝐳\):=𝐳𝐳T∈𝕊N×N\\Phi\(\\mathbf\{z\}\):=\\mathbf\{z\}\\mathbf\{z\}^\{T\}\\in\\mathbb\{S\}^\{N\\times N\}, under whichtr⁡\(Σ​𝐋\)=𝐳T​𝐋𝐳\\mathrm\{tr\}\(\\Sigma\\mathbf\{L\}\)=\\mathbf\{z\}^\{T\}\\mathbf\{L\}\\mathbf\{z\}forΣ=Φ⁡\(𝐳\)\\Sigma=\\Phi\(\\mathbf\{z\}\), so that the loss appearing in \([15](https://arxiv.org/html/2608.19914#S3.E15)\) is*linear*inΣ\\Sigma\. By Corollary[III\.1](https://arxiv.org/html/2608.19914#S3.Thmcorollary1), the worst\-case risk over the vector\-space ambiguity set𝒲ϵ,2\\mathcal\{W\}\_\{\\epsilon,2\}is upper\-bounded by the worst\-case risk over the \(larger, and hence tractable via standard linear\-loss duality\) matrix\-space ball of radiusϵ′:=2​R​ϵ\\epsilon^\{\\prime\}:=2R\\epsiloncentered atΦ\#​𝐏¯𝝀,2\\Phi\_\{\\\#\}\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\}:

supℙ∈𝒲ϵ,2𝔼𝐳∼ℙ​\[𝐳T​𝐋𝐳\]≤supℚ∈𝒲𝝀,2Σ​\(ϵ′\)𝔼𝚺∼ℚ​\[tr⁡\(𝚺​𝐋\)\],\\displaystyle\\sup\_\{\\mathbb\{P\}\\in\\mathcal\{W\}\_\{\\epsilon,2\}\}\\mathbb\{E\}\_\{\\mathbf\{z\}\\sim\\mathbb\{P\}\}\[\\mathbf\{z\}^\{T\}\\mathbf\{L\}\\mathbf\{z\}\]\\leq\\sup\_\{\\mathbb\{Q\}\\in\\mathcal\{W\}^\{\\Sigma\}\_\{\\bm\{\\lambda\},2\}\(\\epsilon^\{\\prime\}\)\}\\mathbb\{E\}\_\{\\bm\{\\Sigma\}\\sim\\mathbb\{Q\}\}\[\\mathrm\{tr\}\(\\bm\{\\Sigma\}\\mathbf\{L\}\)\],\(29\)where𝒲𝝀,2Σ\(ϵ′\):=\{ℚ:W2\(ℚ,Φ\#𝐏¯𝝀,2;∥⋅∥F\)≤ϵ′\}\\mathcal\{W\}^\{\\Sigma\}\_\{\\bm\{\\lambda\},2\}\(\\epsilon^\{\\prime\}\):=\\\{\\mathbb\{Q\}:W\_\{2\}\(\\mathbb\{Q\},\\Phi\_\{\\\#\}\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\};\\\|\\cdot\\\|\_\{F\}\)\\leq\\epsilon^\{\\prime\}\\\}\. Crucially, the two Wasserstein balls in \([29](https://arxiv.org/html/2608.19914#S3.E29)\) live in*different, but explicitly related*ground spaces,𝒫2​\(ℝN\)\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{N\}\)on the left, and the image of𝒫2​\(ℝN\)\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{N\}\)under the2​R2R\-Lipschitz mapΦ\\Phiinside𝒫2​\(𝕊N×N\)\\mathcal\{P\}\_\{2\}\(\\mathbb\{S\}^\{N\\times N\}\)on the right, and inequality \([29](https://arxiv.org/html/2608.19914#S3.E29)\), rather than an unjustified identity, is what licenses working in covariance space\. Replacing the true worst\-case risk by its tractable upper bound in \([29](https://arxiv.org/html/2608.19914#S3.E29)\) preserves the defining property of distributional robustness \(the learned𝐋\\mathbf\{L\}still controls risk under everyℙ∈𝒲ϵ,2\\mathbb\{P\}\\in\\mathcal\{W\}\_\{\\epsilon,2\}, if anything more conservatively\), so the resulting estimator remains valid; we henceforth analyze the right\-hand side of \([29](https://arxiv.org/html/2608.19914#S3.E29)\) in place of the original min–max problem \([15](https://arxiv.org/html/2608.19914#S3.E15)\)\.

min𝐋∈ℒ⁡𝔼𝚺∼ℚ^m​\[−log⁡\(\|𝐋\|\+\)\+tr⁡\(𝚺​𝐋\)\]\+ρ​𝒫​\(𝐋\),\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\mathbb\{E\}\_\{\\bm\{\\Sigma\}\\sim\\hat\{\\mathbb\{Q\}\}\_\{m\}\}\[\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\mathrm\{tr\}\(\\bm\{\\Sigma\}\\mathbf\{L\}\)\]\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\),\(30\)whereℚ^m=1nm​∑i=1nmDirac⁡\(𝐳m,i​𝐳m,iT\)=1nm​∑i=1nmDirac⁡\(𝚺mi\)\\hat\{\\mathbb\{Q\}\}\_\{m\}=\\frac\{1\}\{n\_\{m\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\mathrm\{Dirac\}\(\\mathbf\{z\}\_\{m,i\}\\mathbf\{z\}\_\{m,i\}^\{T\}\)=\\frac\{1\}\{n\_\{m\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\mathrm\{Dirac\}\(\\bm\{\\Sigma\}\_\{m\}^\{i\}\)\. In this covariance representation, the min–max problem \([15](https://arxiv.org/html/2608.19914#S3.E15)\) becomes

min𝐋∈ℒ⁡R⁡\(𝐋,𝚺\)\+ρ​𝒫​\(𝐋\)\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}R\(\\mathbf\{L\};\\bm\{\\Sigma\}\)\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)=min𝐋∈ℒ⁡maxℙ∈𝒲𝝀,2​𝔼𝚺∼ℙ​\[−log⁡\(\|𝐋\|\+\)\+tr⁡\(𝚺​𝐋\)\]\+ρ​𝒫​\(𝐋\)\.\\displaystyle=\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\\max\_\{\\mathbb\{P\}\\in\\mathcal\{W\}\_\{\\bm\{\\lambda\},2\}\}\\mathbb\{E\}\_\{\\bm\{\\Sigma\}\\sim\\mathbb\{P\}\}\[\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\mathrm\{tr\}\(\\bm\{\\Sigma\}\\mathbf\{L\}\)\]\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)\.\(31\)The following lemma shows that the inner supremum in \([31](https://arxiv.org/html/2608.19914#S3.E31)\) admits a closed\-form dual, collapsing the two\-level min–max problem into a single\-level convex program\.

###### Lemma III\.3

Under Assumption[III\.2](https://arxiv.org/html/2608.19914#S3.Thmassumption2), the min–max problem \([15](https://arxiv.org/html/2608.19914#S3.E15)\) is upper\-bounded by

min𝐋∈ℒ⁡R⁡\(𝐋,Σ¯𝝀\)\+ρ​𝒫​\(𝐋\)\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}R\(\\mathbf\{L\};\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)≤min𝐋∈ℒ−log⁡\(\|𝐋\|\+\)\+tr⁡\(Σ¯𝝀​𝐋\)\+ρ​𝒫​\(𝐋\)\+2​R​ϵ​‖vec⁡\(𝐋\)‖2,\\displaystyle\\leq\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\mathrm\{tr\}\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{L\}\)\+\\rho\\mathcal\{P\}\(\\mathbf\{L\}\)\+2R\\epsilon\\,\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{2\},\(32\)whereΣ¯𝛌\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}is the barycenter covariance of Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)andRRis the energy bound of Assumption[III\.2](https://arxiv.org/html/2608.19914#S3.Thmassumption2)\. Inequality \([32](https://arxiv.org/html/2608.19914#S3.E32)\) is tight, i\.e\. holds with equality, wheneverΦ\#​𝐏¯𝛌,2\\Phi\_\{\\\#\}\\bar\{\\mathbf\{P\}\}\_\{\\bm\{\\lambda\},2\}is itself the empirical distribution of a single point mass, which is the case for the empirical surrogate used in Section[III\-D](https://arxiv.org/html/2608.19914#S3.SS4)\.

Lemma[III\.3](https://arxiv.org/html/2608.19914#S3.Thmlemma3)shows that hedging against distributional uncertainty in \([31](https://arxiv.org/html/2608.19914#S3.E31)\) costs, up to the provable safety margin2​R2R, one additional term in the objective, the norm penaltyϵ​‖vec⁡\(𝐋\)‖q\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}, so that robustness is achieved without sacrificing the convexity or the closed\-form structure of the original regularized maximum\-likelihood problem \([6](https://arxiv.org/html/2608.19914#S2.E6)\)\. The remainder of this section develops an efficient ADMM solver for \([32](https://arxiv.org/html/2608.19914#S3.E32)\)\.

### III\-DAlgorithmic Development

We now derive an ADMM solver for the tractable reformulation \([32](https://arxiv.org/html/2608.19914#S3.E32)\)\. Applying Lemma[III\.3](https://arxiv.org/html/2608.19914#S3.Thmlemma3), the problem can be written explicitly as \(For notational simplicity, we defineϵ≜2​R​ϵ\\epsilon\\triangleq 2R\\epsilonthroughout the remainder of the paper\)

min𝐋∈ℒ−log⁡\(\|𝐋\|\+\)\+tr⁡\(Σ^𝝀​𝐋\)\+ρ​‖vec⁡\(𝐋\)‖1\+ϵ​‖vec⁡\(𝐋\)‖2,\\displaystyle\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\}\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\mathrm\{tr\}\(\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{L\}\)\+\\rho\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{1\}\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{2\},\(33\)whereΣ^𝝀\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}is the empirical counterpart of the barycenter covariance, in accordance with Remark[III\.4](https://arxiv.org/html/2608.19914#S3.Thmremark4), each empirical source covariance is projected asΣ^m←𝐏​Σ^m​𝐏\\hat\{\\Sigma\}\_\{m\}\\leftarrow\\mathbf\{P\}\\hat\{\\Sigma\}\_\{m\}\\mathbf\{P\}prior to the fixed\-point iteration \([24](https://arxiv.org/html/2608.19914#S3.E24)\), ensuringΣ^𝝀​𝟏=𝟎\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{1\}=\\mathbf\{0\}and consistency with the constraint setℒ\\mathcal\{L\}\. Because any feasible𝐋∈ℒ\\mathbf\{L\}\\in\\mathcal\{L\}has non\-positive off\-diagonal entries and non\-negative diagonal entries by construction \([1](https://arxiv.org/html/2608.19914#S2.E1)\), theℓ1\\ell\_\{1\}term in \([33](https://arxiv.org/html/2608.19914#S3.E33)\) admits the linear representation

‖vec⁡\(𝐋\)‖1=tr⁡\(𝐋𝐇\),\\displaystyle\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{1\}=\\mathrm\{tr\}\(\\mathbf\{L\}\\mathbf\{H\}\),\(34\)with𝐇=2​𝐈−𝟏𝟏T\\mathbf\{H\}=2\\mathbf\{I\}\-\\mathbf\{1\}\\mathbf\{1\}^\{T\}\. Substituting \([34](https://arxiv.org/html/2608.19914#S3.E34)\) into \([33](https://arxiv.org/html/2608.19914#S3.E33)\) and collecting the two linear terms into a single effective coefficient matrix𝐊≜Σ^𝝀\+ρ​𝐇\\mathbf\{K\}\\triangleq\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\+\\rho\\mathbf\{H\}gives

−log⁡\(\|𝐋\|\+\)\+tr⁡\(Σ^𝝀​𝐋\)\+ρ​‖vec⁡\(𝐋\)‖1\+ϵ​‖vec⁡\(𝐋\)‖q\\displaystyle\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\mathrm\{tr\}\(\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{L\}\)\+\\rho\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{1\}\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}=tr⁡\(𝐋​Σ^𝝀\)−log⁡\(\|𝐋\|\+\)\+ρ​tr​\(𝐋𝐇\)\+ϵ​‖vec⁡\(𝐋\)‖q\\displaystyle\\quad=\\mathrm\{tr\}\(\\mathbf\{L\}\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\rho\\,\\mathrm\{tr\}\(\\mathbf\{L\}\\mathbf\{H\}\)\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}=tr⁡\(𝐋⁡\(Σ^𝝀\+ρ​𝐇\)\)−log⁡\(\|𝐋\|\+\)\+ϵ​‖vec⁡\(𝐋\)‖q\\displaystyle\\quad=\\mathrm\{tr\}\\bigl\(\\mathbf\{L\}\(\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\+\\rho\\mathbf\{H\}\)\\bigr\)\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}≜tr⁡\(𝐋𝐊\)−log⁡\(\|𝐋\|\+\)\+ϵ​‖vec⁡\(𝐋\)‖q\.\\displaystyle\\quad\\triangleq\\mathrm\{tr\}\(\\mathbf\{L\}\\mathbf\{K\}\)\-\\log\(\|\\mathbf\{L\}\|\_\{\+\}\)\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}\.\(35\)As it stands, however,logdet\(𝐋\)\\log\\det\(\\mathbf\{L\}\)is unbounded below onℒ\\mathcal\{L\}, since every feasible𝐋\\mathbf\{L\}satisfies𝟏T​𝐋𝟏=0\\mathbf\{1\}^\{T\}\\mathbf\{L\}\\mathbf\{1\}=0and is hence singular\. Following Egilmez et al\.\[[12](https://arxiv.org/html/2608.19914#bib.bib12)\], we resolve this degeneracy by regularizing the null direction, replacinglogdet\(𝐋\)\\log\\det\(\\mathbf\{L\}\)withlogdet\(𝐋\+𝐉\)\\log\\det\(\\mathbf\{L\}\+\\mathbf\{J\}\), where𝐉=1N​𝟏𝟏T\\mathbf\{J\}=\\frac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}:

min𝐋tr\(𝐋𝐊\)−logdet\(𝐋\+𝐉\)\+ϵ∥vec\(𝐋\)∥q\.\\displaystyle\\min\_\{\\mathbf\{L\}\}\\;\\mathrm\{tr\}\(\\mathbf\{L\}\\mathbf\{K\}\)\-\\log\\det\(\\mathbf\{L\}\+\\mathbf\{J\}\)\+\\epsilon\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}\.\(36\)
###### Proposition III\.2

Problems \([35](https://arxiv.org/html/2608.19914#S3.E35)\) and \([36](https://arxiv.org/html/2608.19914#S3.E36)\) are equivalent\.

Proof\. Writeλi​\(𝐋\)\\lambda\_\{i\}\(\\mathbf\{L\}\)for theii\-th eigenvalue of𝐋\\mathbf\{L\}in ascending order,λ1​\(𝐋\)≤⋯≤λN​\(𝐋\)\\lambda\_\{1\}\(\\mathbf\{L\}\)\\leq\\cdots\\leq\\lambda\_\{N\}\(\\mathbf\{L\}\), so that

logdet\(𝐋\+1N𝟏𝟏T\)=log\(∏i=1Nλi\(𝐋\+1N𝟏𝟏T\)\)\.\\displaystyle\\log\\det\\bigl\(\\mathbf\{L\}\+\\tfrac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\\bigr\)=\\log\\Bigl\(\\prod\_\{i=1\}^\{N\}\\lambda\_\{i\}\\bigl\(\\mathbf\{L\}\+\\tfrac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\\bigr\)\\Bigr\)\.\(37\)The constraint𝐋𝟏=𝟎\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}impliesλ1​\(𝐋\)=0\\lambda\_\{1\}\(\\mathbf\{L\}\)=0with eigenvector𝐮1=𝟏/N\\mathbf\{u\}\_\{1\}=\\mathbf\{1\}/\\sqrt\{N\}, so

𝐋\+1N​𝟏𝟏T=\(λ1​\(𝐋\)⏟=0\+1\)​𝐮1​𝐮1T\+∑i=2Nλi​\(𝐋\)​𝐮i​𝐮iT\.\\displaystyle\\mathbf\{L\}\+\\frac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}=\(\\underbrace\{\\lambda\_\{1\}\(\\mathbf\{L\}\)\}\_\{=0\}\+1\)\\mathbf\{u\}\_\{1\}\\mathbf\{u\}\_\{1\}^\{T\}\+\\sum\_\{i=2\}^\{N\}\\lambda\_\{i\}\(\\mathbf\{L\}\)\\mathbf\{u\}\_\{i\}\\mathbf\{u\}\_\{i\}^\{T\}\.\(38\)Since the determinant equals the product of the eigenvalues, \([38](https://arxiv.org/html/2608.19914#S3.E38)\) gives

logdet\(𝐋\+1N𝟏𝟏T\)=log\(1⋅∏i=2Nλi\(𝐋\)\)=log\|𝐋\|\+,\\displaystyle\\log\\det\\bigl\(\\mathbf\{L\}\+\\tfrac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\\bigr\)=\\log\\Bigl\(1\\cdot\\prod\_\{i=2\}^\{N\}\\lambda\_\{i\}\(\\mathbf\{L\}\)\\Bigr\)=\\log\|\\mathbf\{L\}\|\_\{\+\},\(39\)so the objectives of \([35](https://arxiv.org/html/2608.19914#S3.E35)\) and \([36](https://arxiv.org/html/2608.19914#S3.E36)\) coincide onℒ\\mathcal\{L\}\.□\\square

With Proposition[III\.2](https://arxiv.org/html/2608.19914#S3.Thmproposition2)in place, it remains to solve \([36](https://arxiv.org/html/2608.19914#S3.E36)\) efficiently under the combinatorial structure ofℒ\\mathcal\{L\}\. We proceed in two steps: first re\-expressing the feasible setℒ\\mathcal\{L\}in a form amenable to variable splitting, and then deriving the resulting ADMM updates in closed form\.

Reformulating the constraint set\.Assuming no self\-loops \(so the diagonal of the adjacency structure is zero\), the feasible setℒ\\mathcal\{L\}of \([1](https://arxiv.org/html/2608.19914#S2.E1)\) is compactly rewritten as

\{𝐋⪰𝟎,𝐋𝟏=𝟎𝐋−𝐂=𝟎𝐂∈𝒞\\displaystyle\\left\\\{\\begin\{aligned\} &\\mathbf\{L\}\\succeq\\mathbf\{0\},\\ \\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}\\\\ &\\mathbf\{L\}\-\\mathbf\{C\}=\\mathbf\{0\}\\\\ &\\mathbf\{C\}\\in\\mathcal\{C\}\\end\{aligned\}\\right\.\(40\)where𝐂∈𝒞\\mathbf\{C\}\\in\\mathcal\{C\}encodes

\{𝐈⊙𝐂≥𝟎𝐁⊙𝐂=𝟎𝐀⊙𝐂≤𝟎\\displaystyle\\left\\\{\\begin\{aligned\} &\\mathbf\{I\}\\odot\\mathbf\{C\}\\geq\\mathbf\{0\}\\\\ &\\mathbf\{B\}\\odot\\mathbf\{C\}=\\mathbf\{0\}\\\\ &\\mathbf\{A\}\\odot\\mathbf\{C\}\\leq\\mathbf\{0\}\\end\{aligned\}\\right\.\(41\)with𝐁=𝟏𝟏T−𝐈−𝐀\\mathbf\{B\}=\\mathbf\{1\}\\mathbf\{1\}^\{T\}\-\\mathbf\{I\}\-\\mathbf\{A\}; the constraint𝐈⊙𝐂≥𝟎\\mathbf\{I\}\\odot\\mathbf\{C\}\\geq\\mathbf\{0\}is already implied by𝐋⪰𝟎\\mathbf\{L\}\\succeq\\mathbf\{0\}and is retained for clarity\. The constraint pair𝐋⪰𝟎\\mathbf\{L\}\\succeq\\mathbf\{0\},𝐋𝟏=𝟎\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}admits, in turn, an equivalent low\-dimensional parametrization,

𝐋⪰𝟎,𝐋𝟏=𝟎⟺𝐋=𝐏​𝚵​𝐏T,𝚵⪰𝟎,\\displaystyle\\mathbf\{L\}\\succeq\\mathbf\{0\},\\;\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}\\;\\Longleftrightarrow\\;\\mathbf\{L\}=\\mathbf\{P\}\\bm\{\\Xi\}\\mathbf\{P\}^\{T\},\\ \\bm\{\\Xi\}\\succeq\\mathbf\{0\},\(42\)where𝐏∈ℝN×\(N−1\)\\mathbf\{P\}\\in\\mathbb\{R\}^\{N\\times\(N\-1\)\}is any orthogonal complement of𝟏\\mathbf\{1\}, i\.e\.,𝐏T​𝐏=𝐈\\mathbf\{P\}^\{T\}\\mathbf\{P\}=\\mathbf\{I\}and𝐏T​𝟏=𝟎\\mathbf\{P\}^\{T\}\\mathbf\{1\}=\\mathbf\{0\}\(the choice of𝐏\\mathbf\{P\}is non\-unique: if𝐏0\\mathbf\{P\}\_\{0\}satisfies both conditions, so does𝐏0​𝐔\\mathbf\{P\}\_\{0\}\\mathbf\{U\}for any unitary𝐔∈ℝ\(N−1\)×\(N−1\)\\mathbf\{U\}\\in\\mathbb\{R\}^\{\(N\-1\)\\times\(N\-1\)\}\)\. Substituting \([42](https://arxiv.org/html/2608.19914#S3.E42)\) into the objective of \([36](https://arxiv.org/html/2608.19914#S3.E36)\) eliminates the equality constraint𝐋𝟏=𝟎\\mathbf\{L\}\\mathbf\{1\}=\\mathbf\{0\}altogether: writing𝐊~=𝐏T​𝐊𝐏\\tilde\{\\mathbf\{K\}\}=\\mathbf\{P\}^\{T\}\\mathbf\{K\}\\mathbf\{P\},

tr⁡\(𝐋𝐊\)=tr⁡\(𝚵​𝐊~\),\\displaystyle\\mathrm\{tr\}\(\\mathbf\{L\}\\mathbf\{K\}\)=\\mathrm\{tr\}\(\\bm\{\\Xi\}\\tilde\{\\mathbf\{K\}\}\),\(43\)while the barrier term simplifies via

logdet\(𝐋\+𝐉\)=logdet\(𝐏T𝚵𝐏\+1N𝟏𝟏T\)\\displaystyle\\log\\det\(\\mathbf\{L\}\+\\mathbf\{J\}\)=\\log\\det\\Bigl\(\\mathbf\{P\}^\{T\}\\bm\{\\Xi\}\\mathbf\{P\}\+\\tfrac\{1\}\{N\}\\mathbf\{1\}\\mathbf\{1\}^\{T\}\\Bigr\)=logdet\(\[𝐏,1N𝟏\]\[𝚵𝟎𝟎1\]\[𝐏,1N𝟏\]T\)\\displaystyle=\\log\\det\\Bigl\(\\bigl\[\\mathbf\{P\},\\tfrac\{1\}\{\\sqrt\{N\}\}\\mathbf\{1\}\\bigr\]\\begin\{bmatrix\}\\bm\{\\Xi\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&1\\end\{bmatrix\}\\bigl\[\\mathbf\{P\},\\tfrac\{1\}\{\\sqrt\{N\}\}\\mathbf\{1\}\\bigr\]^\{T\}\\Bigr\)=logdet\(𝚵\)\.\\displaystyle=\\log\\det\(\\bm\{\\Xi\}\)\.\(44\)Problem \([36](https://arxiv.org/html/2608.19914#S3.E36)\) thus reduces to an unconstrained\-domain problem in the pair\(𝚵,𝐂\)\(\\bm\{\\Xi\},\\mathbf\{C\}\),

min𝚵,𝐂\\displaystyle\\min\_\{\\bm\{\\Xi\},\\mathbf\{C\}\}tr\(𝚵𝐊~\)−logdet\(𝚵\)\+ϵ∥𝐂∥F\\displaystyle\\quad\\mathrm\{tr\}\(\\bm\{\\Xi\}\\tilde\{\\mathbf\{K\}\}\)\-\\log\\det\(\\bm\{\\Xi\}\)\+\\epsilon\\\|\\mathbf\{C\}\\\|\_\{F\}s\.t\.\\displaystyle\\mathrm\{s\.t\.\}𝚵⪰𝟎\\displaystyle\\quad\\bm\{\\Xi\}\\succeq\\mathbf\{0\}𝐏​𝚵​𝐏T−𝐂=𝟎,𝐂∈𝒞,\\displaystyle\\quad\\mathbf\{P\}\\bm\{\\Xi\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}=\\mathbf\{0\},\\;\\mathbf\{C\}\\in\\mathcal\{C\},\(45\)which we solve via ADMM, treating𝚵\\bm\{\\Xi\}and𝐂\\mathbf\{C\}as primal variables and introducing a dual variable𝐘\\mathbf\{Y\}for the consensus constraint𝐏​𝚵​𝐏T−𝐂=𝟎\\mathbf\{P\}\\bm\{\\Xi\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}=\\mathbf\{0\}\.

Deriving the ADMM updates\.The augmented Lagrangian of \([45](https://arxiv.org/html/2608.19914#S3.E45)\) is

ℒ\(𝚵,𝐂,𝐘\)=tr\(𝚵𝐊~\)−logdet\(𝚵\)\+ϵ∥𝐂∥F\\displaystyle\\mathcal\{L\}\(\\bm\{\\Xi\},\\mathbf\{C\},\\mathbf\{Y\}\)=\\mathrm\{tr\}\(\\bm\{\\Xi\}\\tilde\{\\mathbf\{K\}\}\)\-\\log\\det\(\\bm\{\\Xi\}\)\+\\epsilon\\\|\\mathbf\{C\}\\\|\_\{F\}\+tr⁡\(𝐘T​\(𝐏T​𝚵​𝐏−𝐂\)\)\+ϱ2​‖𝐏T​𝚵​𝐏−𝐂‖F2,\\displaystyle\\quad\+\\mathrm\{tr\}\\bigl\(\\mathbf\{Y\}^\{T\}\(\\mathbf\{P\}^\{T\}\\bm\{\\Xi\}\\mathbf\{P\}\-\\mathbf\{C\}\)\\bigr\)\+\\frac\{\\varrho\}\{2\}\\\|\\mathbf\{P\}^\{T\}\\bm\{\\Xi\}\\mathbf\{P\}\-\\mathbf\{C\}\\\|\_\{F\}^\{2\},\(46\)where the constraints𝚵⪰𝟎\\bm\{\\Xi\}\\succeq\\mathbf\{0\}and𝐂∈𝒞\\mathbf\{C\}\\in\\mathcal\{C\}are enforced directly on their respective minimization domains rather than relaxed via a multiplier\. Each ADMM iteration alternates a minimization over𝚵\\bm\{\\Xi\}, a minimization over𝐂\\mathbf\{C\}, and a dual ascent step on𝐘\\mathbf\{Y\}; writing\(⋅\)\+\(\\cdot\)^\{\+\}for the updated iterate,

𝚵\+\\displaystyle\\bm\{\\Xi\}^\{\+\}=arg​min𝚵⪰𝟎⁡ℒ​\(𝚵,𝐂,𝐘\),\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\bm\{\\Xi\}\\succeq\\mathbf\{0\}\}\\mathcal\{L\}\(\\bm\{\\Xi\},\\mathbf\{C\},\\mathbf\{Y\}\),𝐂\+\\displaystyle\\mathbf\{C\}^\{\+\}=arg​min𝐂∈𝒞⁡ℒ​\(𝚵\+,𝐂,𝐘\)\.\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\mathbf\{C\}\\in\\mathcal\{C\}\}\\mathcal\{L\}\(\\bm\{\\Xi\}^\{\+\},\\mathbf\{C\},\\mathbf\{Y\}\)\.\(47\)
1\) Update of𝚵\\bm\{\\Xi\}:Completing the square inℒ⁡\(𝚵,𝐂,𝐘\)\\mathcal\{L\}\(\\bm\{\\Xi\},\\mathbf\{C\},\\mathbf\{Y\}\)with respect to𝚵\\bm\{\\Xi\}yields

𝚵\+\\displaystyle\\bm\{\\Xi\}^\{\+\}=arg​min𝚵⪰𝟎tr\(𝚵𝐊~\)−logdet\(𝚵\)\+tr\(𝐏T𝐘T𝐏𝚵\)\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\bm\{\\Xi\}\\succeq\\mathbf\{0\}\}\\mathrm\{tr\}\(\\bm\{\\Xi\}\\tilde\{\\mathbf\{K\}\}\)\-\\log\\det\(\\bm\{\\Xi\}\)\+\\mathrm\{tr\}\(\\mathbf\{P\}^\{T\}\\mathbf\{Y\}^\{T\}\\mathbf\{P\}\\bm\{\\Xi\}\)\+ϱ2​‖𝐏​𝚵​𝐏T−𝐂‖F2\\displaystyle\\quad\+\\frac\{\\varrho\}\{2\}\\\|\\mathbf\{P\}\\bm\{\\Xi\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}\\\|\_\{F\}^\{2\}=arg​min𝚵⪰𝟎ϱ2∥𝚵\+1ϱ\(𝐊~\+𝐘~−ρ𝐂~\)∥F2−logdet\(𝚵\),\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\bm\{\\Xi\}\\succeq\\mathbf\{0\}\}\\frac\{\\varrho\}\{2\}\\Bigl\\\|\\bm\{\\Xi\}\+\\frac\{1\}\{\\varrho\}\(\\tilde\{\\mathbf\{K\}\}\+\\tilde\{\\mathbf\{Y\}\}\-\\rho\\tilde\{\\mathbf\{C\}\}\)\\Bigr\\\|\_\{F\}^\{2\}\-\\log\\det\(\\bm\{\\Xi\}\),\(48\)with𝐘~=𝐏T​𝐘𝐏\\tilde\{\\mathbf\{Y\}\}=\\mathbf\{P\}^\{T\}\\mathbf\{Y\}\\mathbf\{P\}and𝐂~=𝐏T​𝐂𝐏\\tilde\{\\mathbf\{C\}\}=\\mathbf\{P\}^\{T\}\\mathbf\{C\}\\mathbf\{P\}\. The minimizer of \([48](https://arxiv.org/html/2608.19914#S3.E48)\) follows from a standard log\-determinant proximal identity, stated here for completeness\.

###### Lemma III\.4

The minimizer ofminΘ⪰𝟎ϱ2∥Θ\+𝐗∥F2−logdet\(Θ\)\\min\_\{\\Theta\\succeq\\mathbf\{0\}\}\\frac\{\\varrho\}\{2\}\\\|\\Theta\+\\mathbf\{X\}\\\|\_\{F\}^\{2\}\-\\log\\det\(\\Theta\)isΘ∗=𝐔𝐃𝐔T\\Theta^\{\\ast\}=\\mathbf\{U\}\\mathbf\{D\}\\mathbf\{U\}^\{T\}, where𝐗=𝐔​Λ​𝐔T\\mathbf\{X\}=\\mathbf\{U\}\\Lambda\\mathbf\{U\}^\{T\}is the eigendecomposition of𝐗\\mathbf\{X\}and𝐃\\mathbf\{D\}is diagonal with

Di​i=−ϱ​Λi​i\+ϱ2​Λi​i2\+4​ϱ2​ϱ\.\\displaystyle D\_\{ii\}=\\frac\{\-\\varrho\\Lambda\_\{ii\}\+\\sqrt\{\\varrho^\{2\}\\Lambda\_\{ii\}^\{2\}\+4\\varrho\}\}\{2\\varrho\}\.

Applying Lemma[III\.4](https://arxiv.org/html/2608.19914#S3.Thmlemma4)to \([48](https://arxiv.org/html/2608.19914#S3.E48)\) gives the closed\-formΞ\\Xi\-update

𝚵\+=𝐔𝐃𝐔T,\\displaystyle\\bm\{\\Xi\}^\{\+\}=\\mathbf\{U\}\\mathbf\{D\}\\mathbf\{U\}^\{T\},\(49\)where𝐔\\mathbf\{U\}and𝚲\\bm\{\\Lambda\}come from the eigendecomposition1ϱ​𝐏T​\(𝐊\+𝐘−ϱ​𝐂\)​𝐏=𝐔​𝚲​𝐔T\\frac\{1\}\{\\varrho\}\\mathbf\{P\}^\{T\}\(\\mathbf\{K\}\+\\mathbf\{Y\}\-\\varrho\\mathbf\{C\}\)\\mathbf\{P\}=\\mathbf\{U\}\\bm\{\\Lambda\}\\mathbf\{U\}^\{T\}, and𝐃\\mathbf\{D\}is diagonal withDi​i=\(−ϱ​Λi​i\+ϱ2​Λi​i2\+4​ϱ\)/\(2​ϱ\)D\_\{ii\}=\\bigl\(\-\\varrho\\Lambda\_\{ii\}\+\\sqrt\{\\varrho^\{2\}\\Lambda\_\{ii\}^\{2\}\+4\\varrho\}\\bigr\)/\(2\\varrho\)\.

2\) Update of𝐂\\mathbf\{C\}:With𝚵\+\\bm\{\\Xi\}^\{\+\}fixed, the𝐂\\mathbf\{C\}\-update solves

𝐂\+=arg​min𝐂∈𝒞⁡ℒ​\(𝚵\+,𝐂,𝐘\)\\displaystyle\\mathbf\{C\}^\{\+\}=\\operatorname\*\{arg\\,min\}\_\{\\mathbf\{C\}\\in\\mathcal\{C\}\}\\mathcal\{L\}\(\\bm\{\\Xi\}^\{\+\},\\mathbf\{C\},\\mathbf\{Y\}\)=arg​min𝐂∈𝒞−tr⁡\(𝐘T​𝐂\)\+ϵ​‖𝐂‖F\+ϱ2​‖𝐏​𝚵\+​𝐏T−𝐂‖F2,\\displaystyle=\\operatorname\*\{arg\\,min\}\_\{\\mathbf\{C\}\\in\\mathcal\{C\}\}\-\\mathrm\{tr\}\(\\mathbf\{Y\}^\{T\}\\mathbf\{C\}\)\+\\epsilon\\\|\\mathbf\{C\}\\\|\_\{F\}\+\\frac\{\\varrho\}\{2\}\\\|\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}\\\|\_\{F\}^\{2\},\(50\)whose minimizer is given by the following block\-shrinkage lemma\.

###### Lemma III\.5

The minimizer ofmin𝐂∈𝒞−tr⁡\(𝐘T​𝐂\)\+ϵ​‖𝐂‖F\+ϱ2​‖𝐏​𝚵\+​𝐏T−𝐂‖F2\\min\_\{\\mathbf\{C\}\\in\\mathcal\{C\}\}\-\\mathrm\{tr\}\(\\mathbf\{Y\}^\{T\}\\mathbf\{C\}\)\+\\epsilon\\\|\\mathbf\{C\}\\\|\_\{F\}\+\\frac\{\\varrho\}\{2\}\\\|\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}\\\|\_\{F\}^\{2\}is

𝐂∗\\displaystyle\\mathbf\{C\}^\{\\ast\}=𝐈⊙\[s⋅\(1ϱ​𝐘\+𝐏​𝚵\+​𝐏T\)\]\+\\displaystyle=\\mathbf\{I\}\\odot\\Bigl\[s\\cdot\\bigl\(\\tfrac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\bigr\)\\Bigr\]\_\{\+\}\+𝐀⊙\[s⋅\(1ϱ𝐘\+𝐏𝚵\+𝐏T\)\]−,\\displaystyle\\quad\+\\mathbf\{A\}\\odot\\Bigl\[s\\cdot\\bigl\(\\tfrac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\bigr\)\\Bigr\]\_\{\-\},wheres=max⁡\(1−ϵ/ϱ‖1ϱ​𝐘\+𝐏​𝚵\+​𝐏T‖,0\)s=\\max\\Bigl\(1\-\\dfrac\{\\epsilon/\\varrho\}\{\\bigl\\\|\\frac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\bigr\\\|\},0\\Bigr\)is the shrinkage factor\.

Applying Lemma[III\.5](https://arxiv.org/html/2608.19914#S3.Thmlemma5)gives the𝐂\\mathbf\{C\}\-update

𝐂\+=𝐈⊙\[max⁡\(1−ϵ/ϱ‖1ϱ​𝐘\+𝐏​𝚵\+​𝐏T‖,0\)​\(1ϱ​𝐘\+𝐏​𝚵\+​𝐏T\)\]\+\\displaystyle\\mathbf\{C\}^\{\+\}=\\mathbf\{I\}\\odot\\left\[\\max\\Bigl\(1\-\\frac\{\\epsilon/\\varrho\}\{\\\|\\frac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\\|\},0\\Bigr\)\\Bigl\(\\frac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\Bigr\)\\right\]\_\{\+\}\+𝐀⊙\[max\(1−ϵ/ϱ‖1ϱ​𝐘\+𝐏​𝚵\+​𝐏T‖,0\)\(1ϱ𝐘\+𝐏𝚵\+𝐏T\)\]−\.\\displaystyle\+\\mathbf\{A\}\\odot\\left\[\\max\\Bigl\(1\-\\frac\{\\epsilon/\\varrho\}\{\\\|\\frac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\\|\},0\\Bigr\)\\Bigl\(\\frac\{1\}\{\\varrho\}\\mathbf\{Y\}\+\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\\Bigr\)\\right\]\_\{\-\}\.\(51\)
3\) Update of𝐘\\mathbf\{Y\}:The dual variable is updated by standard gradient ascent on the augmented Lagrangian,

𝐘\+=𝐘\+ϱ⁡\(𝐏​𝚵\+​𝐏T−𝐂\+\)\.\\displaystyle\\mathbf\{Y\}^\{\+\}=\\mathbf\{Y\}\+\\varrho\(\\mathbf\{P\}\\bm\{\\Xi\}^\{\+\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}^\{\+\}\)\.\(52\)The complete procedure, alternating the closed\-form updates \([49](https://arxiv.org/html/2608.19914#S3.E49)\), \([51](https://arxiv.org/html/2608.19914#S3.E51)\), and \([52](https://arxiv.org/html/2608.19914#S3.E52)\) until convergence, is summarized in Algorithm[1](https://arxiv.org/html/2608.19914#alg1)\.

Algorithm 1ADMM\-Based Algorithm for MS\-WDRO0:

𝐊\\mathbf\{K\}; ambiguity radius

ϵ\\epsilon; penalty parameters

ρ,ϱ\>0\\rho,\\varrho\>0; symmetric initialization

𝐘\(0\)=𝟎\\mathbf\{Y\}^\{\(0\)\}=\\mathbf\{0\}and symmetric

𝐂\(0\)∈𝒞\\mathbf\{C\}^\{\(0\)\}\\in\\mathcal\{C\};

k=0k=0\.

0:

𝐋∗\\mathbf\{L\}^\{\\ast\}
1:repeat

2:update

𝚵\(k\+1\)\\bm\{\\Xi\}^\{\(k\+1\)\}according to \([49](https://arxiv.org/html/2608.19914#S3.E49)\);

3:update

𝐂\(k\+1\)\\mathbf\{C\}^\{\(k\+1\)\}according to \([51](https://arxiv.org/html/2608.19914#S3.E51)\);

4:update

𝐘\(k\+1\)\\mathbf\{Y\}^\{\(k\+1\)\}according to \([52](https://arxiv.org/html/2608.19914#S3.E52)\);

5:

k=k\+1k=k\+1;

6:untilconvergence

7:return

𝐋∗\\mathbf\{L\}^\{\\ast\}

### III\-EConvergence Analysis

We now establish the global convergence of Algorithm[1](https://arxiv.org/html/2608.19914#alg1)\. Introduce the linear operator𝒯:ℝ\(N−1\)×\(N−1\)→ℝN×N\\mathcal\{T\}:\\mathbb\{R\}^\{\(N\-1\)\\times\(N\-1\)\}\\to\\mathbb\{R\}^\{N\\times N\},𝒯⁡\(𝚵\)≜𝐏​𝚵​𝐏T\\mathcal\{T\}\(\\bm\{\\Xi\}\)\\triangleq\\mathbf\{P\}\\bm\{\\Xi\}\\mathbf\{P\}^\{T\}, with adjoint𝒯∗​\(𝐙\)=𝐏T​𝐙𝐏\\mathcal\{T\}^\{\*\}\(\\mathbf\{Z\}\)=\\mathbf\{P\}^\{T\}\\mathbf\{Z\}\\mathbf\{P\}\. Since𝐏T​𝐏=𝐈\\mathbf\{P\}^\{T\}\\mathbf\{P\}=\\mathbf\{I\},𝒯\\mathcal\{T\}is injective:𝒯⁡\(𝚵\)=𝟎⇒𝚵=𝒯∗​\(𝒯⁡\(𝚵\)\)=𝟎\\mathcal\{T\}\(\\bm\{\\Xi\}\)=\\mathbf\{0\}\\Rightarrow\\bm\{\\Xi\}=\\mathcal\{T\}^\{\*\}\(\\mathcal\{T\}\(\\bm\{\\Xi\}\)\)=\\mathbf\{0\}\. Problem \([45](https://arxiv.org/html/2608.19914#S3.E45)\) is then the canonical two\-block convex program

min𝚵,𝐂⁡f⁡\(𝚵\)\+g⁡\(𝐂\)s\.t\.𝒯⁡\(𝚵\)−𝐂=𝟎,\\displaystyle\\min\_\{\\bm\{\\Xi\},\\mathbf\{C\}\}\\;f\(\\bm\{\\Xi\}\)\+g\(\\mathbf\{C\}\)\\quad\\mathrm\{s\.t\.\}\\quad\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}=\\mathbf\{0\},\(53\)withf\(𝚵\)=tr\(𝚵𝐊~\)−logdet\(𝚵\)f\(\\bm\{\\Xi\}\)=\\mathrm\{tr\}\(\\bm\{\\Xi\}\\tilde\{\\mathbf\{K\}\}\)\-\\log\\det\(\\bm\{\\Xi\}\)ondomf=\{𝚵≻𝟎\}\\mathrm\{dom\}\\,f=\\\{\\bm\{\\Xi\}\\succ\\mathbf\{0\}\\\}\(and\+∞\+\\inftyotherwise\), andg⁡\(𝐂\)=ϵ​‖𝐂‖F\+ι𝒞​\(𝐂\)g\(\\mathbf\{C\}\)=\\epsilon\\\|\\mathbf\{C\}\\\|\_\{F\}\+\\iota\_\{\\mathcal\{C\}\}\(\\mathbf\{C\}\)\. Here,ι𝒞​\(𝐂\)\\iota\_\{\\mathcal\{C\}\}\(\\mathbf\{C\}\)is the indicator function\. Bothffandggare closed, proper, and convex:ffis the sum of a linear term and the standard log\-determinant barrier, strictly convex on the open PD cone;ggis the sum of the \(continuous, hence closed\) Frobenius norm and the indicator of the polyhedral set𝒞\\mathcal\{C\}in \([41](https://arxiv.org/html/2608.19914#S3.E41)\), which is nonempty and closed\.

###### Assumption III\.3

There exists a Laplacian𝐋0∈ℒ\\mathbf\{L\}\_\{0\}\\in\\mathcal\{L\}, corresponding to a connected candidate topology compatible with the sparsity pattern encoded by𝐀\\mathbf\{A\}in \([41](https://arxiv.org/html/2608.19914#S3.E41)\), whose weights are strictly negative on every edge admitted by𝐀\\mathbf\{A\}; equivalently,𝚵0=𝐏T​𝐋0​𝐏≻𝟎\\bm\{\\Xi\}\_\{0\}=\\mathbf\{P\}^\{T\}\\mathbf\{L\}\_\{0\}\\mathbf\{P\}\\succ\\mathbf\{0\}and𝐂0=𝐋0\\mathbf\{C\}\_\{0\}=\\mathbf\{L\}\_\{0\}lies in the relative interior of𝒞\\mathcal\{C\}\.

Assumption[III\.3](https://arxiv.org/html/2608.19914#S3.Thmassumption3)is a mild, generically satisfied regularity condition \(any densely connected weighted candidate graph provides such a point\) and furnishes a point ofri⁡\(dom​f\)×ri⁡\(dom​g\)\\mathrm\{ri\}\(\\mathrm\{dom\}\\,f\)\\times\\mathrm\{ri\}\(\\mathrm\{dom\}\\,g\)\(ri⁡\(⋅\)\\mathrm\{ri\}\(\\cdot\)denotes the relative interior\) satisfying the coupling constraint of \([53](https://arxiv.org/html/2608.19914#S3.E53)\)\. By the Fenchel–Rockafellar duality theorem for linearly constrained convex programs\[[50](https://arxiv.org/html/2608.19914#bib.bib50)\], this guarantees strong duality and the existence of a saddle point\(𝚵⋆,𝐂⋆,𝐘⋆\)\(\\bm\{\\Xi\}^\{\\star\},\\mathbf\{C\}^\{\\star\},\\mathbf\{Y\}^\{\\star\}\)of the unaugmented Lagrangian of \([53](https://arxiv.org/html/2608.19914#S3.E53)\), i\.e\.

−𝒯∗​\(𝐘⋆\)=∇f​\(𝚵⋆\),𝐘⋆∈∂g⁡\(𝐂⋆\),𝒯⁡\(𝚵⋆\)=𝐂⋆\.\\displaystyle\-\\mathcal\{T\}^\{\*\}\(\\mathbf\{Y\}^\{\\star\}\)=\\nabla f\(\\bm\{\\Xi\}^\{\\star\}\),\\mathbf\{Y\}^\{\\star\}\\in\\partial g\(\\mathbf\{C\}^\{\\star\}\),\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\\star\}\)=\\mathbf\{C\}^\{\\star\}\.\(54\)Sinceffis strictly convex on\{𝚵≻𝟎\}\\\{\\bm\{\\Xi\}\\succ\\mathbf\{0\}\\\}, the primal minimizer𝚵⋆\\bm\{\\Xi\}^\{\\star\}is unique, and by injectivity of𝒯\\mathcal\{T\}, so is𝐂⋆=𝒯⁡\(𝚵⋆\)\\mathbf\{C\}^\{\\star\}=\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\\star\}\)\.

###### Theorem III\.2

Under Assumption[III\.3](https://arxiv.org/html/2608.19914#S3.Thmassumption3), for any penaltyϱ\>0\\varrho\>0the sequence\{\(𝚵\(k\),𝐂\(k\),𝐘\(k\)\)\}\\\{\(\\bm\{\\Xi\}^\{\(k\)\},\\mathbf\{C\}^\{\(k\)\},\\mathbf\{Y\}^\{\(k\)\}\)\\\}generated by Algorithm[1](https://arxiv.org/html/2608.19914#alg1)satisfies, writing𝐫\(k\)≜𝒯⁡\(𝚵\(k\)\)−𝐂\(k\)\\mathbf\{r\}^\{\(k\)\}\\triangleq\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\)\}\)\-\\mathbf\{C\}^\{\(k\)\}:

1. \(i\)*\(Consensus\)\.*𝐫\(k\)→𝟎\\mathbf\{r\}^\{\(k\)\}\\to\\mathbf\{0\}and𝐂\(k\+1\)−𝐂\(k\)→𝟎\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\}\\to\\mathbf\{0\};
2. \(ii\)*\(Objective convergence\)\.*f⁡\(𝚵\(k\)\)\+g⁡\(𝐂\(k\)\)→f⁡\(𝚵⋆\)\+g⁡\(𝐂⋆\)f\(\\bm\{\\Xi\}^\{\(k\)\}\)\+g\(\\mathbf\{C\}^\{\(k\)\}\)\\to f\(\\bm\{\\Xi\}^\{\\star\}\)\+g\(\\mathbf\{C\}^\{\\star\}\);
3. \(iii\)*\(Iterate convergence\)\.*𝚵\(k\)→𝚵⋆\\bm\{\\Xi\}^\{\(k\)\}\\to\\bm\{\\Xi\}^\{\\star\},𝐂\(k\)→𝐂⋆\\mathbf\{C\}^\{\(k\)\}\\to\\mathbf\{C\}^\{\\star\}, and𝐘\(k\)→𝐘⋆\\mathbf\{Y\}^\{\(k\)\}\\to\\mathbf\{Y\}^\{\\star\}for some dual optimal𝐘⋆\\mathbf\{Y\}^\{\\star\};
4. \(iv\)*\(Lyapunov monotonicity\)\.*The primal\-dual potential V\(k\)≜ϱ​‖𝐂\(k\)−𝐂⋆‖F2\+1ϱ​‖𝐘\(k\)−𝐘⋆‖F2\\displaystyle V^\{\(k\)\}\\triangleq\\varrho\\bigl\\\|\\mathbf\{C\}^\{\(k\)\}\-\\mathbf\{C\}^\{\\star\}\\bigr\\\|\_\{F\}^\{2\}\+\\frac\{1\}\{\\varrho\}\\bigl\\\|\\mathbf\{Y\}^\{\(k\)\}\-\\mathbf\{Y\}^\{\\star\}\\bigr\\\|\_\{F\}^\{2\}\(55\)is monotonically non\-increasing, and in fact V\(k\+1\)\\displaystyle V^\{\(k\+1\)\}≤V\(k\)−‖𝐫\(k\+1\)‖F2\\displaystyle\\leq V^\{\(k\)\}\-\\bigl\\\|\\mathbf\{r\}^\{\(k\+1\)\}\\bigr\\\|\_\{F\}^\{2\}−ϱ​‖𝐂\(k\+1\)−𝐂\(k\)‖F2\.\\displaystyle\\quad\-\\varrho\\bigl\\\|\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\}\\bigr\\\|\_\{F\}^\{2\}\.\(56\)

\[A proof is given in Appendix[A](https://arxiv.org/html/2608.19914#A1)\.\]

The Lyapunov analysis above certifies asymptotic convergence but not a rate\. We now show that a simple ergodic averaging of the iterates enjoys an explicitO⁡\(1/K\)O\(1/K\)rate, in the sense customary for first\-order primal\-dual methods under mere convexity \(no strong convexity or Lipschitz\-gradient assumption is invoked\)\.

###### Corollary III\.2

ForK≥1K\\geq 1, define the ergodic averages

𝚵¯K\\displaystyle\\bar\{\\bm\{\\Xi\}\}^\{K\}=1K​∑k=1K𝚵\(k\),𝐂¯K=1K​∑k=1K𝐂\(k\),\\displaystyle=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\bm\{\\Xi\}^\{\(k\)\},\\bar\{\\mathbf\{C\}\}^\{K\}=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathbf\{C\}^\{\(k\)\},𝐘¯K\\displaystyle\\bar\{\\mathbf\{Y\}\}^\{K\}=1K​∑k=1K𝐘\(k\)\.\\displaystyle=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathbf\{Y\}^\{\(k\)\}\.\(57\)For a radiusD≥ϱ​‖𝐂\(0\)−𝐂⋆‖F2\+ϱ−1​‖𝐘\(0\)−𝐘⋆‖F2=V\(0\)D\\geq\\varrho\\\|\\mathbf\{C\}^\{\(0\)\}\-\\mathbf\{C\}^\{\\star\}\\\|\_\{F\}^\{2\}\+\\varrho^\{\-1\}\\\|\\mathbf\{Y\}^\{\(0\)\}\-\\mathbf\{Y\}^\{\\star\}\\\|\_\{F\}^\{2\}=V^\{\(0\)\}, letℬD≜\{\(𝐂,𝐘\):ϱ​‖𝐂−𝐂\(0\)‖F2\+ϱ−1​‖𝐘−𝐘\(0\)‖F2≤D\}\\mathcal\{B\}\_\{D\}\\triangleq\\bigl\\\{\(\\mathbf\{C\},\\mathbf\{Y\}\):\\varrho\\\|\\mathbf\{C\}\-\\mathbf\{C\}^\{\(0\)\}\\\|\_\{F\}^\{2\}\+\\varrho^\{\-1\}\\\|\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(0\)\}\\\|\_\{F\}^\{2\}\\leq D\\bigr\\\}, which by Assumption[III\.3](https://arxiv.org/html/2608.19914#S3.Thmassumption3)contains\(𝐂⋆,𝐘⋆\)\(\\mathbf\{C\}^\{\\star\},\\mathbf\{Y\}^\{\\star\}\)\. Define the primal–dual gap function

GapD\(𝚵¯K,𝐂¯K,𝐘¯K\)≜sup\(𝐂,𝐘\)∈ℬD\{f\(𝚵¯K\)\+g\(𝐂¯K\)−g\(𝐂\)\\displaystyle\\mathrm\{Gap\}\_\{D\}\\bigl\(\\bar\{\\bm\{\\Xi\}\}^\{K\},\\bar\{\\mathbf\{C\}\}^\{K\},\\bar\{\\mathbf\{Y\}\}^\{K\}\\bigr\)\\triangleq\\sup\_\{\(\\mathbf\{C\},\\mathbf\{Y\}\)\\in\\mathcal\{B\}\_\{D\}\}\\Bigl\\\{f\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\+g\(\\bar\{\\mathbf\{C\}\}^\{K\}\)\-g\(\\mathbf\{C\}\)\+⟨𝐘,𝒯⁡\(𝚵¯K\)−𝐂¯K⟩\\displaystyle\+\\bigl\\langle\\mathbf\{Y\},\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\bar\{\\mathbf\{C\}\}^\{K\}\\bigr\\rangle−⟨𝐘¯K,𝒯\(𝚵¯K\)−𝐂⟩−f\(𝚵⋆\)−g\(𝐂⋆\)\}\.\\displaystyle\-\\bigl\\langle\\bar\{\\mathbf\{Y\}\}^\{K\},\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\mathbf\{C\}\\bigr\\rangle\-f\(\\bm\{\\Xi\}^\{\\star\}\)\-g\(\\mathbf\{C\}^\{\\star\}\)\\Bigr\\\}\.\(58\)GapD≥0\\mathrm\{Gap\}\_\{D\}\\geq 0always, andGapD=0\\mathrm\{Gap\}\_\{D\}=0characterizes a saddle point of the Lagrangian restricted toℬD\\mathcal\{B\}\_\{D\}\. Then, under Assumption[III\.3](https://arxiv.org/html/2608.19914#S3.Thmassumption3),

GapD\(𝚵¯K,𝐂¯K,𝐘¯K\)≤2​DK,K=1,2,…\.\\displaystyle\\mathrm\{Gap\}\_\{D\}\\bigl\(\\bar\{\\bm\{\\Xi\}\}^\{K\},\\bar\{\\mathbf\{C\}\}^\{K\},\\bar\{\\mathbf\{Y\}\}^\{K\}\\bigr\)\\;\\leq\\;\\frac\{2D\}\{K\},\\qquad K=1,2,\\dots\.\(59\)In particular, both the ergodic constraint violation and the ergodic objective sub\-optimality vanish at rateO⁡\(1/K\)O\(1/K\):

‖𝒯⁡\(𝚵¯K\)−𝐂¯K‖F≤2​D/Kmax⁡\(1,sup\(𝐂,𝐘\)∈ℬD‖𝐘−𝐘¯K‖F\),\\displaystyle\\bigl\\\|\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\bar\{\\mathbf\{C\}\}^\{K\}\\bigr\\\|\_\{F\}\\leq\\frac\{2D/K\}\{\\max\\bigl\(1,\\sup\_\{\(\\mathbf\{C\},\\mathbf\{Y\}\)\\in\\mathcal\{B\}\_\{D\}\}\\\|\\mathbf\{Y\}\-\\bar\{\\mathbf\{Y\}\}^\{K\}\\\|\_\{F\}\\bigr\)\},\(60\)\|f⁡\(𝚵¯K\)\+g⁡\(𝐂¯K\)−f⁡\(𝚵⋆\)−g⁡\(𝐂⋆\)\|=O⁡\(1/K\)\.\\displaystyle\\Bigl\|f\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\+g\(\\bar\{\\mathbf\{C\}\}^\{K\}\)\-f\(\\bm\{\\Xi\}^\{\\star\}\)\-g\(\\mathbf\{C\}^\{\\star\}\)\\Bigr\|=O\(1/K\)\.\(61\)

Proof\.We first establish the*one\-step*variational inequality specific to the\(𝚵,𝐂\)\(\\bm\{\\Xi\},\\mathbf\{C\}\)\-updates of Algorithm[1](https://arxiv.org/html/2608.19914#alg1), then invoke the standard ergodic\-averaging argument for monotone\-operator splitting to obtain \([59](https://arxiv.org/html/2608.19914#S3.E59)\)\.

*Step 1 \(one\-step inequality\)\.*Fixk≥0k\\geq 0and let\(𝚵,𝐂\)∈dom​f×𝒞\(\\bm\{\\Xi\},\\mathbf\{C\}\)\\in\\mathrm\{dom\}\\,f\\times\\mathcal\{C\}and𝐘∈ℝN×N\\mathbf\{Y\}\\in\\mathbb\{R\}^\{N\\times N\}be arbitrary\. By convexity offfand the first\-order optimality condition \([116](https://arxiv.org/html/2608.19914#A1.E116)\) of the𝚵\\bm\{\\Xi\}\-update,

f⁡\(𝚵\(k\+1\)\)−f⁡\(𝚵\)≤⟨∇f​\(𝚵\(k\+1\)\),𝚵\(k\+1\)−𝚵⟩\\displaystyle f\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-f\(\\bm\{\\Xi\}\)\\leq\\bigl\\langle\\nabla f\(\\bm\{\\Xi\}^\{\(k\+1\)\}\),\\,\\bm\{\\Xi\}^\{\(k\+1\)\}\-\\bm\{\\Xi\}\\bigr\\rangle=−⟨𝐘\(k\+1\),𝒯⁡\(𝚵\(k\+1\)\)−𝒯⁡\(𝚵\)⟩\\displaystyle=\-\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\},\\,\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathcal\{T\}\(\\bm\{\\Xi\}\)\\bigr\\rangle−ϱ⁡⟨𝐂\(k\+1\)−𝐂\(k\),𝒯⁡\(𝚵\(k\+1\)\)−𝒯⁡\(𝚵\)⟩\.\\displaystyle\\quad\-\\varrho\\bigl\\langle\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\},\\,\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathcal\{T\}\(\\bm\{\\Xi\}\)\\bigr\\rangle\.\(62\)By convexity ofggand the optimality condition \([117](https://arxiv.org/html/2608.19914#A1.E117)\) of the𝐂\\mathbf\{C\}\-update,

g⁡\(𝐂\(k\+1\)\)−g⁡\(𝐂\)≤⟨𝐘\(k\+1\),𝐂\(k\+1\)−𝐂⟩\.\\displaystyle g\(\\mathbf\{C\}^\{\(k\+1\)\}\)\-g\(\\mathbf\{C\}\)\\leq\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\},\\,\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}\\bigr\\rangle\.\(63\)Adding \([62](https://arxiv.org/html/2608.19914#S3.E62)\)–\([63](https://arxiv.org/html/2608.19914#S3.E63)\), substituting𝒯⁡\(𝚵\(k\+1\)\)=𝐂\(k\+1\)\+𝐫\(k\+1\)\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)=\\mathbf\{C\}^\{\(k\+1\)\}\+\\mathbf\{r\}^\{\(k\+1\)\}, and adding the algebraic identity⟨𝐘−𝐘\(k\+1\),𝐫\(k\+1\)⟩\+⟨𝐘\(k\+1\),𝒯⁡\(𝚵\)−𝐂⟩−⟨𝐘,𝒯⁡\(𝚵\)−𝐂⟩=0\\langle\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(k\+1\)\},\\mathbf\{r\}^\{\(k\+1\)\}\\rangle\+\\langle\\mathbf\{Y\}^\{\(k\+1\)\},\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}\\rangle\-\\langle\\mathbf\{Y\},\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}\\rangle=0to both sides yields, after simplification,

θ⁡\(𝚵\(k\+1\),𝐂\(k\+1\)\)−θ⁡\(𝚵,𝐂\)\+⟨𝐘,𝒯⁡\(𝚵\(k\+1\)\)−𝐂\(k\+1\)⟩\\displaystyle\\theta\(\\bm\{\\Xi\}^\{\(k\+1\)\},\\mathbf\{C\}^\{\(k\+1\)\}\)\-\\theta\(\\bm\{\\Xi\},\\mathbf\{C\}\)\+\\bigl\\langle\\mathbf\{Y\},\\,\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathbf\{C\}^\{\(k\+1\)\}\\bigr\\rangle−⟨𝐘\(k\+1\),𝒯⁡\(𝚵\)−𝐂⟩\\displaystyle\-\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\},\\,\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}\\bigr\\rangle≤−ϱ⁡⟨𝐂\(k\+1\)−𝐂\(k\),𝒯⁡\(𝚵\(k\+1\)\)−𝒯⁡\(𝚵\)⟩\\displaystyle\\leq\-\\varrho\\bigl\\langle\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\},\\,\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathcal\{T\}\(\\bm\{\\Xi\}\)\\bigr\\rangle\+⟨𝐘−𝐘\(k\+1\),𝐫\(k\+1\)⟩,\\displaystyle\+\\bigl\\langle\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(k\+1\)\},\\,\\mathbf\{r\}^\{\(k\+1\)\}\\bigr\\rangle,\(64\)whereθ⁡\(𝚵,𝐂\)≜f⁡\(𝚵\)\+g⁡\(𝐂\)\\theta\(\\bm\{\\Xi\},\\mathbf\{C\}\)\\triangleq f\(\\bm\{\\Xi\}\)\+g\(\\mathbf\{C\}\)\. Using𝐫\(k\+1\)=ϱ−1​\(𝐘\(k\+1\)−𝐘\(k\)\)\\mathbf\{r\}^\{\(k\+1\)\}=\\varrho^\{\-1\}\(\\mathbf\{Y\}^\{\(k\+1\)\}\-\\mathbf\{Y\}^\{\(k\)\}\)and𝒯⁡\(𝚵\(k\+1\)\)−𝒯⁡\(𝚵\)=\(𝐂\(k\+1\)−𝐂\)\+𝐫\(k\+1\)\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathcal\{T\}\(\\bm\{\\Xi\}\)=\(\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}\)\+\\mathbf\{r\}^\{\(k\+1\)\}, the right\-hand side of \([64](https://arxiv.org/html/2608.19914#S3.E64)\) rearranges, exactly as in the standard splitting\-based ergodic\-rate argument for two\-block ADMM \(He & Yuan\[[51](https://arxiv.org/html/2608.19914#bib.bib51)\], proof of Thm\. 4\.1; see also\[[52](https://arxiv.org/html/2608.19914#bib.bib52)\], App\. A\), into the telescoping form

ϱ2​\(‖𝐂−𝐂\(k\)‖F2−‖𝐂−𝐂\(k\+1\)‖F2\)\\displaystyle\\frac\{\\varrho\}\{2\}\\bigl\(\\\|\\mathbf\{C\}\-\\mathbf\{C\}^\{\(k\)\}\\\|\_\{F\}^\{2\}\-\\\|\\mathbf\{C\}\-\\mathbf\{C\}^\{\(k\+1\)\}\\\|\_\{F\}^\{2\}\\bigr\)\+12​ϱ​\(‖𝐘−𝐘\(k\)‖F2−‖𝐘−𝐘\(k\+1\)‖F2\)\.\\displaystyle\+\\frac\{1\}\{2\\varrho\}\\bigl\(\\\|\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(k\)\}\\\|\_\{F\}^\{2\}\-\\\|\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(k\+1\)\}\\\|\_\{F\}^\{2\}\\bigr\)\.\(65\)
*Step 2 \(telescoping and averaging\)\.*Summing \([64](https://arxiv.org/html/2608.19914#S3.E64)\) \(with right\-hand side replaced by \([65](https://arxiv.org/html/2608.19914#S3.E65)\)\) overk=0,…,K−1k=0,\\dots,K\-1, the right\-hand side telescopes toϱ2​‖𝐂−𝐂\(0\)‖F2\+12​ϱ​‖𝐘−𝐘\(0\)‖F2≤D/2\\frac\{\\varrho\}\{2\}\\\|\\mathbf\{C\}\-\\mathbf\{C\}^\{\(0\)\}\\\|\_\{F\}^\{2\}\+\\frac\{1\}\{2\\varrho\}\\\|\\mathbf\{Y\}\-\\mathbf\{Y\}^\{\(0\)\}\\\|\_\{F\}^\{2\}\\leq D/2for\(𝐂,𝐘\)∈ℬD\(\\mathbf\{C\},\\mathbf\{Y\}\)\\in\\mathcal\{B\}\_\{D\}, while by convexity off,gf,gand Jensen’s inequality applied to the left\-hand side,

K\[θ\(𝚵¯K,𝐂¯K\)−θ\(𝚵,𝐂\)\\displaystyle K\\Bigl\[\\theta\(\\bar\{\\bm\{\\Xi\}\}^\{K\},\\bar\{\\mathbf\{C\}\}^\{K\}\)\-\\theta\(\\bm\{\\Xi\},\\mathbf\{C\}\)\+⟨𝐘,𝒯\(𝚵¯K\)−𝐂¯K⟩−⟨𝐘¯K,𝒯\(𝚵\)−𝐂⟩\]\\displaystyle\\quad\+\\bigl\\langle\\mathbf\{Y\},\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\bar\{\\mathbf\{C\}\}^\{K\}\\bigr\\rangle\-\\bigl\\langle\\bar\{\\mathbf\{Y\}\}^\{K\},\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}\\bigr\\rangle\\Bigr\]≤∑k=0K−1\[θ\(𝚵\(k\+1\),𝐂\(k\+1\)\)−θ\(𝚵,𝐂\)\\displaystyle\\leq\\sum\_\{k=0\}^\{K\-1\}\\Bigl\[\\theta\(\\bm\{\\Xi\}^\{\(k\+1\)\},\\mathbf\{C\}^\{\(k\+1\)\}\)\-\\theta\(\\bm\{\\Xi\},\\mathbf\{C\}\)\+⟨𝐘,𝒯⁡\(𝚵\(k\+1\)\)−𝐂\(k\+1\)⟩\\displaystyle\\quad\+\\langle\\mathbf\{Y\},\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\mathbf\{C\}^\{\(k\+1\)\}\\rangle−⟨𝐘\(k\+1\),𝒯\(𝚵\)−𝐂⟩\]≤D\.\\displaystyle\\quad\-\\langle\\mathbf\{Y\}^\{\(k\+1\)\},\\mathcal\{T\}\(\\bm\{\\Xi\}\)\-\\mathbf\{C\}\\rangle\\Bigr\]\\leq D\.\(66\)Dividing byKKand taking the supremum over\(𝐂,𝐘\)∈ℬD\(\\mathbf\{C\},\\mathbf\{Y\}\)\\in\\mathcal\{B\}\_\{D\}givesGapD​\(𝚵¯K,𝐂¯K,𝐘¯K\)≤D/K\\mathrm\{Gap\}\_\{D\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\},\\bar\{\\mathbf\{C\}\}^\{K\},\\bar\{\\mathbf\{Y\}\}^\{K\}\)\\leq D/K; evaluating at\(𝚵,𝐂\)=\(𝚵⋆,𝐂⋆\)\(\\bm\{\\Xi\},\\mathbf\{C\}\)=\(\\bm\{\\Xi\}^\{\\star\},\\mathbf\{C\}^\{\\star\}\)and usingf⁡\(𝚵⋆\)\+g⁡\(𝐂⋆\)=p⋆f\(\\bm\{\\Xi\}^\{\\star\}\)\+g\(\\mathbf\{C\}^\{\\star\}\)=p^\{\\star\}tightens the constant to2​D/K2D/Konce the \(non\-negative\) sup over𝐘\\mathbf\{Y\}is separated from the fixed evaluation at\(𝚵⋆,𝐂⋆\)\(\\bm\{\\Xi\}^\{\\star\},\\mathbf\{C\}^\{\\star\}\), giving \([59](https://arxiv.org/html/2608.19914#S3.E59)\)\. Bounds \([60](https://arxiv.org/html/2608.19914#S3.E60)\)–\([61](https://arxiv.org/html/2608.19914#S3.E61)\) follow by taking𝐘=𝐘¯K\+t⁡\(𝒯⁡\(𝚵¯K\)−𝐂¯K\)/‖𝒯⁡\(𝚵¯K\)−𝐂¯K‖F\\mathbf\{Y\}=\\bar\{\\mathbf\{Y\}\}^\{K\}\+t\\,\\bigl\(\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\bar\{\\mathbf\{C\}\}^\{K\}\\bigr\)/\\\|\\mathcal\{T\}\(\\bar\{\\bm\{\\Xi\}\}^\{K\}\)\-\\bar\{\\mathbf\{C\}\}^\{K\}\\\|\_\{F\}for a suitablet\>0t\>0with\(𝐂,𝐘\)∈ℬD\(\\mathbf\{C\},\\mathbf\{Y\}\)\\in\\mathcal\{B\}\_\{D\}at𝐂=𝐂¯K\\mathbf\{C\}=\\bar\{\\mathbf\{C\}\}^\{K\}in \([58](https://arxiv.org/html/2608.19914#S3.E58)\), and\(𝐂,𝐘\)=\(𝐂⋆,𝐘⋆\)\(\\mathbf\{C\},\\mathbf\{Y\}\)=\(\\mathbf\{C\}^\{\\star\},\\mathbf\{Y\}^\{\\star\}\)respectively, which is the standard extraction of individualO⁡\(1/K\)O\(1/K\)feasibility and objective bounds from a bounded primal–dual gap \(cf\.\[[51](https://arxiv.org/html/2608.19914#bib.bib51)\], Cor\. 4\.1\)\.□\\square

### III\-FOut\-of\-Sample Excess Risk via Rademacher Complexity

The coverage theorems established in Theorems[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1)and[IV\.3](https://arxiv.org/html/2608.19914#S4.Thmtheorem3)certify that the barycentric ambiguity set contains the population barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}with prescribed probability, yet they do not quantify how rapidly the WDRO\-GL estimator𝐋^∗\\hat\{\\mathbf\{L\}\}^\{\*\}converges to the oracle population minimizer as the total sample size grows\. We close this gap through a uniform Rademacher complexity argument, yielding an explicitO\(nmin−1/2\)O\(n\_\{\\min\}^\{\-1/2\}\)excess\-risk bound that holds simultaneously for all𝐋∈ℒB\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}and reveals the precise role of inter\-source heterogeneity in the generalization rate\.

Working with zero\-mean observations for notational simplicity\(𝝁=𝟎\)\(\\bm\{\\mu\}=\\mathbf\{0\}\), write𝚺m,i=𝐳m,i​𝐳m,i⊤\\bm\{\\Sigma\}\_\{m,i\}=\\mathbf\{z\}\_\{m,i\}\\mathbf\{z\}\_\{m,i\}^\{\\top\}for the rank\-one outer product from sourcekkand observationii\. FixB\>0B\>0and letℒB:=\{𝐋∈ℒ:‖𝐋‖F≤B\}\\mathcal\{L\}\_\{B\}:=\\\{\\mathbf\{L\}\\in\\mathcal\{L\}:\\\|\\mathbf\{L\}\\\|\_\{F\}\\leq B\\\}\. Define the population barycentric risk and the empirical WDRO risk \(from Lemma[III\.3](https://arxiv.org/html/2608.19914#S3.Thmlemma3)\) as

R⁡\(𝐋\)\\displaystyle R\(\\mathbf\{L\}\):=−log⁡\|𝐋\|\+\+tr⁡\(Σ¯𝝀∗​𝐋\),\\displaystyle:=\-\\log\|\\mathbf\{L\}\|\_\{\+\}\+\\mathrm\{tr\}\\\!\\bigl\(\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\\mathbf\{L\}\\bigr\),\(67\)R^ϵn​\(𝐋\)\\displaystyle\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}\):=−log⁡\|𝐋\|\+\+tr⁡\(Σ^𝝀​𝐋\)\+ϵn​‖vec⁡\(𝐋\)‖q,\\displaystyle:=\-\\log\|\\mathbf\{L\}\|\_\{\+\}\+\\mathrm\{tr\}\\\!\\bigl\(\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\mathbf\{L\}\\bigr\)\+\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\},\(68\)whereΣ¯𝝀∗\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}andΣ^𝝀\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}are the covariances of the population and empirical Wasserstein barycenters, respectively, computed via the fixed\-point iteration \([23](https://arxiv.org/html/2608.19914#S3.E23)\)\. The WDRO\-GL estimator and the oracle minimizer are

𝐋^∗:=arg⁡min𝐋∈ℒB​R^ϵn​\(𝐋\),𝐋∗:=arg⁡min𝐋∈ℒB⁡R⁡\(𝐋\)\.\\displaystyle\\hat\{\\mathbf\{L\}\}^\{\*\}:=\\arg\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}\}\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}\),\\qquad\\mathbf\{L\}^\{\*\}:=\\arg\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}\}R\(\\mathbf\{L\}\)\.
###### Assumption III\.4

Fix a confidence levelδ∈\(0,1\)\\delta\\in\(0,1\)and let𝒵:=𝒵R⁡\(δ\)\\mathcal\{Z\}:=\\mathcal\{Z\}\_\{R\(\\delta\)\}andℰtrunc​\(δ\)\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta\)be as in Corollary[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmcorollary1)\. Conditionally onℰtrunc​\(δ\)\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta\), every source distributionℙm∗\\mathbb\{P\}\_\{m\}^\{\*\},m∈\[M\]m\\in\[M\], is absolutely continuous with respect to Lebesgue measure and has support contained in the common convex compact set𝒵⊂𝕊\+N\\mathcal\{Z\}\\subset\\mathbb\{S\}\_\{\+\}^\{N\}, with diameterD𝒵≜supΣ,Σ′∈𝒵‖Σ−Σ′‖F≤2​R​\(δ\)2D\_\{\\mathcal\{Z\}\}\\triangleq\\sup\_\{\\Sigma,\\Sigma^\{\\prime\}\\in\\mathcal\{Z\}\}\\\|\\Sigma\-\\Sigma^\{\\prime\}\\\|\_\{F\}\\leq 2R\(\\delta\)^\{2\}\. The \(conditional\) density of eachℙm∗\\mathbb\{P\}\_\{m\}^\{\*\}is bounded below by a constantm¯\>0\\underline\{m\}\>0on𝒵\\mathcal\{Z\}\.

Beyond Assumptions[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1)and[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4), the following mild stability property of the Bures\-Wasserstein barycenter covariance map is required\.

###### Assumption III\.5

The map\(Σ1,…,ΣM\)↦BCov⁡\(Σ1,…,ΣM,𝛌\)\(\\Sigma\_\{1\},\\ldots,\\Sigma\_\{M\}\)\\mapsto\\mathrm\{BCov\}\(\\Sigma\_\{1\},\\ldots,\\allowbreak\\Sigma\_\{M\};\\bm\{\\lambda\}\), which assigns to anyMMsource covariances their𝛌\\bm\{\\lambda\}\-weighted Bures\-Wasserstein barycenter covariance via the fixed\-point equation \([23](https://arxiv.org/html/2608.19914#S3.E23)\), is Lipschitz in the weightedℓ1\\ell\_\{1\}sense: for any two families of positive\-definite matrices\{Σm\}\\\{\\Sigma\_\{m\}\\\}and\{Σm′\}\\\{\\Sigma\_\{m\}^\{\\prime\}\\\},

‖BCov⁡\(Σ1,…,ΣM,𝝀\)−BCov⁡\(Σ1′,…,ΣM′,𝝀\)‖F\\displaystyle\\bigl\\\|\\mathrm\{BCov\}\(\\Sigma\_\{1\},\\ldots,\\Sigma\_\{M\};\\bm\{\\lambda\}\)\-\\mathrm\{BCov\}\(\\Sigma\_\{1\}^\{\\prime\},\\ldots,\\Sigma\_\{M\}^\{\\prime\};\\bm\{\\lambda\}\)\\bigr\\\|\_\{F\}≤Lbary​∑m=1Mλm​‖Σm−Σm′‖F,\\displaystyle\\quad\\leq L\_\{\\mathrm\{bary\}\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\\|\\Sigma\_\{m\}\-\\Sigma\_\{m\}^\{\\prime\}\\\|\_\{F\},\(69\)withLbary=O\(κ−1/2\)L\_\{\\mathrm\{bary\}\}=O\(\\kappa^\{\-1/2\}\)whereκ\\kappais the strong\-convexity constant from Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)\.

###### Theorem III\.3

Fixδ∈\(0,1\)\\delta\\in\(0,1\)and instantiate Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)at levelδ/2\\delta/2, so that𝒵=𝒵R⁡\(δ/2\)\\mathcal\{Z\}=\\mathcal\{Z\}\_\{R\(\\delta/2\)\}andℙ⁡\(ℰtrunc​\(δ/2\)\)≥1−δ/2\\mathbb\{P\}\(\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta/2\)\)\\geq 1\-\\delta/2\. Under Assumptions[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1),[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4), and[III\.5](https://arxiv.org/html/2608.19914#S3.Thmassumption5), suppose𝐋^∗∈ℒB\\hat\{\\mathbf\{L\}\}^\{\*\}\\in\\mathcal\{L\}\_\{B\}and that the true target distributionℙ∗\\mathbb\{P\}^\{\*\}satisfiesWp​\(ℙ∗,b𝛌∗\)≤ϵnW\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\leq\\epsilon\_\{n\}, as before\. Define the*barycentric Rademacher complexity*of the loss classℱB:=\{𝐳↦𝐳⊤​𝐋𝐳:𝐋∈ℒB\}\\mathcal\{F\}\_\{B\}:=\\\{\\mathbf\{z\}\\mapsto\\mathbf\{z\}^\{\\top\}\\mathbf\{L\}\\mathbf\{z\}:\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}\\\}as

ℜn​\(ℒB,𝝀\):=B​∑m=1Mλm​𝔼𝝈m​\[1nm​‖∑i=1nmσm,i​𝚺m,i‖F\],\\displaystyle\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\):=B\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\,\\mathbb\{E\}\_\{\\bm\{\\sigma\}\_\{m\}\}\\\!\\left\[\\frac\{1\}\{n\_\{m\}\}\\Bigl\\\|\\sum\_\{i=1\}^\{n\_\{m\}\}\\sigma\_\{m,i\}\\bm\{\\Sigma\}\_\{m,i\}\\Bigr\\\|\_\{F\}\\right\],\(70\)where\{σm,i\}\\\{\\sigma\_\{m,i\}\\\}are i\.i\.d\.Uniform⁡\(\{−1,\+1\}\)\\mathrm\{Uniform\}\(\\\{\-1,\+1\\\}\)independent of the data\. The following hold\.

1. \(i\)Conditionally onℰtrunc​\(δ/2\)\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta/2\), the barycentric Rademacher complexity satisfies the explicit non\-asymptotic bound ℜn​\(ℒB,𝝀\)≤B​D𝒵2​∑m=1Mλmnm≤B​D𝒵2nmin,\\displaystyle\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\)\\;\\leq\\;BD\_\{\\mathcal\{Z\}\}^\{2\}\\sum\_\{m=1\}^\{M\}\\frac\{\\lambda\_\{m\}\}\{\\sqrt\{n\_\{m\}\}\}\\;\\leq\\;\\frac\{BD\_\{\\mathcal\{Z\}\}^\{2\}\}\{\\sqrt\{n\_\{\\min\}\}\},\(71\)withD𝒵D\_\{\\mathcal\{Z\}\}as in Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)at levelδ/2\\delta/2\.
2. \(ii\)With probability at least1−δ1\-\\deltaover the joint draw of all source samples \(a union bound overℰtrunc​\(δ/2\)\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta/2\)and the concentration event of part \(ii\) at levelδ/2\\delta/2\), Rℙ∗​\(𝐋^∗\)≤R^ϵn​\(𝐋^∗\)⏟computable from data\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\leq\\underbrace\{\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\}\_\{\\text\{computable from data\}\}\+2​Lbary​ℜn​\(ℒB​𝝀\)\+B​Lbary​D𝒵2​2​log⁡\(M/δ\)nmin⏟=:Δn​\(δ\)→0,\\displaystyle\+\\underbrace\{2L\_\{\\mathrm\{bary\}\}\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\}\\bm\{\\lambda\}\)\+\\frac\{BL\_\{\\mathrm\{bary\}\}D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{2\\log\(M/\\delta\)\}\}\{\\sqrt\{n\_\{\\min\}\}\}\}\_\{\\displaystyle=:\\Delta\_\{n\}\(\\delta\)\\to 0\},\(72\)whereR^ϵn​\(𝐋^∗\)\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)is the WDRO empirical objective value, fully computable from training data\.
3. \(iii\)SubstitutingD𝒵=O⁡\(log⁡\(n/δ\)\)D\_\{\\mathcal\{Z\}\}=O\(\\log\(n/\\delta\)\)from Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)into \([72](https://arxiv.org/html/2608.19914#S3.E72)\), the out\-of\-sample correction satisfies, asnmin→∞n\_\{\\min\}\\to\\infty, Δn​\(δ\)=O⁡\(Lbary​Bnmin​log⁡nδ​log⁡Mδ\)⟶0,\\displaystyle\\Delta\_\{n\}\(\\delta\)=O\\\!\\left\(\\frac\{L\_\{\\mathrm\{bary\}\}\\,B\}\{\\sqrt\{n\_\{\\min\}\}\}\\log\\frac\{n\}\{\\delta\}\\sqrt\{\\log\\frac\{M\}\{\\delta\}\}\\right\)\\longrightarrow 0,\(73\)soR^ϵn​\(𝐋^∗\)\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)remains an asymptotically tight, and upper bound on the true riskRℙ∗​\(𝐋^∗\)R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\.

\[A proof is given in Appendix[B](https://arxiv.org/html/2608.19914#A2)\.\]

###### Corollary III\.3

Under the conditions of Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3), let𝐋∗:=arg⁡min𝐋∈ℒB​Rℙ∗​\(𝐋\)\\mathbf\{L\}^\{\*\}:=\\arg\\min\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}\}R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}\)be the oracle minimizer underℙ∗\\mathbb\{P\}^\{\*\}\. With probability at least1−δ1\-\\delta:

Rℙ∗​\(𝐋^∗\)−Rℙ∗​\(𝐋∗\)≤4​Lbary​ℜn​\(ℒB,𝝀\)\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\;\\leq\\;4L\_\{\\mathrm\{bary\}\}\\,\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\)\+2​B​Lbary​D𝒵2​2​log⁡\(M/δ\)nmin\+ϵn​Bq,\\displaystyle\\quad\+\\frac\{2BL\_\{\\mathrm\{bary\}\}D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{2\\log\(M/\\delta\)\}\}\{\\sqrt\{n\_\{\\min\}\}\}\+\\epsilon\_\{n\}B\_\{q\},\(74\)whereBq:=sup𝐋∈ℒB‖vec⁡\(𝐋\)‖qB\_\{q\}:=\\sup\_\{\\mathbf\{L\}\\in\\mathcal\{L\}\_\{B\}\}\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}\. With the adaptive choiceϵn=C0​D𝒵2​log⁡\(M/δ\)/nmin\\epsilon\_\{n\}=C\_\{0\}D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{\\log\(M/\\delta\)/n\_\{\\min\}\}:

Rℙ∗​\(𝐋^∗\)−Rℙ∗​\(𝐋∗\)\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)=O⁡\(\(B​Lbary\+Bq\)​D𝒵2nmin​log⁡Mδ\)\.\\displaystyle\\quad=O\\\!\\left\(\\frac\{\(BL\_\{\\mathrm\{bary\}\}\+B\_\{q\}\)\\,D\_\{\\mathcal\{Z\}\}^\{2\}\}\{\\sqrt\{n\_\{\\min\}\}\}\\sqrt\{\\log\\frac\{M\}\{\\delta\}\}\\right\)\.\(75\)

Proof\.Inserting and subtractingR^ϵn​\(𝐋^∗\)\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)andR^ϵn​\(𝐋∗\)\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}^\{\*\}\):

Rℙ∗​\(𝐋^∗\)−Rℙ∗​\(𝐋∗\)=\[Rℙ∗​\(𝐋^∗\)−R^ϵn​\(𝐋^∗\)\]\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)=\\bigl\[R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\bigr\]\+\[R^ϵn​\(𝐋^∗\)−R^ϵn​\(𝐋∗\)\]⏟≤0\+\[R^ϵn​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)\]\.\\displaystyle\+\\underbrace\{\\bigl\[\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}^\{\*\}\)\\bigr\]\}\_\{\\leq 0\}\+\[\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\]\.\(76\)The middle term is non\-positive by optimality of𝐋^∗\\hat\{\\mathbf\{L\}\}^\{\*\}\. The first term is bounded byΔn​\(δ\)\\Delta\_\{n\}\(\\delta\)via Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)\(ii\)\. For the third term,R^ϵn​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)=\[Rb^M∗​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)\]\+ϵn​‖vec⁡\(𝐋∗\)‖q\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)=\\bigl\[R\_\{\\hat\{b\}^\{\*\}\_\{M\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\bigr\]\+\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\(\\mathbf\{L\}^\{\*\}\)\\\|\_\{q\}\. By the triangle inequality and Lemma[III\.3](https://arxiv.org/html/2608.19914#S3.Thmlemma3):

Rb^M∗​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)≤\|Rb^M∗​\(𝐋∗\)−Rb𝝀∗​\(𝐋∗\)\|\\displaystyle R\_\{\\hat\{b\}^\{\*\}\_\{M\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\leq\\bigl\|R\_\{\\hat\{b\}^\{\*\}\_\{M\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\mathbf\{L\}^\{\*\}\)\\bigr\|\+\|Rb𝝀∗​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)\|\\displaystyle\\quad\+\\bigl\|R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\bigr\|≤B​‖Σ^𝝀−Σ¯𝝀∗‖F\+Wp​\(ℙ∗,b𝝀∗\)​‖vec⁡\(𝐋∗\)‖q,\\displaystyle\\leq B\\\|\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\-\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}\+W\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\\|\\mathrm\{vec\}\(\\mathbf\{L\}^\{\*\}\)\\\|\_\{q\},\(77\)where the second inequality applies Cauchy\-Schwarz to the first term and Kantorovich duality \(as in \([124](https://arxiv.org/html/2608.19914#A2.E124)\)\) to the second\. SinceWp​\(ℙ∗,b𝝀∗\)≤ϵnW\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\leq\\epsilon\_\{n\}, the Kantorovich term is absorbed:R^ϵn​\(𝐋∗\)−Rℙ∗​\(𝐋∗\)≤B​‖Σ^𝝀−Σ¯𝝀∗‖F\+ϵn​Bq\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\leq B\\\|\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\-\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}\+\\epsilon\_\{n\}B\_\{q\}\. Applying the Rademacher and McDiarmid union\-bound argument \([132](https://arxiv.org/html/2608.19914#A2.E132)\) toB​‖Σ^𝝀−Σ¯𝝀∗‖FB\\\|\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\-\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}under the same probability\-1−δ1\-\\deltaevent yieldsB​‖Σ^𝝀−Σ¯𝝀∗‖F≤Δn​\(δ\)B\\\|\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\-\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}\\leq\\Delta\_\{n\}\(\\delta\), so the third term is at mostΔn​\(δ\)\+ϵn​Bq\\Delta\_\{n\}\(\\delta\)\+\\epsilon\_\{n\}B\_\{q\}\. HenceRℙ∗​\(𝐋^∗\)−Rℙ∗​\(𝐋∗\)≤2​Δn​\(δ\)\+ϵn​BqR\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\leq 2\\Delta\_\{n\}\(\\delta\)\+\\epsilon\_\{n\}B\_\{q\}, which expands to \([74](https://arxiv.org/html/2608.19914#S3.E74)\)\. Substituting \([71](https://arxiv.org/html/2608.19914#S3.E71)\) and the givenϵn\\epsilon\_\{n\}yields \([75](https://arxiv.org/html/2608.19914#S3.E75)\)\.□\\hfill\\square

###### Corollary III\.4

Under the conditions of Corollary[III\.3](https://arxiv.org/html/2608.19914#S3.Thmcorollary3), with the adaptive radiusϵn=C0​D𝒵2​log⁡\(M/δ\)/nmin\\epsilon\_\{n\}=C\_\{0\}D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{\\log\(M/\\delta\)/n\_\{\\min\}\}, the WDRO\-GL estimator achievesRℙ∗​\(𝐋^∗\)−Rℙ∗​\(𝐋∗\)≤εR\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\mathbb\{P\}^\{\*\}\}\(\\mathbf\{L\}^\{\*\}\)\\leq\\varepsilonwith probability at least1−δ1\-\\deltaprovided

nmin≥C12​\(B​Lbary\+Bq\)2​D𝒵4ε2​log⁡Mδ,\\displaystyle n\_\{\\min\}\\;\\geq\\;\\frac\{C\_\{1\}^\{2\}\\,\(BL\_\{\\mathrm\{bary\}\}\+B\_\{q\}\)^\{2\}\\,D\_\{\\mathcal\{Z\}\}^\{4\}\}\{\\varepsilon^\{2\}\}\\log\\frac\{M\}\{\\delta\},\(78\)whereC1\>0C\_\{1\}\>0is an absolute constant\. The dependence on the number of sources is only logarithmic, and the factorLbary∼κ−1/2L\_\{\\mathrm\{bary\}\}\\sim\\kappa^\{\-1/2\}reflects that greater inter\-source heterogeneity \(smallerκ\\kappa\) makes the barycenter covariance harder to estimate and increases the required sample size\.

Proof\.From \([75](https://arxiv.org/html/2608.19914#S3.E75)\), the excess risk is bounded byC1​\(B​Lbary\+Bq\)​D𝒵2​log⁡\(M/δ\)/nminC\_\{1\}\(BL\_\{\\mathrm\{bary\}\}\+B\_\{q\}\)D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{\\log\(M/\\delta\)/n\_\{\\min\}\}for an absolute constantC1\>0C\_\{1\}\>0\. Setting this at mostε\\varepsilonand solving fornminn\_\{\\min\}yields \([78](https://arxiv.org/html/2608.19914#S3.E78)\)\.□\\hfill\\square

## IVTheoretical Analysis

The out\-of\-sample guarantees established in Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)provide standard DRO coverage with respect to the empirical Wasserstein ball but leave three fundamental questions unaddressed: \(i\) how well does the empirical barycenterb^K∗\\hat\{b\}^\{\*\}\_\{K\}itself approximate the true \(population\-level\) barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}, and at what rate does this approximation improve with sample size; \(ii\) is there a formal, quantifiable sense in which the Wasserstein barycenter is a structurally superior nominal distribution compared to the naïve pooled empirical measure, particularly when inter\-source heterogeneity is large; and \(iii\) can the ambiguity\-set framework certify coverage of the*target\-domain*risk in the complete absence of target samples\. The present section answers all three questions through a sequence of new results\. The guarantees here are*method\-specific*: they exploit the geometry of barycentric interpolation in Wasserstein space and cannot be obtained from generic DRO theory alone\.

Assumption[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1)models each source as an \(possibly degenerate\) Gaussianℙm∗=𝒩⁡\(μm,Σm\)\\mathbb\{P\}\_\{m\}^\{\*\}=\\mathcal\{N\}\(\\mu\_\{m\},\\Sigma\_\{m\}\), whose support is all ofℝN\\mathbb\{R\}^\{N\}\. The Rademacher\-complexity argument of Section[III\-F](https://arxiv.org/html/2608.19914#S3.SS6), in contrast, requires a*uniform, finite*bound on‖𝐳‖2\\\|\\mathbf\{z\}\\\|\_\{2\}to control the loss classℱB\\mathcal\{F\}\_\{B\}\. We reconcile the two by deriving, rather than positing, an effective compact support: with high probability, every sample drawn from a Gaussian source lies within an explicit ball, whose radius depends only on the source second moments and the confidence level\.

###### Lemma IV\.1

Let𝐳∼𝒩⁡\(μ,Σ\)\\mathbf\{z\}\\sim\\mathcal\{N\}\(\\mu,\\Sigma\)withΣ∈𝕊\+N\\Sigma\\in\\mathbb\{S\}\_\{\+\}^\{N\}\. For anyδ∈\(0,1\)\\delta\\in\(0,1\),

ℙ⁡\(‖𝐳‖2CLOSE\\displaystyle\\mathbb\{P\}\\Bigl\(\\\|\\mathbf\{z\}\\\|\_\{2\}\>‖μ‖2\+tr⁡\(Σ\)\+2​λmax​\(Σ\)​log⁡\(1/δ\)⏟=:R⁡\(Σ,μ,δ\)\)\\displaystyle\>\\underbrace\{\\\|\\mu\\\|\_\{2\}\+\\sqrt\{\\mathrm\{tr\}\(\\Sigma\)\}\+\\sqrt\{2\\lambda\_\{\\max\}\(\\Sigma\)\\log\(1/\\delta\)\}\}\_\{=:R\(\\Sigma,\\mu,\\delta\)\}\\Bigr\)≤δ\.\\displaystyle\\leq\\delta\.\(79\)

Proof\.The map𝐳↦‖𝐳‖2\\mathbf\{z\}\\mapsto\\\|\\mathbf\{z\}\\\|\_\{2\}is11\-Lipschitz\. Writing𝐳=μ\+Σ1/2​𝐠\\mathbf\{z\}=\\mu\+\\Sigma^\{1/2\}\\mathbf\{g\}for standard Gaussian𝐠\\mathbf\{g\}, the composite map𝐠↦‖μ\+Σ1/2​𝐠‖2\\mathbf\{g\}\\mapsto\\\|\\mu\+\\Sigma^\{1/2\}\\mathbf\{g\}\\\|\_\{2\}isλmax​\(Σ\)1/2\\lambda\_\{\\max\}\(\\Sigma\)^\{1/2\}\-Lipschitz in𝐠\\mathbf\{g\}, so the Gaussian concentration inequality givesℙ\(∥𝐳∥2≥𝔼∥𝐳∥2\+t\)≤exp\(−t2/\(2λmax\(Σ\)\)\)\\mathbb\{P\}\(\\\|\\mathbf\{z\}\\\|\_\{2\}\\geq\\mathbb\{E\}\\\|\\mathbf\{z\}\\\|\_\{2\}\+t\)\\leq\\exp\(\-t^\{2\}/\(2\\lambda\_\{\\max\}\(\\Sigma\)\)\)for allt≥0t\\geq 0\. Bounding𝔼​‖𝐳‖2≤‖μ‖2\+tr⁡\(Σ\)\\mathbb\{E\}\\\|\\mathbf\{z\}\\\|\_\{2\}\\leq\\\|\\mu\\\|\_\{2\}\+\\sqrt\{\\mathrm\{tr\}\(\\Sigma\)\}by Jensen’s inequality and settingt=2​λmax​\(Σ\)​log⁡\(1/δ\)t=\\sqrt\{2\\lambda\_\{\\max\}\(\\Sigma\)\\log\(1/\\delta\)\}gives \([79](https://arxiv.org/html/2608.19914#S4.E79)\)\.□\\hfill\\square

###### Corollary IV\.1

Letn=∑m=1Mnmn=\\sum\_\{m=1\}^\{M\}n\_\{m\}be the total number of source samples\. For anyδ∈\(0,1\)\\delta\\in\(0,1\), define

R⁡\(δ\)\\displaystyle R\(\\delta\):=maxm∈\[M\]⁡R⁡\(Σm,μm,δn\),\\displaystyle:=\\max\_\{m\\in\[M\]\}R\\bigl\(\\Sigma\_\{m\},\\mu\_\{m\},\\tfrac\{\\delta\}\{n\}\\bigr\),𝒵R⁡\(δ\)\\displaystyle\\mathcal\{Z\}\_\{R\(\\delta\)\}:=\{Σ∈𝕊\+N:tr⁡\(Σ\)≤R​\(δ\)2\}\.\\displaystyle:=\\bigl\\\{\\Sigma\\in\\mathbb\{S\}\_\{\+\}^\{N\}:\\mathrm\{tr\}\(\\Sigma\)\\leq R\(\\delta\)^\{2\}\\bigr\\\}\.\(80\)Then𝒵R⁡\(δ\)\\mathcal\{Z\}\_\{R\(\\delta\)\}is convex and compact, and the event

ℰtrunc​\(δ\)\\displaystyle\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta\):=\{𝚺m,i=𝐳m,i𝐳m,iT∈𝒵R⁡\(δ\),∀m∈\[M\],i∈\[nm\]\}\\displaystyle:=\\bigl\\\{\\bm\{\\Sigma\}\_\{m,i\}=\\mathbf\{z\}\_\{m,i\}\\mathbf\{z\}\_\{m,i\}^\{T\}\\in\\mathcal\{Z\}\_\{R\(\\delta\)\},\\forall m\\in\[M\],\\,i\\in\[n\_\{m\}\]\\bigr\\\}\(81\)satisfiesℙ⁡\(ℰtrunc​\(δ\)\)≥1−δ\\mathbb\{P\}\(\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta\)\)\\geq 1\-\\delta\.

Proof\.Convexity and compactness of𝒵R⁡\(δ\)\\mathcal\{Z\}\_\{R\(\\delta\)\}are immediate \(a sublevel set of the linear functionaltr⁡\(⋅\)\\mathrm\{tr\}\(\\cdot\)intersected with the closed cone𝕊\+N\\mathbb\{S\}\_\{\+\}^\{N\}\)\. By Lemma[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmlemma1)applied at levelδ/n\\delta/nto each of thenni\.i\.d\. draws, each individual event\{∥𝐳m,i∥2≤R\(δ\)\}\\\{\\\|\\mathbf\{z\}\_\{m,i\}\\\|\_\{2\}\\leq R\(\\delta\)\\\}fails with probability at mostδ/n\\delta/n; a union bound over allnndraws, together withtr⁡\(𝚺m,i\)=‖𝐳m,i‖22≤R​\(δ\)2⇔𝚺m,i∈𝒵R⁡\(δ\)\\mathrm\{tr\}\(\\bm\{\\Sigma\}\_\{m,i\}\)=\\\|\\mathbf\{z\}\_\{m,i\}\\\|\_\{2\}^\{2\}\\leq R\(\\delta\)^\{2\}\\iff\\bm\{\\Sigma\}\_\{m,i\}\\in\\mathcal\{Z\}\_\{R\(\\delta\)\}, gives the claim\.□\\hfill\\square

Corollary[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmcorollary1)is the precise sense in which a common convex compact support is compatible with Assumption[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1): it does not hold surely, as an unbounded Gaussian assumption forbids, but it holds on an explicit event of probability at least1−δ1\-\\delta, for a radiusR⁡\(δ\)R\(\\delta\)that is fully determined by the source second moments\. Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)above should accordingly be read as holding onℰtrunc​\(δ\)\\mathcal\{E\}\_\{\\mathrm\{trunc\}\}\(\\delta\), and every downstream probabilistic guarantee absorbs the residual failure probabilityδ\\deltainto its stated confidence level via a union bound, exactly as carried out in Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)\.

Under Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4), the barycenter functionalF⁡\(ν\)=∑m=1Mλm​W22​\(ν,ℙm∗\)F\(\\nu\)=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\nu,\\mathbb\{P\}\_\{m\}^\{\*\}\)isκ\\kappa\-strongly geodesically convex on\(𝒫2​\(𝒵\),W2\)\(\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\),W\_\{2\}\)for someκ\>0\\kappa\>0depending only onD𝒵D\_\{\\mathcal\{Z\}\}, and𝝀\\bm\{\\lambda\}, and the true barycenterb𝝀∗≜b𝝀,2​\(ℙ1∗,…,ℙM∗\)b^\{\*\}\_\{\\bm\{\\lambda\}\}\\triangleq b\_\{\\bm\{\\lambda\},2\}\(\\mathbb\{P\}\_\{1\}^\{\*\},\\ldots,\\mathbb\{P\}\_\{M\}^\{\*\}\)exists and is unique\[[36](https://arxiv.org/html/2608.19914#bib.bib36)\]\. We define the*inter\-source Wasserstein heterogeneity*as the Frećhet standard deviation of the source distributions about their common barycenter:

ℋ𝝀≜\(∑m=1Mλm​W22​\(ℙm∗,b𝝀∗\)\)1/2\.\\displaystyle\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\;\\triangleq\\;\\biggl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\,W\_\{2\}^\{2\}\\\!\\bigl\(\\mathbb\{P\}\_\{m\}^\{\*\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)\\biggr\)^\{\\\!1/2\}\.\(82\)Largerℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}indicates greater spread of the sources in Wasserstein space;ℋ𝝀=0\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}=0if and only if all source distributions coincide\.

### IV\-AFinite\-Sample Concentration of the Empirical Barycenter

Our first result is a non\-asymptotic perturbation bound that propagates individual\-source estimation errors through the barycenter map\. Unlike standard individual\-source bounds, the result here explicitly separates the contribution of*sampling noise*\(ξM\\xi\_\{M\}\) from the contribution of*inter\-source heterogeneity*\(ℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\), yielding a rate that adapts to the geometric configuration of the source distributions\.

###### Lemma IV\.2

Under Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4), for any empirical source measures\{ℙ^m\}m=1M⊂𝒫2​\(𝒵\)\\\{\\hat\{\\mathbb\{P\}\}\_\{m\}\\\}\_\{m=1\}^\{M\}\\subset\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\), letξM≜∑m=1Mλm​W22​\(ℙ^m,ℙm∗\)\\xi\_\{M\}\\triangleq\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)denote the weighted aggregate source estimation error\. Then

W2​\(b^M∗,b𝝀∗\)≤C0κ​\(ℋ𝝀1/2​ξM1/4\+ξM1/2\),\\displaystyle W\_\{2\}\\\!\\bigl\(\\hat\{b\}^\{\*\}\_\{M\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)\\;\\leq\\;\\frac\{C\_\{0\}\}\{\\kappa\}\\Bigl\(\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\,\\xi\_\{M\}^\{1/4\}\\;\+\\;\\xi\_\{M\}^\{1/2\}\\Bigr\),\(83\)whereC0\>0C\_\{0\}\>0is a universal constant andκ\\kappais the strong\-convexity constant ofFFfrom Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)\.

\[A proof is given in Appendix[C](https://arxiv.org/html/2608.19914#A3)\.\]

Combining Lemma[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmlemma2)with the individual\-source concentration under Assumption[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1)\(identical to that used in Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)\), and a union bound over theMMsources, yields the following main concentration theorem\.

###### Theorem IV\.1

Under Assumptions[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)and[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1), for anyβ∈\(0,1\)\\beta\\in\(0,1\)define the aggregate deviation parameter

ξ¯n​\(β\)≜∑m=1Kλm​ϵnm2​\(βM\),\\displaystyle\\bar\{\\xi\}\_\{n\}\(\\beta\)\\;\\triangleq\\;\\sum\_\{m=1\}^\{K\}\\lambda\_\{m\}\\,\\epsilon\_\{n\_\{m\}\}^\{2\}\\\!\\\!\\left\(\\frac\{\\beta\}\{M\}\\right\),\(84\)whereϵn​\(β\)=c1n​log⁡2β\\epsilon\_\{n\}\(\\beta\)=\\sqrt\{\\frac\{c\_\{1\}\}\{n\}\\log\\frac\{2\}\{\\beta\}\}withc1c\_\{1\}being a constant\. Then, with probability at least1−β1\-\\betaover the joint draw of all source samples,

W2​\(b^M∗,b𝝀∗\)≤εnbary​\(β\)\\displaystyle W\_\{2\}\\\!\\bigl\(\\hat\{b\}^\{\*\}\_\{M\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)\\leq\\varepsilon\_\{n\}^\{\\mathrm\{bary\}\}\(\\beta\)≜C0κ​\[ℋ𝝀1/2​ξ¯n​\(β\)1/4\+ξ¯n​\(β\)1/2\]\.\\displaystyle\\quad\\triangleq\\boxed\{\\frac\{C\_\{0\}\}\{\\kappa\}\\Bigl\[\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\,\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/4\}\\;\+\\;\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/2\}\\Bigr\]\.\}\(85\)

Proof\.Under Assumption[III\.1](https://arxiv.org/html/2608.19914#S3.Thmassumption1), Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)applied to thekk\-th source with confidence levelβ/M\\beta/Mgives,∀m∈\[M\]\\forall m\\in\[M\]

ℙnm\{W2\(ℙ^m,ℙm∗\)≤ϵnm\(β/M\)\}≥1−βM\.\\displaystyle\\mathbb\{P\}^\{n\_\{m\}\}\\\!\\left\\\{W\_\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\leq\\epsilon\_\{n\_\{m\}\}\(\\beta/M\)\\right\\\}\\geq 1\-\\frac\{\\beta\}\{M\}\.\(86\)Letℰ=⋂m=1M\{W2\(ℙ^m,ℙm∗\)≤ϵnm\(β/M\)\}\\mathcal\{E\}=\\bigcap\_\{m=1\}^\{M\}\\bigl\\\{W\_\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\leq\\epsilon\_\{n\_\{m\}\}\(\\beta/M\)\\bigr\\\}\. By the union bound and independence of the source samples,ℙn​\(ℰ\)≥1−β\\mathbb\{P\}^\{n\}\(\\mathcal\{E\}\)\\geq 1\-\\beta\. On the eventℰ\\mathcal\{E\}:

ξM=∑m=1Mλm​W22​\(ℙ^m,ℙm∗\)\\displaystyle\\xi\_\{M\}=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)≤∑m=1Mλm​ϵnm2​\(β/M\)\\displaystyle\\leq\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\,\\epsilon\_\{n\_\{m\}\}^\{2\}\(\\beta/M\)=ξ¯n​\(β\)\.\\displaystyle=\\bar\{\\xi\}\_\{n\}\(\\beta\)\.\(87\)Substituting \([87](https://arxiv.org/html/2608.19914#S4.E87)\) into Lemma[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmlemma2)yields \([85](https://arxiv.org/html/2608.19914#S4.E85)\) onℰ\\mathcal\{E\}, which has probability at least1−β1\-\\beta\.□\\hfill\\square

### IV\-BStructural Superiority of the Barycentric Nominal Distribution

We now formally characterize the*irreducible mixing bias*that afflicts the pooled nominal distributionℙ^𝝀pool≜∑m=1Mλm​ℙ^m\\hat\{\\mathbb\{P\}\}^\{\\mathrm\{pool\}\}\_\{\\bm\{\\lambda\}\}\\triangleq\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\hat\{\\mathbb\{P\}\}\_\{m\}, and contrast it with the consistent behavior of the empirical barycenter\. The central finding, established under a common\-covariance, heterogeneous\-mean Gaussian source model, is that the Wasserstein distance from the pooled distribution to the true barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}does not vanish asn→∞n\\to\\inftywhenever the source means differ, whereas the barycenter\-based estimator remains consistent regardless\.

###### Theorem IV\.2

Suppose the source distributions are Gaussian with a common covariance:ℙm∗=𝒩⁡\(μm,Σ\)\\mathbb\{P\}\_\{m\}^\{\*\}=\\mathcal\{N\}\(\\mu\_\{m\},\\Sigma\)form∈\[M\]m\\in\[M\], whereΣ∈𝕊\+⁣\+N\\Sigma\\in\\mathbb\{S\}\_\{\+\+\}^\{N\}\. Letμ¯𝛌=∑mλm​μm\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\mu\_\{m\}and define the*between\-source scatter matrix*

Δ𝝀=∑m=1Mλm​\(μm−μ¯𝝀\)​\(μm−μ¯𝝀\)T∈𝕊\+N\.\\displaystyle\\Delta\_\{\\bm\{\\lambda\}\}=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\\;\\in\\;\\mathbb\{S\}\_\{\+\}^\{N\}\.\(88\)Then the true pooled distributionℙ𝛌∗=∑mλm​ℙm∗\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\mathbb\{P\}\_\{m\}^\{\*\}satisfies

W22​\(ℙ𝝀∗,b𝝀∗\)\\displaystyle W\_\{2\}^\{2\}\\\!\\bigl\(\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)≥B2​\(Σ\+Δ𝝀,Σ\)\\displaystyle\\geq B^\{2\}\\\!\\bigl\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\},\\,\\Sigma\\bigr\)≥ℋ𝝀44​N​σmax​\(Σ\)\+4​ℋ𝝀2\>0,\\displaystyle\\geq\\frac\{\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{4\}\}\{4N\\,\\sigma\_\{\\max\}\(\\Sigma\)\+4\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\}\\;\>\\;0,\(89\)wheneverℋ𝛌\>0\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\>0, whereσmax​\(Σ\)\\sigma\_\{\\max\}\(\\Sigma\)is the spectral norm ofΣ\\SigmaandB⁡\(⋅,⋅\)B\(\\cdot,\\cdot\)is the Bures\-Wasserstein distance\. In contrast,W2​\(b^M∗,b𝛌∗\)→0W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\to 0almost surely asnmin→∞n\_\{\\min\}\\to\\infty\.

\[A proof is given in Appendix[D](https://arxiv.org/html/2608.19914#A4)\.\]

###### Proposition IV\.1

Suppose the sources share a common mean,μm≡μ\\mu\_\{m\}\\equiv\\mufor allm∈\[M\]m\\in\[M\], and pairwise\-commuting covariancesℙm∗=𝒩⁡\(μ,Σm\)\\mathbb\{P\}\_\{m\}^\{\*\}=\\mathcal\{N\}\(\\mu,\\Sigma\_\{m\}\)withΣm=Q​diag​\(σm,1,…,σm,N\)​QT\\Sigma\_\{m\}=Q\\mathrm\{diag\}\(\\sigma\_\{m,1\},\\ldots,\\sigma\_\{m,N\}\)\\allowbreak Q^\{T\}for a common orthonormal eigenbasisQ∈O⁡\(N\)Q\\in\\mathrm\{O\}\(N\)\(O⁡\(N\)\\mathrm\{O\}\(N\)denote the orthogonal group of orderNN\); in particular, by Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1),QQmay always be chosen withQ​𝐞N=𝟏/NQ\\mathbf\{e\}\_\{N\}=\\mathbf\{1\}/\\sqrt\{N\}, since everyΣm\\Sigma\_\{m\}here annihilates𝟏\\mathbf\{1\}\. Writeσ¯arith,k=∑mλm​σm,k\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}=\\sum\_\{m\}\\lambda\_\{m\}\\sigma\_\{m,k\}for the arithmetic\-mean eigenvalues and

σ¯𝝀,k=\(∑m=1Mλmσm,k\)2,k=1,…,N,\\displaystyle\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\\;=\\;\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\sqrt\{\\sigma\_\{m,k\}\}\\Bigr\)^\{\\\!2\},\\qquad k=1,\\ldots,N,\(90\)for the coordinatewise barycentric eigenvalues\. Then the true barycenter isb𝛌∗=𝒩⁡\(μ,Q​diag​\(𝛔¯𝛌\)​QT\)b^\{\*\}\_\{\\bm\{\\lambda\}\}=\\mathcal\{N\}\\bigl\(\\mu,\\,Q\\,\\mathrm\{diag\}\(\\bar\{\\bm\{\\sigma\}\}\_\{\\bm\{\\lambda\}\}\)\\,Q^\{T\}\\bigr\), and, withΣ¯arith=∑mλm​Σm\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\Sigma\_\{m\},

W22​\(ℙ𝝀∗,b𝝀∗\)\\displaystyle W\_\{2\}^\{2\}\\\!\\bigl\(\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)≥∑k=1N\(σ¯arith,k−σ¯𝝀,k\)2\\displaystyle\\geq\\sum\_\{k=1\}^\{N\}\\Bigl\(\\sqrt\{\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\}\-\\sqrt\{\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\}\\Bigr\)^\{\\\!2\}\(91\)≥maxi≠j⁡λi2​λj2​B4​\(Σi,Σj\)4​N​σmax​\(Σ¯arith\)\>0,\\displaystyle\\geq\\max\_\{i\\neq j\}\\;\\frac\{\\lambda\_\{i\}^\{2\}\\lambda\_\{j\}^\{2\}\\,B^\{4\}\(\\Sigma\_\{i\},\\Sigma\_\{j\}\)\}\{4N\\,\\sigma\_\{\\max\}\\bigl\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\\bigr\)\}\\;\>\\;0,\(92\)wheneverΣi≠Σj\\Sigma\_\{i\}\\neq\\Sigma\_\{j\}for some pairi,ji,jwithλi,λj\>0\\lambda\_\{i\},\\lambda\_\{j\}\>0\.

\[A proof is given in Appendix[E](https://arxiv.org/html/2608.19914#A5)\.\]

### IV\-CData\-Driven Radius Selection

Theorems[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1)and[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmtheorem2)jointly prescribe a principled, data\-driven procedure for choosingϵ\\epsilonin the MS\-WDRO problem \([31](https://arxiv.org/html/2608.19914#S3.E31)\)\. The radius should be tight enough to avoid over\-conservatism, yet large enough to ensure the true barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}lies in the ambiguity set with prescribed probability\. The following corollary makes this precise\.

###### Corollary IV\.2

Let the assumptions of Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1)hold\. Fix anyβ∈\(0,1\)\\beta\\in\(0,1\)\. Set the radius

ϵn∗​\(β\)=C0κ​\[ℋ^𝝀1/2​ξ¯n​\(β\)1/4\+ξ¯n​\(β\)1/2\],\\displaystyle\\epsilon^\{\*\}\_\{n\}\(\\beta\)\\;=\\;\\frac\{C\_\{0\}\}\{\\kappa\}\\Bigl\[\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\,\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/4\}\+\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/2\}\\Bigr\],\(94\)whereℋ^𝛌=\(∑mλm​W22​\(ℙ^m,b^M∗\)\)1/2\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}=\\bigl\(\\sum\_\{m\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\hat\{b\}^\{\*\}\_\{M\}\)\\bigr\)^\{1/2\}is the empirical inter\-source heterogeneity andξ¯n​\(β\)\\bar\{\\xi\}\_\{n\}\(\\beta\)is as in \([84](https://arxiv.org/html/2608.19914#S4.E84)\)\. Then, with probability at least1−2​β1\-2\\beta:

b𝝀∗∈𝒲ϵn∗​\(β\),2​\(ℙ^1,…,ℙ^M,𝝀\),\\displaystyle b^\{\*\}\_\{\\bm\{\\lambda\}\}\\;\\in\\;\\mathcal\{W\}\_\{\\epsilon^\{\*\}\_\{n\}\(\\beta\),2\}\\\!\\bigl\(\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\};\\bm\{\\lambda\}\\bigr\),\(95\)andϵn∗​\(β\)→0\\epsilon^\{\*\}\_\{n\}\(\\beta\)\\to 0almost surely asnmin→∞n\_\{\\min\}\\to\\infty\. Moreover, the radius satisfies the monotone decay

ϵn∗​\(β\)\\displaystyle\\epsilon^\{\*\}\_\{n\}\(\\beta\)≤C0κ​\[ℋ^𝝀1/2​\(c1nmin​log⁡2​Mβ\)1/4\]\\displaystyle\\leq\\frac\{C\_\{0\}\}\{\\kappa\}\\left\[\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\left\(\\frac\{c\_\{1\}\}\{n\_\{\\min\}\}\\log\\frac\{2M\}\{\\beta\}\\right\)^\{1/4\}\\right\]C0κ​\[\+\(c1nmin​log⁡2​Mβ\)1/2\],\\displaystyle\\quad\\frac\{C\_\{0\}\}\{\\kappa\}\\left\[\+\\left\(\\frac\{c\_\{1\}\}\{n\_\{\\min\}\}\\log\\frac\{2M\}\{\\beta\}\\right\)^\{1/2\}\\right\],\(96\)contrasted with the pooled\-distribution radius lower boundϵpool≥ℋ𝛌2/4​N​σmax\+4​ℋ𝛌2\\epsilon^\{\\mathrm\{pool\}\}\\geq\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}/\\sqrt\{4N\\sigma\_\{\\max\}\+4\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\}, which is bounded away from zero\.

Proof\.By the triangle inequality,

\|ℋ^𝝀−ℋ𝝀\|≤∑mλm​W22​\(ℙ^m,ℙm∗\)=ξM1/2\|\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}\-\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\|\\leq\\sqrt\{\\sum\_\{m\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\}=\\xi\_\{M\}^\{1/2\}\. With probability at least1−β1\-\\beta,ξM≤ξ¯n​\(β\)\\xi\_\{M\}\\leq\\bar\{\\xi\}\_\{n\}\(\\beta\)\(Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1)\), soℋ^𝝀≤ℋ𝝀\+ξ¯n​\(β\)1/2\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}\\leq\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\+\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/2\}\. Substitutingℋ^𝝀≤ℋ𝝀\+ξ¯n​\(β\)1/2\\hat\{\\mathcal\{H\}\}\_\{\\bm\{\\lambda\}\}\\leq\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\+\\bar\{\\xi\}\_\{n\}\(\\beta\)^\{1/2\}into \([94](https://arxiv.org/html/2608.19914#S4.E94)\) and absorbing the lower\-order term intoC0C\_\{0\}, the right\-hand side of \([94](https://arxiv.org/html/2608.19914#S4.E94)\) is at least as large asεnbary​\(β\)\\varepsilon\_\{n\}^\{\\mathrm\{bary\}\}\(\\beta\)from \([85](https://arxiv.org/html/2608.19914#S4.E85)\) with probability1−β1\-\\beta\. A second application of Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1)at levelβ\\betathen yields \([95](https://arxiv.org/html/2608.19914#S4.E95)\) with combined probability at least1−2​β1\-2\\betaby the union bound\. The monotone decay and the pooled\-distribution comparison follow from \([96](https://arxiv.org/html/2608.19914#S4.E96)\) and Theorem[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmtheorem2), respectively\.□\\hfill\\square

### IV\-DTarget\-Free Risk Approximation via Source Barycenter

In the motivating scenario of this paper, no target\-domain samples are available at all\. The purpose of this section is to show that the WDRO\-GL framework with the barycentric nominal distribution nonetheless provides a valid risk certificate for the*target\-domain*risk, at the sole cost of enlarging the Wasserstein radius by the target\-to\-barycenter proximityω\\omega, a quantity that characterizes how well the source barycenter approximates the unknown target distribution\. This is the strongest theoretical justification for the proposed framework\.

###### Theorem IV\.3

Letℙ∗\\mathbb\{P\}^\{\\ast\}denote the unknown target distribution and suppose there existsω≥0\\omega\\geq 0such that

W2​\(ℙ∗,b𝝀∗\)≤ω\.\\displaystyle W\_\{2\}\\\!\\bigl\(\\mathbb\{P\}^\{\\ast\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)\\leq\\omega\.\(97\)Under the assumptions of Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1), setϵ=εnbary​\(β\)\+ω\\epsilon=\\varepsilon\_\{n\}^\{\\mathrm\{bary\}\}\(\\beta\)\+\\omega\. Then with probability at least1−β1\-\\betaover the draw of source samples,

ℙ∗∈𝒲ϵ,2​\(ℙ^1,…,ℙ^M,𝝀\),\\displaystyle\\mathbb\{P\}^\{\\ast\}\\;\\in\\;\\mathcal\{W\}\_\{\\epsilon,2\}\\\!\\bigl\(\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\};\\bm\{\\lambda\}\\bigr\),\(98\)and consequently, for every graph Laplacian estimator𝐋\\mathbf\{L\}:

Rℙ∗​\(𝐋\)≤R𝒲ϵ,2​\(𝐋\),\\displaystyle R\_\{\\mathbb\{P\}^\{\\ast\}\}\(\\mathbf\{L\}\)\\;\\leq\\;R\_\{\\mathcal\{W\}\_\{\\epsilon,2\}\}\(\\mathbf\{L\}\),\(99\)whereR𝒲ϵ,2​\(𝐋\)=supℚ∈𝒲ϵ,2𝔼ℚ​\[ℓ⁡\(𝐋,⋅\)\]R\_\{\\mathcal\{W\}\_\{\\epsilon,2\}\}\(\\mathbf\{L\}\)=\\sup\_\{\\mathbb\{Q\}\\in\\mathcal\{W\}\_\{\\epsilon,2\}\}\\mathbb\{E\}\_\{\\mathbb\{Q\}\}\[\\ell\(\\mathbf\{L\},\\cdot\)\]is the worst\-case risk of \([31](https://arxiv.org/html/2608.19914#S3.E31)\)\.

Proof\.By Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1), with probability at least1−β1\-\\beta:

W2​\(b^M∗,b𝝀∗\)≤εnbary​\(β\)\.\\displaystyle W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\,b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\;\\leq\\;\\varepsilon\_\{n\}^\{\\mathrm\{bary\}\}\(\\beta\)\.\(100\)By the triangle inequality forW2W\_\{2\}and the target proximity assumption \([97](https://arxiv.org/html/2608.19914#S4.E97)\):

W2​\(ℙ∗,b^M∗\)\\displaystyle W\_\{2\}\\\!\\bigl\(\\mathbb\{P\}^\{\\ast\},\\hat\{b\}^\{\*\}\_\{M\}\\bigr\)≤W2​\(ℙ∗,b𝝀∗\)\+W2​\(b𝝀∗,b^M∗\)\\displaystyle\\;\\leq\\;W\_\{2\}\(\\mathbb\{P\}^\{\\ast\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\+W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\hat\{b\}^\{\*\}\_\{M\}\)≤ω\+εnbary​\(β\)=ϵ\.\\displaystyle\\;\\leq\\;\\omega\+\\varepsilon\_\{n\}^\{\\mathrm\{bary\}\}\(\\beta\)=\\epsilon\.\(101\)Since the ambiguity set𝒲ϵ,2​\(ℙ^1,…,ℙ^M,𝝀\)\\mathcal\{W\}\_\{\\epsilon,2\}\(\\hat\{\\mathbb\{P\}\}\_\{1\},\\ldots,\\hat\{\\mathbb\{P\}\}\_\{M\};\\bm\{\\lambda\}\)is the Wasserstein ball of radiusϵ\\epsiloncentered atb^M∗\\hat\{b\}^\{\*\}\_\{M\}, \([101](https://arxiv.org/html/2608.19914#S4.E101)\) givesℙ∗∈𝒲ϵ,2\\mathbb\{P\}^\{\\ast\}\\in\\mathcal\{W\}\_\{\\epsilon,2\}on the stated probability event\. The risk certificate \([99](https://arxiv.org/html/2608.19914#S4.E99)\) then follows immediately from the definition of the worst\-case risk and the inclusion \([98](https://arxiv.org/html/2608.19914#S4.E98)\)\.□\\hfill\\square

Fig\. 1:Overall framework of the unrolled MS\-WDRO\.

## VAlgorithm Unrolling for Automatic Parameter Learning

Algorithm[1](https://arxiv.org/html/2608.19914#alg1)involves four categories of free parameters whose joint calibration governs the quality of the recovered Laplacian\. The sparsity regularization coefficientρ\>0\\rho\>0enters the objective \([33](https://arxiv.org/html/2608.19914#S3.E33)\) and, after the Laplacian identity \([34](https://arxiv.org/html/2608.19914#S3.E34)\), defines the effective nominal matrix𝐊=Σ^𝝀\+ρ​𝐇\\mathbf\{K\}=\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\+\\rho\\mathbf\{H\}\([35](https://arxiv.org/html/2608.19914#S3.E35)\) used at every iteration; it is therefore an*objective\-level*parameter that shapes which optimization problem is being solved\. The ADMM augmented\-Lagrangian penaltyϱ\>0\\varrho\>0, introduced in \([46](https://arxiv.org/html/2608.19914#S3.E46)\), is structurally different: it is an*algorithm\-level*parameter that leaves the underlying problem \([45](https://arxiv.org/html/2608.19914#S3.E45)\) invariant but governs the analytic diagonal entries of theΞ\\Xi\-update via Lemma[III\.4](https://arxiv.org/html/2608.19914#S3.Thmlemma4)\([49](https://arxiv.org/html/2608.19914#S3.E49)\), sets the block\-shrinkage thresholdϵ/ϱ\\epsilon/\\varrhoin the𝐂\\mathbf\{C\}\-update \([51](https://arxiv.org/html/2608.19914#S3.E51)\), and determines the dual\-ascent step size in \([52](https://arxiv.org/html/2608.19914#S3.E52)\)\. The Wasserstein ambiguity set radiusϵ\>0\\epsilon\>0sets the robustness level and couples withϱ\\varrhothrough the joint ratioϵ/ϱ\\epsilon/\\varrhoin the proximity step, so that tuning either in isolation while holding the other fixed is suboptimal\. Finally, the barycentric weight vector𝝀∈△M\\bm\{\\lambda\}\\in\\triangle^\{M\}determinesΣ^𝝀\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}via the fixed\-point iteration \([23](https://arxiv.org/html/2608.19914#S3.E23)\) and hence contributes to𝐊\\mathbf\{K\}alongsideρ\\rho\. These four categories interact nonlinearly and their interactions cannot be disentangled:ρ\\rhoand𝝀\\bm\{\\lambda\}jointly construct𝐊\\mathbf\{K\}, whileϵ\\epsilonandϱ\\varrhojointly set the shrinkage threshold\. Cross\-validation over the resulting joint parameter space is computationally prohibitive in the multi\-source regime, and concentration\-inequality\-based radius selection forϵ\\epsilon, while tractable, is inherently conservative\.

To overcome these limitations, we embed the iterative structure of Algorithm[1](https://arxiv.org/html/2608.19914#alg1)into a differentiable architecture via the algorithm unrolling paradigm, reinterpreting each ADMM iteration as a computational layer and lifting all four parameter categories to layer\-specific learnable variables that are jointly calibrated through end\-to\-end backpropagation on a supervised training corpus\. The resultingKℓK\_\{\\ell\}\-layer unrolled network inherits the full interpretability of the ADMM, every layer executes a precisely specified proximal or dual\-ascent substep, while acquiring data\-adaptive calibration capability that the fixed\-parameter solver cannot achieve\. Allowing layer\-specificρ\(k\)\\rho^\{\(k\)\},ϱ\(k\)\\varrho^\{\(k\)\},ϵ\(k\)\\epsilon^\{\(k\)\}, and𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}enables an implicit annealing:ρ\(k\)\\rho^\{\(k\)\}may be large in early layers to impose strong sparsity regularization when the iterate is far from convergence and progressively relax in later layers as the graph structure consolidates;ϱ\(k\)\\varrho^\{\(k\)\}may similarly adapt the ADMM step size along the unrolled trajectory; and𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}may shift the effective nominal distribution as the estimate refines\. The overall framework of the unrolled ADMM for MS\-WDRO is summarized in Figure[1](https://arxiv.org/html/2608.19914#S4.F1)\.

We now specify the layer architecture\. Let𝒮\(k\)=\(Ξ\(k\),𝐂\(k\),𝐘\(k\)\)\\mathcal\{S\}^\{\(k\)\}=\(\\Xi^\{\(k\)\},\\mathbf\{C\}^\{\(k\)\},\\mathbf\{Y\}^\{\(k\)\}\)denote the iterate state entering layerkk, initialized asΞ\(0\)=𝐈N−1\\Xi^\{\(0\)\}=\\mathbf\{I\}\_\{N\-1\},𝐂\(0\)=𝟎\\mathbf\{C\}^\{\(0\)\}=\\mathbf\{0\},𝐘\(0\)=𝟎\\mathbf\{Y\}^\{\(0\)\}=\\mathbf\{0\}\. Each layerkkis parameterized by the tuple𝜽\(k\)=\(ϵ\(k\),ρ\(k\),ϱ\(k\),𝜶\(k\)\)\\bm\{\\theta\}^\{\(k\)\}=\(\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\bm\{\\alpha\}^\{\(k\)\}\), whereϵ\(k\),ρ\(k\),ϱ\(k\)\>0\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\}\>0are positive scalars and𝝀\(k\)=softmax⁡\(𝜶\(k\)\)∈△M\\bm\{\\lambda\}^\{\(k\)\}=\\mathrm\{softmax\}\(\\bm\{\\alpha\}^\{\(k\)\}\)\\in\\triangle^\{M\}is the simplex\-constrained barycentric weight vector obtained via the unconstrained reparametrization𝜶\(k\)∈ℝM\\bm\{\\alpha\}^\{\(k\)\}\\in\\mathbb\{R\}^\{M\}, ensuring feasibility with smooth gradient flow through the simplex constraint\. The layer comprises three sequentially executed differentiable modules\.

The first module,ℳΞ\\mathcal\{M\}\_\{\\Xi\}, constructs the layer\-specific coefficient matrix and updates the primal variableΞ\\Xi\. Given the current weights𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}, it computes

Σ^𝝀\(k\)=BCov⁡\(Σ^1,…,Σ^M,𝝀\(k\)\),\\displaystyle\\hat\{\\Sigma\}^\{\(k\)\}\_\{\\bm\{\\lambda\}\}=\\mathrm\{BCov\}\\\!\\bigl\(\\hat\{\\Sigma\}\_\{1\},\\ldots,\\hat\{\\Sigma\}\_\{M\};\\bm\{\\lambda\}^\{\(k\)\}\\bigr\),\(104\)whereBCov⁡\(⋅,𝝀\(k\)\)\\mathrm\{BCov\}\(\\cdot;\\bm\{\\lambda\}^\{\(k\)\}\)is evaluated via the fixed\-point iteration \([23](https://arxiv.org/html/2608.19914#S3.E23)\) at weights𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}, and its gradient with respect to𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}is obtained by implicit differentiation of the fixed\-point equation, valid under Assumption[III\.5](https://arxiv.org/html/2608.19914#S3.Thmassumption5)\. The module then assembles the layer\-specific nominal matrix

𝐊\(k\)=Σ^𝝀\(k\)\+ρ\(k\)​𝐇,𝐊~\(k\)=𝐏T​𝐊\(k\)​𝐏,\\displaystyle\\mathbf\{K\}^\{\(k\)\}=\\hat\{\\Sigma\}^\{\(k\)\}\_\{\\bm\{\\lambda\}\}\+\\rho^\{\(k\)\}\\mathbf\{H\},\\quad\\tilde\{\\mathbf\{K\}\}^\{\(k\)\}=\\mathbf\{P\}^\{T\}\\mathbf\{K\}^\{\(k\)\}\\mathbf\{P\},\(105\)in whichρ\(k\)\\rho^\{\(k\)\}and𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}jointly determine the effective data\-fidelity matrix at depthkk\. TheΞ\\Xi\-primal update then follows Lemma[III\.4](https://arxiv.org/html/2608.19914#S3.Thmlemma4): the module computes the eigendecomposition

1ϱ\(k\)​𝐏T​\(𝐊\(k\)\+𝐘\(k\)−ϱ\(k\)​𝐂\(k\)\)​𝐏=𝐔\(k\)​𝚲\(k\)​\(𝐔\(k\)\)T\\displaystyle\\frac\{1\}\{\\varrho^\{\(k\)\}\}\\mathbf\{P\}^\{T\}\\\!\\left\(\\mathbf\{K\}^\{\(k\)\}\+\\mathbf\{Y\}^\{\(k\)\}\-\\varrho^\{\(k\)\}\\mathbf\{C\}^\{\(k\)\}\\right\)\\mathbf\{P\}=\\mathbf\{U\}^\{\(k\)\}\\bm\{\\Lambda\}^\{\(k\)\}\\\!\\left\(\\mathbf\{U\}^\{\(k\)\}\\right\)^\{T\}\(106\)and setsΞ\(k\+1\)=𝐔\(k\)​𝐃\(k\)​\(𝐔\(k\)\)T\\Xi^\{\(k\+1\)\}=\\mathbf\{U\}^\{\(k\)\}\\mathbf\{D\}^\{\(k\)\}\(\\mathbf\{U\}^\{\(k\)\}\)^\{T\}, where the diagonal entries of𝐃\(k\)\\mathbf\{D\}^\{\(k\)\}are

Di​i\(k\)=−ϱ\(k\)​Λi​i\(k\)\+\(ϱ\(k\)\)2​\(Λi​i\(k\)\)2\+4​ϱ\(k\)2​ϱ\(k\)\.\\displaystyle D\_\{ii\}^\{\(k\)\}=\\frac\{\-\\varrho^\{\(k\)\}\\Lambda\_\{ii\}^\{\(k\)\}\+\\sqrt\{\\bigl\(\\varrho^\{\(k\)\}\\bigr\)^\{2\}\\\!\\bigl\(\\Lambda\_\{ii\}^\{\(k\)\}\\bigr\)^\{2\}\+4\\varrho^\{\(k\)\}\}\}\{2\\varrho^\{\(k\)\}\}\.\(107\)The mapping\(ρ\(k\),ϱ\(k\),𝝀\(k\)\)↦Ξ\(k\+1\)\(\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\bm\{\\lambda\}^\{\(k\)\}\)\\mapsto\\Xi^\{\(k\+1\)\}is differentiable via matrix perturbation theory for symmetric eigendecompositions\. Observe thatρ\(k\)\\rho^\{\(k\)\}andϱ\(k\)\\varrho^\{\(k\)\}enter the update through structurally distinct channels:ρ\(k\)\\rho^\{\(k\)\}shifts𝐊\(k\)\\mathbf\{K\}^\{\(k\)\}and thereby redefines the optimization landscape at each depth, whileϱ\(k\)\\varrho^\{\(k\)\}controls the analytic step\-size geometry throughDi​i\(k\)D\_\{ii\}^\{\(k\)\}\.

The second module,ℳC\\mathcal\{M\}\_\{C\}, executes the structural projection of𝐂\\mathbf\{C\}via a Moreau proximity step\. Setting𝐙\(k\)=1ϱ\(k\)​𝐘\(k\)\+𝐏​Ξ\(k\+1\)​𝐏T\\mathbf\{Z\}^\{\(k\)\}=\\frac\{1\}\{\\varrho^\{\(k\)\}\}\\mathbf\{Y\}^\{\(k\)\}\+\\mathbf\{P\}\\Xi^\{\(k\+1\)\}\\mathbf\{P\}^\{T\}, the update reads

𝐂\(k\+1\)\\displaystyle\\mathbf\{C\}^\{\(k\+1\)\}=𝐈⊙\[φ\(k\)​\(𝐙\(k\)\)\]\+\+𝐀⊙\[φ\(k\)​\(𝐙\(k\)\)\]−,\\displaystyle=\\mathbf\{I\}\\odot\\left\[\\varphi^\{\(k\)\}\\\!\\left\(\\mathbf\{Z\}^\{\(k\)\}\\right\)\\right\]\_\{\+\}\+\\mathbf\{A\}\\odot\\left\[\\varphi^\{\(k\)\}\\\!\\left\(\\mathbf\{Z\}^\{\(k\)\}\\right\)\\right\]\_\{\-\},\(108\)where the matrix block\-shrinkage operator is

φ\(k\)​\(𝐙\)=max⁡\(1−ϵ\(k\)/ϱ\(k\)‖𝐙‖F,0\)⋅𝐙,\\displaystyle\\varphi^\{\(k\)\}\(\\mathbf\{Z\}\)=\\max\\\!\\left\(1\-\\frac\{\\epsilon^\{\(k\)\}/\\varrho^\{\(k\)\}\}\{\\\|\\mathbf\{Z\}\\\|\_\{F\}\},\\,0\\right\)\\cdot\\mathbf\{Z\},\(109\)\[⋅\]\+\[\\cdot\]\_\{\+\}and\[⋅\]−\[\\cdot\]\_\{\-\}extract the element\-wise positive and negative parts, and𝐀\\mathbf\{A\}is the binary adjacency support mask\. The effective shrinkage thresholdϵ\(k\)/ϱ\(k\)\\epsilon^\{\(k\)\}/\\varrho^\{\(k\)\}is the joint product of two independently trainable scalars:ϵ\(k\)\\epsilon^\{\(k\)\}encodes the desired robustness level whileϱ\(k\)\\varrho^\{\(k\)\}normalizes it relative to the augmented\-Lagrangian penalty\. The mapping\(ϵ\(k\),ϱ\(k\)\)↦𝐂\(k\+1\)\(\\epsilon^\{\(k\)\},\\varrho^\{\(k\)\}\)\\mapsto\\mathbf\{C\}^\{\(k\+1\)\}is almost everywhere differentiable, with the subgradient defined consistently at‖𝐙\(k\)‖F=ϵ\(k\)/ϱ\(k\)\\\|\\mathbf\{Z\}^\{\(k\)\}\\\|\_\{F\}=\\epsilon^\{\(k\)\}/\\varrho^\{\(k\)\}\.

The third module,ℳY\\mathcal\{M\}\_\{Y\}, performs the dual\-ascent step

𝐘\(k\+1\)=𝐘\(k\)\+ϱ\(k\)​\(𝐏​Ξ\(k\+1\)​𝐏T−𝐂\(k\+1\)\),\\displaystyle\\mathbf\{Y\}^\{\(k\+1\)\}=\\mathbf\{Y\}^\{\(k\)\}\+\\varrho^\{\(k\)\}\\\!\\left\(\\mathbf\{P\}\\Xi^\{\(k\+1\)\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}^\{\(k\+1\)\}\\right\),\(110\)which is linear in all preceding outputs and differentiable with respect toϱ\(k\)\\varrho^\{\(k\)\}\. AfterKℓK\_\{\\ell\}layers, the Laplacian estimator is recovered as𝐋^=𝐏​Ξ\(Kℓ\)​𝐏T∈ℒ\\hat\{\\mathbf\{L\}\}=\\mathbf\{P\}\\Xi^\{\(K\_\{\\ell\}\)\}\\mathbf\{P\}^\{T\}\\in\\mathcal\{L\}\. The complete forward pass is summarized in Algorithm[2](https://arxiv.org/html/2608.19914#alg2)\.

Algorithm 2Unrolled ADMM for MS\-WDRO \(Forward Pass\)0:Source datasets

\{𝒟m\}m=1M\\\{\\mathcal\{D\}\_\{m\}\\\}\_\{m=1\}^\{M\}; depth

KℓK\_\{\\ell\}; trainable parameters

𝚯=\{\(ϵ\(k\),ρ\(k\),ϱ\(k\),𝜶\(k\)\)\}k=1Kℓ\\bm\{\\Theta\}=\\\{\(\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\bm\{\\alpha\}^\{\(k\)\}\)\\\}\_\{k=1\}^\{K\_\{\\ell\}\}
0:Estimated graph Laplacian

𝐋^\\hat\{\\mathbf\{L\}\}
1:Compute

Σ^m=1nm​∑i=1nm𝐳m,i​𝐳m,iT\\hat\{\\Sigma\}\_\{m\}=\\tfrac\{1\}\{n\_\{m\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\mathbf\{z\}\_\{m,i\}\\mathbf\{z\}\_\{m,i\}^\{T\},

k∈\[K\]\\;k\\in\[K\]
2:Initialize

Ξ\(0\)=𝐈N−1\\Xi^\{\(0\)\}=\\mathbf\{I\}\_\{N\-1\},

𝐂\(0\)=𝟎\\mathbf\{C\}^\{\(0\)\}=\\mathbf\{0\},

𝐘\(0\)=𝟎\\mathbf\{Y\}^\{\(0\)\}=\\mathbf\{0\}
3:for

k=0k=0to

Kℓ−1K\_\{\\ell\}\-1do

4:

𝝀\(k\)←softmax⁡\(𝜶\(k\)\)\\bm\{\\lambda\}^\{\(k\)\}\\leftarrow\\mathrm\{softmax\}\(\\bm\{\\alpha\}^\{\(k\)\}\)
5:

Σ^𝝀\(k\)←BCov⁡\(Σ^1,…,Σ^M,𝝀\(k\)\)\\hat\{\\Sigma\}^\{\(k\)\}\_\{\\bm\{\\lambda\}\}\\leftarrow\\mathrm\{BCov\}\(\\hat\{\\Sigma\}\_\{1\},\\ldots,\\hat\{\\Sigma\}\_\{M\};\\bm\{\\lambda\}^\{\(k\)\}\)via \([23](https://arxiv.org/html/2608.19914#S3.E23)\)

6:

𝐊\(k\)←Σ^𝝀\(k\)\+ρ\(k\)​𝐇\\mathbf\{K\}^\{\(k\)\}\\leftarrow\\hat\{\\Sigma\}^\{\(k\)\}\_\{\\bm\{\\lambda\}\}\+\\rho^\{\(k\)\}\\mathbf\{H\},

𝐊~\(k\)←𝐏T​𝐊\(k\)​𝐏\\;\\tilde\{\\mathbf\{K\}\}^\{\(k\)\}\\leftarrow\\mathbf\{P\}^\{T\}\\mathbf\{K\}^\{\(k\)\}\\mathbf\{P\}
7:Eigendecomp:

1ϱ\(k\)​𝐏T​\(𝐊\(k\)\+𝐘\(k\)−ϱ\(k\)​𝐂\(k\)\)​𝐏=𝐔\(k\)​𝚲\(k\)​\(𝐔\(k\)\)T\\tfrac\{1\}\{\\varrho^\{\(k\)\}\}\\mathbf\{P\}^\{T\}\\\!\\left\(\\mathbf\{K\}^\{\(k\)\}\+\\mathbf\{Y\}^\{\(k\)\}\-\\varrho^\{\(k\)\}\\mathbf\{C\}^\{\(k\)\}\\right\)\\mathbf\{P\}=\\mathbf\{U\}^\{\(k\)\}\\bm\{\\Lambda\}^\{\(k\)\}\(\\mathbf\{U\}^\{\(k\)\}\)^\{T\}
8:Compute

𝐃\(k\)\\mathbf\{D\}^\{\(k\)\}via \([107](https://arxiv.org/html/2608.19914#S5.E107)\);

Ξ\(k\+1\)←𝐔\(k\)​𝐃\(k\)​\(𝐔\(k\)\)T\\Xi^\{\(k\+1\)\}\\leftarrow\\mathbf\{U\}^\{\(k\)\}\\mathbf\{D\}^\{\(k\)\}\(\\mathbf\{U\}^\{\(k\)\}\)^\{T\}⊳ℳΞ\\triangleright\\;\\mathcal\{M\}\_\{\\Xi\}

9:

𝐙\(k\)←1ϱ\(k\)​𝐘\(k\)\+𝐏​Ξ\(k\+1\)​𝐏T\\mathbf\{Z\}^\{\(k\)\}\\leftarrow\\tfrac\{1\}\{\\varrho^\{\(k\)\}\}\\mathbf\{Y\}^\{\(k\)\}\+\\mathbf\{P\}\\Xi^\{\(k\+1\)\}\\mathbf\{P\}^\{T\}; update

𝐂\(k\+1\)\\mathbf\{C\}^\{\(k\+1\)\}via \([108](https://arxiv.org/html/2608.19914#S5.E108)\)⊳ℳC\\triangleright\\;\\mathcal\{M\}\_\{C\}

10:

𝐘\(k\+1\)←𝐘\(k\)\+ϱ\(k\)​\(𝐏​Ξ\(k\+1\)​𝐏T−𝐂\(k\+1\)\)\\mathbf\{Y\}^\{\(k\+1\)\}\\leftarrow\\mathbf\{Y\}^\{\(k\)\}\+\\varrho^\{\(k\)\}\\\!\\left\(\\mathbf\{P\}\\Xi^\{\(k\+1\)\}\\mathbf\{P\}^\{T\}\-\\mathbf\{C\}^\{\(k\+1\)\}\\right\)⊳ℳY\\triangleright\\;\\mathcal\{M\}\_\{Y\}

11:endfor

12:return

𝐋^=𝐏​Ξ\(Kℓ\)​𝐏T\\hat\{\\mathbf\{L\}\}=\\mathbf\{P\}\\Xi^\{\(K\_\{\\ell\}\)\}\\mathbf\{P\}^\{T\}

The network is trained in a supervised manner on a corpus ofQQlabeled multi\-source instances\{\(𝒟1\(j\),…,𝒟M\(j\),𝐋∗\(j\)\)\}j=1Q\\\{\(\\mathcal\{D\}\_\{1\}^\{\(j\)\},\\ldots,\\mathcal\{D\}\_\{M\}^\{\(j\)\},\\mathbf\{L\}^\{\*\(j\)\}\\allowbreak\)\\\}\_\{j=1\}^\{Q\}\. Denoting the intermediate estimate at layerkkand samplejjby𝐋^j\(k\)=𝐏​Ξj\(k\)​𝐏T\\hat\{\\mathbf\{L\}\}\_\{j\}^\{\(k\)\}=\\mathbf\{P\}\\Xi\_\{j\}^\{\(k\)\}\\mathbf\{P\}^\{T\}, the squared relative error \(SRE\) on edge weights is

SREj\(k\)=‖𝐋^j\(k\)−𝐋∗\(j\)‖F2‖𝐋∗\(j\)‖F2\.\\displaystyle\\mathrm\{SRE\}\_\{j\}^\{\(k\)\}=\\frac\{\\\|\\hat\{\\mathbf\{L\}\}\_\{j\}^\{\(k\)\}\-\\mathbf\{L\}^\{\*\(j\)\}\\\|\_\{F\}^\{2\}\}\{\\\|\\mathbf\{L\}^\{\*\(j\)\}\\\|\_\{F\}^\{2\}\}\.\(111\)Because every feasible𝐋^j\(k\)∈ℒ\\hat\{\\mathbf\{L\}\}\_\{j\}^\{\(k\)\}\\in\\mathcal\{L\}has non\-positive off\-diagonal entries, the induced adjacency estimate𝐀^j\(k\):=−\(𝐋^j\(k\)−diag⁡\(𝐋^j\(k\)\)\)\\hat\{\\mathbf\{A\}\}\_\{j\}^\{\(k\)\}:=\-\\bigl\(\\hat\{\\mathbf\{L\}\}\_\{j\}^\{\(k\)\}\-\\mathrm\{diag\}\(\\hat\{\\mathbf\{L\}\}\_\{j\}^\{\(k\)\}\)\\bigr\)is entrywise non\-negative, so applying the sigmoid directly to𝐀^j\(k\)\\hat\{\\mathbf\{A\}\}\_\{j\}^\{\(k\)\}would forceσ⁡\(A^j,u​v\(k\)\)≥σ⁡\(0\)=0\.5\\sigma\(\\hat\{A\}\_\{j,uv\}^\{\(k\)\}\)\\geq\\sigma\(0\)=0\.5for every node pair\(u,v\)\(u,v\), precluding any calibrated probability below one\-half even for true non\-edges\. We avoid this by first mapping the non\-negative weight estimate to a signed logit via a per\-layer, trainable affine transform,

sj,u​v\(k\)=γ\(k\)​A^j,u​v\(k\)−η\(k\),γ\(k\)=softplus⁡\(γ~\(k\)\)\>0,\\displaystyle s\_\{j,uv\}^\{\(k\)\}=\\gamma^\{\(k\)\}\\,\\hat\{A\}\_\{j,uv\}^\{\(k\)\}\-\\eta^\{\(k\)\},\\;\\gamma^\{\(k\)\}=\\mathrm\{softplus\}\(\\tilde\{\\gamma\}^\{\(k\)\}\)\>0,\(112\)whereη\(k\)∈ℝ\\eta^\{\(k\)\}\\in\\mathbb\{R\}is a learnable decision threshold and the softplus reparameterization of the slopeγ\(k\)\\gamma^\{\(k\)\}preserves monotonicity ofsj,u​v\(k\)s\_\{j,uv\}^\{\(k\)\}in the edge weight while allowingη\(k\)\\eta^\{\(k\)\}to push small, likely\-spurious weights below the decision boundary \(sj,u​v\(k\)<0⇒pj,u​v\(k\)<0\.5s\_\{j,uv\}^\{\(k\)\}<0\\Rightarrow p\_\{j,uv\}^\{\(k\)\}<0\.5\)\. Topology recovery is then penalized via the binary cross\-entropy on the resulting, properly calibrated probability,

BCEj\(k\)=BCE⁡\(sgn⁡\(𝐀∗\(j\)\),σ⁡\(𝐬j\(k\)\)\),\\displaystyle\\mathrm\{BCE\}\_\{j\}^\{\(k\)\}=\\mathrm\{BCE\}\\\!\\left\(\\mathrm\{sgn\}\(\\mathbf\{A\}^\{\*\(j\)\}\),\\,\\sigma\\bigl\(\\mathbf\{s\}\_\{j\}^\{\(k\)\}\\bigr\)\\right\),\(113\)whereσ⁡\(⋅\)\\sigma\(\\cdot\)is the element\-wise sigmoid applied to the logit matrix𝐬j\(k\)\\mathbf\{s\}\_\{j\}^\{\(k\)\}of \([112](https://arxiv.org/html/2608.19914#S5.E112)\), andsgn⁡\(𝐀∗\(j\)\)\\mathrm\{sgn\}\(\\mathbf\{A\}^\{\*\(j\)\}\)is the binary edge indicator of the ground truth\. The two additional scalars\(γ~\(k\),η\(k\)\)\(\\tilde\{\\gamma\}^\{\(k\)\},\\eta^\{\(k\)\}\)are appended to𝚯\\bm\{\\Theta\}and trained jointly with all other layer parameters; they add𝒪⁡\(Kℓ\)\\mathcal\{O\}\(K\_\{\\ell\}\)parameters in total and leave the forward pass of Algorithm[2](https://arxiv.org/html/2608.19914#alg2), which never invokesBCEj\(k\)\\mathrm\{BCE\}\_\{j\}^\{\(k\)\}at inference time, unchanged\. The aggregate layer\-discounted training loss is

𝒥⁡\(𝚯\)=1Q​∑j=1Q∑k=1KℓτKℓ−k​\[SREj\(k\)\+ζ​BCEj\(k\)\],\\displaystyle\\mathcal\{J\}\(\\bm\{\\Theta\}\)=\\frac\{1\}\{Q\}\\sum\_\{j=1\}^\{Q\}\\sum\_\{k=1\}^\{K\_\{\\ell\}\}\\tau^\{K\_\{\\ell\}\-k\}\\\!\\left\[\\mathrm\{SRE\}\_\{j\}^\{\(k\)\}\+\\zeta\\,\\mathrm\{BCE\}\_\{j\}^\{\(k\)\}\\right\],\(114\)where𝚯=\{\(ϵ\(k\),ρ\(k\),ϱ\(k\),𝜶\(k\)\)\}k=1Kℓ\\bm\{\\Theta\}=\\\{\(\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\bm\{\\alpha\}^\{\(k\)\}\)\\\}\_\{k=1\}^\{K\_\{\\ell\}\}collects all trainable parameters,τ∈\(0,1\]\\tau\\in\(0,1\]is a discounting factor that reduces the contribution of early\-layer intermediates which are farther from convergence, andζ\>0\\zeta\>0balances reconstruction fidelity against topological accuracy\. The network is trained by minimizing \([114](https://arxiv.org/html/2608.19914#S5.E114)\) via the Adam optimizer with a decaying learning rate, initializingϵ\(k\)=1\\epsilon^\{\(k\)\}=1,ρ\(k\)=0\.5\\rho^\{\(k\)\}=0\.5,ϱ\(k\)=0\.5\\varrho^\{\(k\)\}=0\.5,𝜶\(k\)=𝟎\\bm\{\\alpha\}^\{\(k\)\}=\\mathbf\{0\}\(uniform barycentric weights\) for allkk, and projectingϵ\(k\),ρ\(k\),ϱ\(k\)\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\}onto\(0,\+∞\)\(0,\+\\infty\)after each gradient step to maintain feasibility\.

Several structural properties of the proposed architecture merit discussion\. The four parameter categories\(ϵ\(k\),ρ\(k\),ϱ\(k\),𝝀\(k\)\)\(\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\bm\{\\lambda\}^\{\(k\)\}\)are not interchangeable:ρ\(k\)\\rho^\{\(k\)\}and𝝀\(k\)\\bm\{\\lambda\}^\{\(k\)\}are objective\-level quantities that alter the optimization problem solved at layerkkthrough𝐊\(k\)\\mathbf\{K\}^\{\(k\)\}, whileϱ\(k\)\\varrho^\{\(k\)\}is a purely algorithmic quantity that affects the iterative updates without changing the objective\. Conflating them, as would occur ifϱ\\varrhowere set equal toρ\\rho, would destroy this structural separation and preclude the network from independently optimizing the sparsity\-robustness trade\-off and the ADMM convergence rate\. The joint ratioϵ\(k\)/ϱ\(k\)\\epsilon^\{\(k\)\}/\\varrho^\{\(k\)\}, which governs the shrinkage threshold inℳC\\mathcal\{M\}\_\{C\}, illustrates another benefit of separate learning: the network can adjust the effective robustness \(throughϵ\(k\)\\epsilon^\{\(k\)\}\) and the proximity step scale \(throughϱ\(k\)\\varrho^\{\(k\)\}\) in tandem, achieving any desired threshold through a continuum of\(ϵ\(k\),ϱ\(k\)\)\(\\epsilon^\{\(k\)\},\\varrho^\{\(k\)\}\)combinations that a single conflated parameter cannot express\. The total number of trainable parameters isKℓ⋅\(5\+M\)K\_\{\\ell\}\\cdot\(5\+M\), three positive scalars\(ϵ\(k\),ρ\(k\),ϱ\(k\),γ~\(k\),η\(k\)\)\(\\epsilon^\{\(k\)\},\\rho^\{\(k\)\},\\varrho^\{\(k\)\},\\tilde\{\\gamma\}^\{\(k\)\},\\eta^\{\(k\)\}\)andMMsimplex\-space coordinates𝜶\(k\)\\bm\{\\alpha\}^\{\(k\)\}per layer, growing only linearly in network depth and source\-domain count\. This compactness, combined with the domain knowledge encoded in the ADMM structure, allows the unrolled network to generalize from substantially fewer labeled training graphs than a comparably expressive generic deep architecture, which is particularly valuable in the heterogeneous multi\-source regime where ground\-truth graphs at the target site are typically scarce\.

## VIExperimental Results

We evaluate the proposed multi\-source Wasserstein distributionally robust graph learning framework \(MS\-WDRO\) against seven representative baselines spanning three methodological families: smooth\-signal optimization methods, deep\-learning\-based structure\-learning methods, and distributionally robust graph learning methods\. Section[VI\-A](https://arxiv.org/html/2608.19914#S6.SS1)reports controlled experiments on synthetic multi\-source networks, designed to isolate the effect of target\-domain sample scarcity, source\-target heterogeneity, and the number of available source domains under known ground truth\. Section[VI\-B](https://arxiv.org/html/2608.19914#S6.SS2)validates the framework on a real multi\-site resting\-state fMRI cohort, where no ground\-truth connectivity graph is available and performance must instead be assessed through held\-out signal reconstruction and a downstream clinical classification task\. Throughout, we report means and standard errors over independent trials and refrain from selectively favorable comparisons\.

### VI\-AExperiments on Synthetic Data

#### VI\-A1Multi\-Source Network Generation

Each trial instantiates a ground\-truth target network𝒢⋆=\(𝒱,ℰ⋆\)\\mathcal\{G\}^\{\\star\}=\(\\mathcal\{V\},\\mathcal\{E\}^\{\\star\}\)onNNnodes, drawn from one of three canonical random\-graph families used throughout the graph signal processing literature: an Erdős–Rényi graph with connection probabilityp=0\.18p=0\.18, a Barabási–Albert graph with attachment parameterm=2m=2, and a four\-block stochastic block model with assortative intra\-block connectivity\. Edge weights are drawn independently fromUnif⁡\(0\.3,1\.0\)\\mathrm\{Unif\}\(0\.3,1\.0\), and the resulting adjacency matrix is normalized to the combinatorial graph Laplacian𝐋⋆\\mathbf\{L\}^\{\\star\}\. Unless otherwise stated, results are reported forN=20N=20nodes to permit dense Monte Carlo averaging; the scalability study in Section[VI\-A5](https://arxiv.org/html/2608.19914#S6.SS1.SSS5)extends this toNNup to500500\.

To emulate a realistic multi\-source deployment in whichMMauxiliary domains are structurally related to, but not identical to, the target, each source graph𝒢m\\mathcal\{G\}\_\{m\},m=1,…,Mm=1,\\dots,M, is obtained by applying independent random edge rewiring to𝒢⋆\\mathcal\{G\}^\{\\star\}at rater∈\[0,0\.5\]r\\in\[0,0\.5\]: with probabilityrr, each edge inℰ⋆\\mathcal\{E\}^\{\\star\}is deleted and reconnected to a uniformly sampled node pair, after which edge weights are redrawn fromUnif⁡\(0\.3,1\.0\)\\mathrm\{Unif\}\(0\.3,1\.0\)\. The rewiring raterrthus serves as a single, interpretable knob for source\-target heterogeneity, withr=0r=0recovering an idealized homogeneous multi\-source setting and largerrrproducing source domains that are topologically related to, but increasingly divergent from, the target\. Unless swept explicitly, we fixM=5M=5source domains andr=0\.3r=0\.3, values chosen to reflect a moderately heterogeneous federation of related but non\-identical domains\.

#### VI\-A2Graph Signal Generation

Signals are generated to match the generative law of \([4](https://arxiv.org/html/2608.19914#S2.E4)\) exactly\. For every domaind∈\{target,1,…,M\}d\\in\\\{\\text\{target\},1,\\ldots,M\\\}with Laplacian𝐋d\\mathbf\{L\}\_\{d\}, we draw

𝐱=𝐋d†1/2​𝐳,𝐳∼𝒩⁡\(𝟎,𝐈N\),\\displaystyle\\mathbf\{x\}=\\mathbf\{L\}\_\{d\}^\{\\dagger 1/2\}\\mathbf\{z\},\\qquad\\mathbf\{z\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{N\}\),\(115\)where𝐋d†1/2:=𝐔d​\(𝚲d†\)1/2​𝐔dT\\mathbf\{L\}\_\{d\}^\{\\dagger 1/2\}:=\\mathbf\{U\}\_\{d\}\(\\bm\{\\Lambda\}\_\{d\}^\{\\dagger\}\)^\{1/2\}\\mathbf\{U\}\_\{d\}^\{T\}is obtained from the eigendecomposition𝐋d=𝐔d​𝚲d​𝐔dT\\mathbf\{L\}\_\{d\}=\\mathbf\{U\}\_\{d\}\\bm\{\\Lambda\}\_\{d\}\\mathbf\{U\}\_\{d\}^\{T\}, with\(Λd†\)i​i=1/λd,i\(\\Lambda\_\{d\}^\{\\dagger\}\)\_\{ii\}=1/\\lambda\_\{d,i\}forλd,i\>0\\lambda\_\{d,i\}\>0and00on the null direction𝟏\\mathbf\{1\}\. By construction,Cov⁡\(𝐱\)=𝐋d†\\mathrm\{Cov\}\(\\mathbf\{x\}\)=\\mathbf\{L\}\_\{d\}^\{\\dagger\}exactly, so the precision matrix of the generated signals coincides with𝐋d\\mathbf\{L\}\_\{d\}without approximation, and graph recovery under this protocol directly instantiates the statistical model underlying Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)and Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)\.

Each sample is further corrupted by i\.i\.d\. observation noise,𝐲=𝐱\+𝐧\\mathbf\{y\}=\\mathbf\{x\}\+\\mathbf\{n\},𝐧∼𝒩⁡\(𝟎,σ2​𝐈N\)\\mathbf\{n\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\sigma^\{2\}\\mathbf\{I\}\_\{N\}\),σ=0\.05\\sigma=0\.05, which is not part of the generative law of Section[II](https://arxiv.org/html/2608.19914#S2)but represents the residual per\-sample uncertainty the Wasserstein ambiguity set is designed to absorb\. All methods observe only𝐲\\mathbf\{y\}and are evaluated against the ground\-truth𝐋d\\mathbf\{L\}\_\{d\}\. Each source domain contributesnsrc=200n\_\{\\text\{src\}\}=200i\.i\.d\. samples; the target domain contributes onlyntgt∈\{5,…,100\}n\_\{\\text\{tgt\}\}\\in\\\{5,\\ldots,100\\\}samples, reflecting the small\-sample regime that motivates this work\.

#### VI\-A3Baseline Methods and Adaptation Protocol

Table[I](https://arxiv.org/html/2608.19914#S6.T1)summarizes the seven baselines\. None of them natively supports multiple heterogeneous source domains, so a consistent and explicitly stated adaptation protocol is required for a fair comparison\. We group the baselines into two protocols according to their native design: \(i\)*naive pooling*, applied to methods with no training/meta\-training phase \(SigRep, GLE\-ADMM, MUGL, WDRO\-GL\), where all source and target samples are concatenated into a single empirical distribution prior to estimation; and \(ii\)*source meta\-training*, applied to methods that are explicitly designed to learn a solver on a distribution of training instances and generalize to a new one \(DeepGraph, GLAD, L2G\), where theMMsource domains serve as meta\-training tasks and the learned solver is evaluated zero\-shot on the scarce target data\. MS\-WDRO uses neither protocol, since it retains the per\-source empirical distributions and combines them through a Wasserstein barycenter ambiguity set, which is the central methodological distinction evaluated throughout this section\. We stress that DeepGraph and GLAD were originally developed for sparse precision\-matrix recovery from i\.i\.d\. Gaussian samples rather than Laplacian estimation from smooth graph signals; their inclusion probes robustness to this task mismatch under an otherwise favorable adaptation protocol \(meta\-training\), rather than penalizing them through a disadvantageous one\.

TABLE I:Baseline methods, their methodological family, and the protocol used to adapt them to the multi\-source setting\.
#### VI\-A4Evaluation Metrics

Graph recovery accuracy is measured by the edge\-detectionFF\-score, computed by thresholding the estimated weighted adjacency matrix at zero and comparing the resulting support to that of𝐋⋆\\mathbf\{L\}^\{\\star\}, and by the relative Frobenius error‖𝐋^−𝐋⋆‖F/‖𝐋⋆‖F\\\|\\hat\{\\mathbf\{L\}\}\-\\mathbf\{L\}^\{\\star\}\\\|\_\{F\}/\\\|\\mathbf\{L\}^\{\\star\}\\\|\_\{F\}\. Both metrics are averaged over5050independent Monte Carlo trials with independently resampled graphs, signals, and noise; shaded bands and error bars throughout report the resulting standard error of the mean\. Wall\-clock time is measured per problem instance on identical hardware and is used to assess the practical benefit of algorithm unrolling independently of estimation accuracy\.

#### VI\-A5Results and Analysis

Fig\. 2:EdgeFF\-score and relative Frobenius error versus target sample sizentgtn\_\{\\mathrm\{tgt\}\}, averaged over 50 trials \(N=20N=20,M=5M=5,r=0\.3r=0\.3\)\. Dashed: naive\-pooled baselines; dotted: source\-meta\-trained baselines; solid: proposed\.##### Sample efficiency\.

Figure[2](https://arxiv.org/html/2608.19914#S6.F2)reportsFF\-score and relative error as a function of the number of target\-domain samplesntgtn\_\{\\text\{tgt\}\}\. MS\-WDRO dominates all baselines across the entire sampled range, with the largest relative margin in the most sample\-starved regime: atntgt=5n\_\{\\text\{tgt\}\}=5, MS\-WDRO attains anFF\-score of0\.750\.75, compared to0\.720\.72for WDRO\-GL,0\.600\.60–0\.670\.67for the DRO\- and meta\-trained baselines, and below0\.530\.53for the smooth\-signal and precision\-matrix baselines\. Asntgtn\_\{\\text\{tgt\}\}grows to100100, all methods improve and the gap to the strongest baseline \(WDRO\-GL\) narrows to under twoFF\-score points, consistent with the expectation that the value of multi\-source information diminishes once the target domain is itself well sampled\. We note that SigRep and GLE\-ADMM cross over nearntgt=30n\_\{\\text\{tgt\}\}=30: SigRep’s simpler alternating\-minimization scheme reaches a useful solution sooner under extreme scarcity, but GLE\-ADMM’s convergence\-guaranteed ADMM/MM iteration eventually yields a better\-conditioned estimate once sufficient samples are pooled, an interaction between optimizer design and sample size that would be obscured by reporting a single operating point\.

Fig\. 3:EdgeFF\-score and relative Frobenius error versus source\-target heterogeneity \(rewiring rate\)rr\(N=20N=20,M=5M=5,ntgt=20n\_\{\\mathrm\{tgt\}\}=20\)\.
##### Robustness to source\-target heterogeneity\.

Figure[3](https://arxiv.org/html/2608.19914#S6.F3)sweeps the rewiring raterrfrom00\(homogeneous sources\) to0\.50\.5\(substantially divergent sources\) at fixedntgt=20n\_\{\\text\{tgt\}\}=20andM=5M=5\. All methods degrade monotonically asrrincreases, but at markedly different rates: MS\-WDRO’sFF\-score falls by14%14\\%relative \(0\.945→0\.8150\.945\\to 0\.815\), WDRO\-GL’s by17%17\\%, while the naive\-pooling classical baselines lose over30%30\\%\. This differential is the direct empirical signature of the barycentric ambiguity set: because MS\-WDRO retains each source’s empirical distribution and lets the learned radii and barycentric weights adapt to inter\-source discrepancy, it is far less sensitive to the injected heterogeneity than methods that collapse all sources into one pooled empirical distribution before estimation\.

Fig\. 4:EdgeFF\-score versus number of source domainsMM\(N=20N=20,r=0\.3r=0\.3,ntgt=20n\_\{\\mathrm\{tgt\}\}=20\)\. Naive\-pooling baselines exhibit negative transfer beyondM≈5M\\approx 5\.
##### Scalability in the number of source domains\.

Figure[4](https://arxiv.org/html/2608.19914#S6.F4)varies the number of source domainsM∈\{1,2,3,5,8,12\}M\\in\\\{1,2,3,5,8,12\\\}at fixedr=0\.3r=0\.3\. Naive\-pooling baselines exhibit a rise\-then\-fall pattern, improving as additional sources are added up toM≈5M\\approx 5and then*degrading*as further, more heterogeneous sources dilute the pooled distribution with conflicting structure, a form of negative transfer\. In contrast, MS\-WDRO and the source\-meta\-trained baselines improve monotonically or plateau, since additional sources are either explicitly reweighted \(MS\-WDRO\) or contribute additional meta\-training diversity \(DeepGraph, GLAD, L2G\) rather than being blindly pooled\. This experiment provides direct evidence for the practical necessity of the barycentric formulation whenever the number and heterogeneity of available sources cannot be controlled a priori\.

Fig\. 5:Ablation study isolating the contribution of barycentric fusion and algorithm unrolling, measured byFF\-score, relative error, and per\-instance estimation time\.
##### Ablation study\.

Figure[5](https://arxiv.org/html/2608.19914#S6.F5)isolates the contribution of each architectural component by comparing the full MS\-WDRO model against four reduced variants: \(i\)*No\-Unroll*, which replaces the learned per\-layer parameters with a single set of hyperparameters selected by cross\-validation; \(ii\)*Linear\-Pooling*, which replaces the Wasserstein barycenter with a simple arithmetic mean of the source covariances as the ambiguity\-set center; \(iii\)*Naive\-Pooled*, which discards the multi\-source structure entirely and pools all samples as in the baseline protocol; and \(iv\)*Target\-Only*, which uses no source information whatsoever\. Each simplification incurs a measurable cost: removing the barycenter in favor of linear pooling costs10\.610\.6FF\-score points, and discarding multi\-source structure entirely costs22\.922\.9points relative to the full model\. The unrolling architecture itself contributes a comparatively modest4\.14\.1\-point accuracy gain over its cross\-validated, non\-unrolled counterpart, but converts a850​ms850\\,\\mathrm\{ms\}per\-instance hyperparameter search into a4\.2​ms4\.2\\,\\mathrm\{ms\}forward pass, a two\-orders\-of\-magnitude reduction in inference cost that is orthogonal to, and compounds with, the accuracy benefit of barycentric fusion\.

Fig\. 6:Layer\-wise evolution of the four learned parameters overKℓ=10K\_\{\\ell\}=10unrolled layers: ambiguity radiusϵ\(k\)\\epsilon^\{\(k\)\}\(objective\-level; decreasing\), sparsity regularisation coefficientρ\(k\)\\rho^\{\(k\)\}\(objective\-level; decreasing\), ADMM penaltyϱ\(k\)\\varrho^\{\(k\)\}\(algorithm\-level; increasing\), and barycentric weightsλm\(k\)\\lambda\_\{m\}^\{\(k\)\},m=1,…,Mm=1,\\ldots,M, forM=5M=5sources\.
##### Interpretability of the unrolled parameters\.

Figure[6](https://arxiv.org/html/2608.19914#S6.F6)tracks the layer\-wise evolution of all four learned parameter classes, the ambiguity radiusϵ\(k\)\\epsilon^\{\(k\)\}, the sparsity regularization coefficientρ\(k\)\\rho^\{\(k\)\}, the ADMM penaltyϱ\(k\)\\varrho^\{\(k\)\}, and the barycentric weightsλm\(k\)\\lambda\_\{m\}^\{\(k\)\},m=1,…,Mm=1,\\dots,M, across theKℓ=10K\_\{\\ell\}=10unrolled layers\. The ambiguity radiusϵ\(k\)\\epsilon^\{\(k\)\}contracts monotonically from1\.371\.37to0\.240\.24: as successive layers refine an increasingly confident Laplacian estimate, the network requires a progressively smaller Wasserstein robustness margin, reflecting a form of learned distributional annealing\. The sparsity regularization coefficientρ\(k\)\\rho^\{\(k\)\}follows a distinct but also decreasing trajectory, declining from1\.081\.08to0\.400\.40at a more gradual pace\. Its role is to modulate the effective nominal matrix𝐊\(k\)=Σ^𝝀\(k\)\+ρ\(k\)​𝐇\\mathbf\{K\}^\{\(k\)\}=\\hat\{\\Sigma\}^\{\(k\)\}\_\{\\bm\{\\lambda\}\}\+\\rho^\{\(k\)\}\\mathbf\{H\}: large values in early layers impose strongℓ1\\ell\_\{1\}regularization that promotes sparse graph recovery, while the later relaxation allows fine\-grained edge\-weight estimation once the topological structure has been identified, implementing a coarse\-to\-fine refinement across the unrolled depth\. The ADMM penaltyϱ\(k\)\\varrho^\{\(k\)\}increases monotonically from0\.550\.55to1\.781\.78, consistent with standard adaptive\-penalty schedules that tighten the consensus constraint as the primal iterates stabilize\. Crucially, the joint block\-shrinkage thresholdϵ\(k\)/ϱ\(k\)\\epsilon^\{\(k\)\}/\\varrho^\{\(k\)\}in theℳC\\mathcal\{M\}\_\{C\}update contracts from2\.492\.49at layer11to0\.130\.13at layerKℓK\_\{\\ell\}, a reduction of roughly19×19\\times, because the simultaneous decrease ofϵ\(k\)\\epsilon^\{\(k\)\}and increase ofϱ\(k\)\\varrho^\{\(k\)\}compound in the denominator of the proximity operator, progressively hardening the structural projection as the estimate matures\. The barycentric weights, initialized uniformly at1/M1/M, diverge substantially over the unrolled trajectory: the network concentrates mass on the two structurally closest sources \(λ1\(Kℓ\)=0\.34\\lambda\_\{1\}^\{\(K\_\{\\ell\}\)\}=0\.34,λ3\(Kℓ\)=0\.30\\lambda\_\{3\}^\{\(K\_\{\\ell\}\)\}=0\.30\) while down\-weighting the two most dissimilar ones toλ4\(Kℓ\)=0\.11\\lambda\_\{4\}^\{\(K\_\{\\ell\}\)\}=0\.11andλ5\(Kℓ\)=0\.06\\lambda\_\{5\}^\{\(K\_\{\\ell\}\)\}=0\.06\. Taken together, these four trajectories provide qualitative confirmation that the unrolled network learns a physically interpretable and structurally motivated parameter schedule, simultaneous relaxation of distributional robustness and sparsity regularization, adaptive ADMM step\-size growth, and distributionally aware source reweighting, rather than an opaque black\-box transformation\.

Fig\. 7:Empirical out\-of\-sample excess risk versus target sample sizenn, compared against the theoretical𝒪⁡\(1/n\)\\mathcal\{O\}\(1/\\sqrt\{n\}\)bound of Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)\(log\-log scale\)\.
##### Validation of the theoretical generalization bound\.

Figure[7](https://arxiv.org/html/2608.19914#S6.F7)compares the empirical out\-of\-sample excess risk against the𝒪⁡\(1/n\)\\mathcal\{O\}\(1/\\sqrt\{n\}\)Rademacher\-complexity\-based bound derived in Section[IV](https://arxiv.org/html/2608.19914#S4)\(out\-of\-sample performance theorem\), as a function of target sample sizen∈\[10,1000\]n\\in\[10,1000\]on a log\-log scale\. The empirical curve tracks the theoretical rate closely, with an ordinary\-least\-squares fit yielding a log\-log slope of−0\.50\-0\.50, matching the bound’s predicted exponent to two decimal places and confirming, on this synthetic testbed, that the derived finite\-sample guarantee is not merely asymptotically valid but quantitatively tight\.

![Refer to caption](https://arxiv.org/html/2608.19914v1/synth2_exp7_graph_recovery_visualization.png)Fig\. 8:Recovered adjacency matrices for all eight methods on a representative trial, alongside the ground\-truth graph\.
##### Qualitative graph recovery\.

Figure[8](https://arxiv.org/html/2608.19914#S6.F8)visualizes the recovered adjacency matrices of all eight methods against the ground\-truth graph for one representative trial \(N=20N=20,ntgt=20n\_\{\\text\{tgt\}\}=20,r=0\.3r=0\.3,M=5M=5\)\. The visual pattern corroborates the quantitative results: DeepGraph and GLAD recover a visibly sparser and noisier structure that omits several true edges, the smooth\-signal baselines recover the dominant edges but with substantial spurious activity in low\-weight regions, and MS\-WDRO’s reconstruction is the closest visual match to the ground truth, including in the recovery of several low\-weight true edges missed by all baselines\.

Fig\. 9:Wall\-clock time per instance versus number of nodesNN\(log\-log scale\)\.
##### Computational scalability\.

Figure[9](https://arxiv.org/html/2608.19914#S6.F9)reports wall\-clock time as a function of network sizeN∈\[10,500\]N\\in\[10,500\]\. Methods with a fixed\-depth forward pass at inference time, the meta\-trained baselines and the two unrolled DRO methods \(WDRO\-GL, MS\-WDRO\), scale sub\-cubically in practice \(empirical exponent≈2\.2\\approx 2\.2–2\.32\.3\), while the naive\-pooling iterative optimizers \(SigRep, GLE\-ADMM, MUGL\), which must repeat full eigendecomposition steps until convergence at test time, scale closer toN2\.9N^\{2\.9\}–N3N^\{3\}\. AtN=500N=500, MS\-WDRO completes in under1010seconds versus several minutes for MUGL, the slowest baseline\. Critically, this scalability advantage is not unique to MS\-WDRO: WDRO\-GL, which shares the unrolling backbone, scales comparably, indicating that the computational benefit stems from algorithm unrolling per se, while the accuracy benefit documented above stems specifically from the barycentric multi\-source formulation\. MS\-WDRO is, to our knowledge, the only method evaluated here that obtains both benefits simultaneously\.

### VI\-BExperiments on Real\-World Data

#### VI\-B1Dataset and Preprocessing

We evaluate on the Autism Brain Imaging Data Exchange I \(ABIDE I\) repository111https://fcon\_1000\.projects\.nitrc\.org/indi/abide, a multi\-site consortium of resting\-state functional MRI acquisitions that provides a naturally occurring, rather than synthetically constructed, instance of heterogeneous multi\-source data: each acquisition site employs a distinct scanner vendor, field strength, and imaging protocol, inducing site\-specific distribution shift that is well documented in the neuroimaging literature\. We designate seven sites with comparatively abundant subjects, NYU \(n=172n=172\), UM\_1 \(n=86n=86\), USM \(n=71n=71\), Yale \(n=56n=56\), Pitt \(n=50n=50\), Stanford \(n=39n=39\), and KKI \(n=33n=33\), as source domains, and the smallest available site, CMU \(n=14n=14\), as the target domain, directly instantiating the scarce\-target, abundant\-heterogeneous\-source scenario that motivates this work\.

All functional volumes are preprocessed with a standard resting\-state pipeline: slice\-timing and motion correction, nuisance regression against the six rigid\-body motion parameters together with white\-matter and cerebrospinal\-fluid signals, band\-pass temporal filtering to0\.010\.01–0\.1​Hz0\.1\\,\\mathrm\{Hz\}, and spatial normalization to MNI152 space\. Regional time series are extracted over116116regions of interest using the AAL atlas and treated as graph signals on a node set of sizeN=116N=116; each subject’s time series constitutes one signal sample\. Because functional connectivity graphs are unavailable as ground truth in vivo, we depart from theFF\-score and relative\-error metrics used in Section[VI\-A](https://arxiv.org/html/2608.19914#S6.SS1)and instead adopt two metrics standard in the graph\-signal\-processing and neuroimaging literature: \(i\)*held\-out reconstruction error*, the normalized mean\-squared error of predicting left\-out subjects’ regional time series from the learned graph under the smoothness objective, evaluated by2424\-fold cross\-validation within the target site; and \(ii\)*downstream classification performance*, the leave\-one\-out area under the ROC curve \(AUC\) of a linear classifier operating on spectral features of the learned target\-site graph, applied to the autism\-versus\-control diagnostic label provided by ABIDE I\. The seven baselines and their adaptation protocols follow Section[VI\-A3](https://arxiv.org/html/2608.19914#S6.SS1.SSS3)unchanged, with naive pooling now concatenating all source\-site and target\-site subjects and source meta\-training using the seven source sites as meta\-training tasks\.

#### VI\-B2Results and Analysis

Fig\. 10:Held\-out reconstruction NMSE on ABIDE I \(target site: CMU,N=14N=14\) for all eight methods\. Hatched bars: naive\-pooled baselines; solid bars: source\-meta\-trained baselines; blue: proposed\.##### Main comparison\.

Figure[10](https://arxiv.org/html/2608.19914#S6.F10)reports held\-out reconstruction NMSE for all eight methods\. MS\-WDRO achieves the lowest error \(0\.3340\.334\), followed by WDRO\-GL \(0\.3780\.378\); the gap between them \(11\.6%11\.6\\%relative\) is the smallest among all pairwise comparisons, consistent with the synthetic\-data finding that the two methods share an architecture differing only in single\- versus multi\-source ambiguity\-set construction\. The task\-mismatched baselines \(DeepGraph, GLAD\), despite being meta\-trained on the same seven source sites available to MS\-WDRO, trail even the basic naive\-pooled classical methods \(SigRep, GLE\-ADMM\), confirming that a favorable adaptation protocol cannot fully compensate for a generative mismatch between the precision\-matrix assumption underlying these methods and the smooth\-signal structure of resting\-state fMRI connectivity\.

Fig\. 11:Held\-out reconstruction NMSE versus number of target\-site \(CMU\) subjects used for estimation\.
##### Sample efficiency at the target site\.

Figure[11](https://arxiv.org/html/2608.19914#S6.F11)sweeps the number of CMU subjects used for target\-domain estimation from33to the full1414\. The ranking established in the main comparison is preserved across the entire range, and the relative advantage of MS\-WDRO over WDRO\-GL is largest at the smallest sample size \(n=3n=3\), narrowing as more target data becomes available, mirroring the synthetic sample\-efficiency result and confirming that the practical benefit of multi\-source fusion is concentrated precisely in the small\-sample regime that real clinical neuroimaging cohorts routinely face\.

Fig\. 12:Leave\-one\-out AUC for autism\-versus\-control classification using graph\-spectral features of the learned target\-site connectivity graph\.
##### Downstream diagnostic classification\.

Figure[12](https://arxiv.org/html/2608.19914#S6.F12)reports leave\-one\-out AUC for autism\-versus\-control classification using graph\-spectral features derived from each method’s learned target\-site graph\. MS\-WDRO attains an AUC of0\.7690\.769, compared to0\.7320\.732for WDRO\-GL and0\.5590\.559–0\.6480\.648for the task\-mismatched and classical baselines; all methods exceed the chance level of0\.50\.5\. We caution that, with only1414target\-site subjects, these AUC estimates carry substantial sampling variance \(bootstrap standard errors of0\.040\.04–0\.060\.06\), and the absolute classification performance should be interpreted as a proof of concept that improved connectivity estimation translates into improved downstream utility, rather than as a claim of clinical\-grade diagnostic accuracy\.

Fig\. 13:Statistical significance \(−log10⁡p\-\\log\_\{10\}p, paired Wilcoxon signed\-rank test, 24 cross\-validation folds\) of MS\-WDRO’s improvement over each baseline\.
##### Statistical significance\.

Figure[13](https://arxiv.org/html/2608.19914#S6.F13)reports paired Wilcoxon signed\-rank tests, computed across the2424cross\-validation folds, comparing MS\-WDRO’s reconstruction error against each baseline\. All seven comparisons reach significance at the0\.050\.05level; the margin is overwhelming against the task\-mismatched and classical baselines \(p<10−6p<10^\{\-6\}\) and comparatively narrow but still significant against WDRO\-GL \(p=0\.016p=0\.016\), the only baseline sharing MS\-WDRO’s distributionally robust, unrolled architecture\.

Fig\. 14:Barycentric weights learned by MS\-WDRO across the seven ABIDE I source sites when estimating the CMU target graph, annotated by acquisition scanner vendor\.
##### Interpretability: learned source contributions\.

Figure[14](https://arxiv.org/html/2608.19914#S6.F14)shows the barycentric weights that MS\-WDRO assigns to the seven source sites when estimating the CMU target graph\. The two largest weights are assigned to NYU \(λ=0\.26\\lambda=0\.26\) and USM \(λ=0\.23\\lambda=0\.23\), both of which, like CMU, were acquired on Siemens scanners, while the two GE\-acquired sites \(UM\_1, Stanford\) and the single Philips\-acquired site \(KKI\) receive markedly lower weights \(λ≤0\.07\\lambda\\leq 0\.07\)\. This alignment between the learned weights and scanner vendor, a known confound in multi\-site fMRI studies that was never provided to the model as a label, offers external, domain\-grounded evidence that the learned barycentric weights capture genuine cross\-site distributional similarity rather than an uninterpretable statistical artifact\.

Fig\. 15:Held\-out reconstruction NMSE versus per\-instance wall\-clock time on ABIDE I; MS\-WDRO and WDRO\-GL jointly define the Pareto frontier\.
##### Runtime–accuracy trade\-off\.

Figure[15](https://arxiv.org/html/2608.19914#S6.F15)plots reconstruction NMSE against per\-instance wall\-clock time\. MS\-WDRO and WDRO\-GL jointly form the Pareto frontier, combining sub\-0\.250\.25\-second inference with the lowest reconstruction error; the naive\-pooled iterative optimizers require0\.850\.85–2\.12\.1seconds per instance for comparatively worse accuracy, while the meta\-trained precision\-matrix baselines are fast but inaccurate\. No baseline is simultaneously fast and accurate\.

Fig\. 16:Reconstruction NMSE aggregated by methodological family\.
##### Category\-level comparison\.

Figure[16](https://arxiv.org/html/2608.19914#S6.F16)aggregates reconstruction error by methodological family \(task\-mismatched precision\-matrix models; task\-aligned single\-source optimization methods; task\-aligned single\-source DRO methods; the proposed multi\-source DRO method\)\. The between\-family variance dominates the within\-family variance, reinforcing that task alignment with the smooth\-signal generative assumption, and subsequently distributional robustness, are the primary drivers of performance on this dataset, more so than the classical\-versus\-deep\-learning distinction that is often treated as the primary axis of comparison in the graph\-learning literature\.

Fig\. 17:Circular connectograms of the recovered CMU functional network for a representative classical baseline \(GLE\-ADMM\), the strongest single\-source baseline \(WDRO\-GL\), and MS\-WDRO, alongside a structural\-proximity reference graph\.
##### Qualitative connectome visualization\.

Figure[17](https://arxiv.org/html/2608.19914#S6.F17)displays circular connectograms of the recovered CMU functional network for a representative classical baseline \(GLE\-ADMM\), the strongest single\-source baseline \(WDRO\-GL\), and MS\-WDRO, alongside a structural\-proximity reference graph over sixteen coarse anatomical regions\. MS\-WDRO’s recovered connectogram most closely reproduces the reference graph’s dominant inter\-hemispheric and homologous\-region connections, while retaining several finer intra\-hemis\-pheric edges that GLE\-ADMM omits\.

## VIIConclusion

This paper addressed the problem of network topology inference from smooth graph signals in the practically important regime where target\-domain observations are scarce or entirely unavailable, yet abundant heterogeneous data from multiple related source domains can be leveraged\. We proposed MS\-WDRO, a framework that resolves this challenge through three tightly integrated components: a Wasserstein barycentric nominal distribution that aggregates heterogeneous source statistics in a geometrically principled manner; a distributionally robust minimax formulation that guards against residual uncertainty between the barycenter and the true target distribution; and an algorithm\-unrolling architecture that jointly learns all four framework hyperparameters, the ambiguity set radius, the sparsity regularization coefficient, the augmented Lagrangian penalty, and the barycentric fusion weights, end\-to\-end from labeled training data\. Together, these components form a unified approach that is simultaneously grounded in statistical optimality theory, computationally tractable, and practically self\-calibrating\.

The experimental evidence across both controlled synthetic benchmarks and the real\-world ABIDE I multi\-site neuroimaging dataset revealed several insights that go beyond a straightforward accuracy comparison\. First, the barycentric formulation is the dominant contributor to overall performance: replacing the Wasserstein barycenter with an arithmetic average of source covariances, a natural and computationally cheaper alternative, cost more than ten F\-score points in the ablation study, confirming that the geometric structure of the Wasserstein metric space is not a theoretical nicety but a practical necessity\. Second, naive\-pooling baselines exhibited negative transfer as the number of source domains grew beyond a critical threshold, a phenomenon absent in MS\-WDRO, whose learned barycentric weights automatically downweighted structurally dissimilar sources rather than allowing them to dilute the nominal distribution\. Third, the algorithm unrolling architecture contributed a comparatively modest accuracy gain over its cross\-validated fixed\-parameter counterpart, but delivered a two\-orders\-of\-magnitude reduction in per\-instance inference time, transforming the framework from a tool suitable for offline analysis into one compatible with the throughput demands of large\-scale neuroimaging studies\. Fourth, and perhaps most tellingly, the barycentric weights learned on the neuroimaging dataset aligned spontaneously with scanner vendor, a known confound that was never provided as a label, offering externally grounded evidence that the learned representations encode genuine distributional similarity rather than statistical artifacts\.

### Future Directions

The present work opens several directions for future investigation\. On the signal modeling side, extending the framework beyond smooth Gaussian signals to heavy\-tailed, non\-Gaussian, or time\-varying graph settings is a promising direction, as the WDRO formulation provides a flexible basis for such extensions\. On the learning side, developing self\-supervised training strategies without ground\-truth topology labels and adaptive mechanisms for selecting the unrolling depth could further improve practical applicability and computational efficiency\. From a theoretical and practical perspective, establishing generalization guarantees for the end\-to\-end unrolled predictor and extending MS\-WDRO to decentralized federated settings would provide important steps toward robust and privacy\-preserving network learning\.

## Appendix AProof of Theorem[III\.2](https://arxiv.org/html/2608.19914#S3.Thmtheorem2)

Proof\.By Remark[III\.6](https://arxiv.org/html/2608.19914#S3.Thmremark6),ffis differentiable at every𝚵\(k\+1\)\\bm\{\\Xi\}^\{\(k\+1\)\},k≥0k\\geq 0, so the first\-order optimality condition of the𝚵\\bm\{\\Xi\}\-update \([48](https://arxiv.org/html/2608.19914#S3.E48)\) reads, using the scaled form of \([46](https://arxiv.org/html/2608.19914#S3.E46)\) and the dual update \([52](https://arxiv.org/html/2608.19914#S3.E52)\),

∇f​\(𝚵\(k\+1\)\)\+𝒯∗​\(𝐘\(k\+1\)\)=ϱ​𝒯∗​\(𝐂\(k\+1\)−𝐂\(k\)\)\.\\displaystyle\\nabla f\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\+\\mathcal\{T\}^\{\*\}\(\\mathbf\{Y\}^\{\(k\+1\)\}\)=\\varrho\\,\\mathcal\{T\}^\{\*\}\\bigl\(\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\}\\bigr\)\.\(116\)Likewise, the first\-order \(subgradient\) optimality condition of the𝐂\\mathbf\{C\}\-update \([50](https://arxiv.org/html/2608.19914#S3.E50)\) together with \([52](https://arxiv.org/html/2608.19914#S3.E52)\) gives

𝐘\(k\+1\)∈∂g⁡\(𝐂\(k\+1\)\)\.\\displaystyle\\mathbf\{Y\}^\{\(k\+1\)\}\\in\\partial g\(\\mathbf\{C\}^\{\(k\+1\)\}\)\.\(117\)
Since∇f\\nabla fis monotone on the convex set\{𝚵≻𝟎\}\\\{\\bm\{\\Xi\}\\succ\\mathbf\{0\}\\\}\(ffconvex\) and∂g\\partial gis monotone \(ggconvex\), we have

⟨∇f​\(𝚵\(k\+1\)\)−∇f​\(𝚵⋆\),𝚵\(k\+1\)−𝚵⋆⟩\\displaystyle\\bigl\\langle\\nabla f\(\\bm\{\\Xi\}^\{\(k\+1\)\}\)\-\\nabla f\(\\bm\{\\Xi\}^\{\\star\}\),\\,\\bm\{\\Xi\}^\{\(k\+1\)\}\-\\bm\{\\Xi\}^\{\\star\}\\bigr\\rangle≥0,\\displaystyle\\geq 0,\(118\)⟨𝐘\(k\+1\)−𝐘⋆,𝐂\(k\+1\)−𝐂⋆⟩\\displaystyle\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\}\-\\mathbf\{Y\}^\{\\star\},\\,\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\\star\}\\bigr\\rangle≥0\.\\displaystyle\\geq 0\.\(119\)Substituting the KKT identity∇f​\(𝚵⋆\)=−𝒯∗​\(𝐘⋆\)\\nabla f\(\\bm\{\\Xi\}^\{\\star\}\)=\-\\mathcal\{T\}^\{\*\}\(\\mathbf\{Y\}^\{\\star\}\)and \([116](https://arxiv.org/html/2608.19914#A1.E116)\) into \([118](https://arxiv.org/html/2608.19914#A1.E118)\), and using𝒯⁡\(𝚵\(k\+1\)−𝚵⋆\)=𝐫\(k\+1\)\+𝐂\(k\+1\)−𝐂⋆\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\+1\)\}\-\\bm\{\\Xi\}^\{\\star\}\)=\\mathbf\{r\}^\{\(k\+1\)\}\+\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\\star\}\(since𝒯⁡\(𝚵⋆\)=𝐂⋆\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\\star\}\)=\\mathbf\{C\}^\{\\star\}\), we obtain

⟨𝐘\(k\+1\)−𝐘⋆,𝐫\(k\+1\)⟩\+⟨𝐘\(k\+1\)−𝐘⋆,𝐂\(k\+1\)−𝐂⋆⟩\\displaystyle\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\}\-\\mathbf\{Y\}^\{\\star\},\\,\\mathbf\{r\}^\{\(k\+1\)\}\\bigr\\rangle\+\\bigl\\langle\\mathbf\{Y\}^\{\(k\+1\)\}\-\\mathbf\{Y\}^\{\\star\},\\,\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\\star\}\\bigr\\rangle≤−ϱ⁡⟨𝐂\(k\+1\)−𝐂\(k\),𝐫\(k\+1\)\+𝐂\(k\+1\)−𝐂⋆⟩\.\\displaystyle\\leq\-\\varrho\\bigl\\langle\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\},\\,\\mathbf\{r\}^\{\(k\+1\)\}\+\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\\star\}\\bigr\\rangle\.\(120\)Combining \([120](https://arxiv.org/html/2608.19914#A1.E120)\) with \([119](https://arxiv.org/html/2608.19914#A1.E119)\) to drop the non\-negative cross term, then using𝐘\(k\+1\)−𝐘⋆=\(𝐘\(k\)−𝐘⋆\)\+ϱ​𝐫\(k\+1\)\\mathbf\{Y\}^\{\(k\+1\)\}\-\\mathbf\{Y\}^\{\\star\}=\(\\mathbf\{Y\}^\{\(k\)\}\-\\mathbf\{Y\}^\{\\star\}\)\+\\varrho\\,\\mathbf\{r\}^\{\(k\+1\)\}\(scaled dual update \([52](https://arxiv.org/html/2608.19914#S3.E52)\)\) and completing the square exactly as in the standard two\-block ADMM convergence proof \(cf\.\[[52](https://arxiv.org/html/2608.19914#bib.bib52)\], App\. A;\[[51](https://arxiv.org/html/2608.19914#bib.bib51)\]\)222The final algebraic combination is problem\-agnostic once \([116](https://arxiv.org/html/2608.19914#A1.E116)\)–\([120](https://arxiv.org/html/2608.19914#A1.E120)\) are established, and is reproduced here for completeness\.yields exactly the claimed descent inequality \([56](https://arxiv.org/html/2608.19914#S3.E56)\)\.

Non\-negativity and monotonic non\-increase ofV\(k\)V^\{\(k\)\}in \([56](https://arxiv.org/html/2608.19914#S3.E56)\) imply: \(a\)V\(k\)V^\{\(k\)\}converges to a finite limit, hence\{𝐂\(k\)\}\\\{\\mathbf\{C\}^\{\(k\)\}\\\}and\{𝐘\(k\)\}\\\{\\mathbf\{Y\}^\{\(k\)\}\\\}are bounded; \(b\) summing \([56](https://arxiv.org/html/2608.19914#S3.E56)\) overkkgives∑k\(ϱ​‖𝐂\(k\+1\)−𝐂\(k\)‖F2\+‖𝐫\(k\+1\)‖F2\)≤V\(0\)<∞\\sum\_\{k\}\\bigl\(\\varrho\\\|\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\}\\\|\_\{F\}^\{2\}\+\\\|\\mathbf\{r\}^\{\(k\+1\)\}\\\|\_\{F\}^\{2\}\\bigr\)\\leq V^\{\(0\)\}<\\infty, so𝐂\(k\+1\)−𝐂\(k\)→𝟎\\mathbf\{C\}^\{\(k\+1\)\}\-\\mathbf\{C\}^\{\(k\)\}\\to\\mathbf\{0\}and𝐫\(k\)→𝟎\\mathbf\{r\}^\{\(k\)\}\\to\\mathbf\{0\}, proving \(i\)\. Boundedness of𝐂\(k\)\\mathbf\{C\}^\{\(k\)\}together with𝐫\(k\)→𝟎\\mathbf\{r\}^\{\(k\)\}\\to\\mathbf\{0\}gives boundedness of𝒯⁡\(𝚵\(k\)\)=𝐂\(k\)\+𝐫\(k\)\\mathcal\{T\}\(\\bm\{\\Xi\}^\{\(k\)\}\)=\\mathbf\{C\}^\{\(k\)\}\+\\mathbf\{r\}^\{\(k\)\}; since𝒯\\mathcal\{T\}is injective linear between finite\-dimensional spaces, its restriction to any bounded\-image sequence has a bounded pre\-image, so\{𝚵\(k\)\}\\\{\\bm\{\\Xi\}^\{\(k\)\}\\\}is bounded\. Every subsequential limit\(𝚵¯,𝐂¯,𝐘¯\)\(\\bar\{\\bm\{\\Xi\}\},\\bar\{\\mathbf\{C\}\},\\bar\{\\mathbf\{Y\}\}\)therefore satisfies, by closedness of∇f\\nabla fand∂g\\partial gand passing to the limit in \([116](https://arxiv.org/html/2608.19914#A1.E116)\)–\([117](https://arxiv.org/html/2608.19914#A1.E117)\) using \(i\), exactly the KKT system \([54](https://arxiv.org/html/2608.19914#S3.E54)\); by uniqueness of𝚵⋆\\bm\{\\Xi\}^\{\\star\}and𝐂⋆\\mathbf\{C\}^\{\\star\}, every subsequential limit of𝚵\(k\)\\bm\{\\Xi\}^\{\(k\)\}\(resp\.𝐂\(k\)\\mathbf\{C\}^\{\(k\)\}\) equals𝚵⋆\\bm\{\\Xi\}^\{\\star\}\(resp\.𝐂⋆\\mathbf\{C\}^\{\\star\}\), so the full sequences converge, proving \(iii\); continuity off,gf,gat the limit gives \(ii\); and a standard argument \(monotonicity ofV\(k\)V^\{\(k\)\}defined with any dual\-optimal𝐘¯\\bar\{\\mathbf\{Y\}\}forces𝐘\(k\)\\mathbf\{Y\}^\{\(k\)\}to converge to some dual optimum\) completes the proof of \(iii\) for𝐘\(k\)\\mathbf\{Y\}^\{\(k\)\}\.□\\hfill\\square

## Appendix BProof of Theorem[III\.3](https://arxiv.org/html/2608.19914#S3.Thmtheorem3)

Proof\.We first prove part \(i\)\. By the mutual independence ofσm,i\\sigma\_\{m,i\}andσm,j\\sigma\_\{m,j\}fori≠ji\\neq j,𝔼𝝈m​‖∑iσm,i​𝚺m,i‖F2=∑i‖𝚺m,i‖F2\\mathbb\{E\}\_\{\\bm\{\\sigma\}\_\{m\}\}\\\|\\sum\_\{i\}\\sigma\_\{m,i\}\\bm\{\\Sigma\}\_\{m,i\}\\\|\_\{F\}^\{2\}=\\sum\_\{i\}\\\|\\bm\{\\Sigma\}\_\{m,i\}\\\|\_\{F\}^\{2\}\. Jensen’s inequality applied to the concave square\-root and‖𝚺m,i‖F=‖𝐳m,i‖22≤D𝒵2\\\|\\bm\{\\Sigma\}\_\{m,i\}\\\|\_\{F\}=\\\|\\mathbf\{z\}\_\{m,i\}\\\|\_\{2\}^\{2\}\\leq D\_\{\\mathcal\{Z\}\}^\{2\}\(Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)\) give \(SetD𝒵=D𝒵R⁡\(δ/2\)D\_\{\\mathcal\{Z\}\}=D\_\{\\mathcal\{Z\}\_\{R\(\\delta/2\)\}\}as specified by Corollary[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmcorollary1)\.\)

𝔼𝝈m​\[1nm​‖∑i=1nmσm,i​𝚺m,i‖F\]\\displaystyle\\mathbb\{E\}\_\{\\bm\{\\sigma\}\_\{m\}\}\\\!\\left\[\\frac\{1\}\{n\_\{m\}\}\\Bigl\\\|\\sum\_\{i=1\}^\{n\_\{m\}\}\\sigma\_\{m,i\}\\bm\{\\Sigma\}\_\{m,i\}\\Bigr\\\|\_\{F\}\\right\]≤1nm2​∑i=1nm‖𝚺m,i‖F2\\displaystyle\\leq\\sqrt\{\\frac\{1\}\{n\_\{m\}^\{2\}\}\\sum\_\{i=1\}^\{n\_\{m\}\}\\\|\\bm\{\\Sigma\}\_\{m,i\}\\\|\_\{F\}^\{2\}\}≤D𝒵2nm\.\\displaystyle\\leq\\frac\{D\_\{\\mathcal\{Z\}\}^\{2\}\}\{\\sqrt\{n\_\{m\}\}\}\.\(121\)Multiplying byB​λmB\\lambda\_\{m\}, summing overmm, and using∑mλm/nm≤1/nmin\\sum\_\{m\}\\lambda\_\{m\}/\\sqrt\{n\_\{m\}\}\\allowbreak\\leq 1/\\sqrt\{n\_\{\\min\}\}yields \([71](https://arxiv.org/html/2608.19914#S3.E71)\)\.

We now prove part \(ii\)\. Introduce the*population WDRO risk*centered at the true barycenter asRϵn∗​\(𝐋\):=Rb𝝀∗​\(𝐋\)\+ϵn​‖vec⁡\(𝐋\)‖qR^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\mathbf\{L\}\):=R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\mathbf\{L\}\)\+\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}, and decompose:

Rℙ∗​\(𝐋^∗\)−R^ϵn​\(𝐋^∗\)\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)=\[Rℙ∗​\(𝐋^∗\)−Rϵn∗​\(𝐋^∗\)\]⏟\(I\)\\displaystyle=\\underbrace\{\\bigl\[R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\bigr\]\}\_\{\\mathrm\{\(I\)\}\}\+\[Rϵn∗​\(𝐋^∗\)−R^ϵn​\(𝐋^∗\)\]⏟\(II\)\.\\displaystyle\+\\underbrace\{\\bigl\[R^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\bigr\]\}\_\{\\mathrm\{\(II\)\}\}\.\(122\)For term \(I\), since−log⁡\|𝐋^∗\|\+\-\\log\|\\hat\{\\mathbf\{L\}\}^\{\*\}\|\_\{\+\}cancels:

Rℙ∗​\(𝐋^∗\)−Rϵn∗​\(𝐋^∗\)\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)=\[Rℙ∗​\(𝐋^∗\)−Rb𝝀∗​\(𝐋^∗\)\]\\displaystyle=\\bigl\[R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\bigr\]−ϵn​‖vec⁡\(𝐋^∗\)‖q\.\\displaystyle\\quad\-\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\\|\_\{q\}\.\(123\)By Lemma[III\.3](https://arxiv.org/html/2608.19914#S3.Thmlemma3), the lossℓ⁡\(𝐋,⋅\)\\ell\(\\mathbf\{L\},\\cdot\)is‖vec⁡\(𝐋\)‖q\\\|\\mathrm\{vec\}\(\\mathbf\{L\}\)\\\|\_\{q\}\-Lipschitz with respect to thepp\-Wasserstein metric, so Kantorovich\-Rubinstein duality gives

Rℙ∗​\(𝐋^∗\)−Rb𝝀∗​\(𝐋^∗\)≤Wp​\(ℙ∗,b𝝀∗\)​‖vec⁡\(𝐋^∗\)‖q\.\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\;\\leq\\;W\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\,\\\|\\mathrm\{vec\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\\|\_\{q\}\.\(124\)Substituting \([124](https://arxiv.org/html/2608.19914#A2.E124)\) into \([123](https://arxiv.org/html/2608.19914#A2.E123)\) and invoking the hypothesisWp​\(ℙ∗,b𝝀∗\)≤ϵnW\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\leq\\epsilon\_\{n\}:

Rℙ∗​\(𝐋^∗\)−Rϵn∗​\(𝐋^∗\)\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)≤‖vec⁡\(𝐋^∗\)‖q​\[Wp​\(ℙ∗,b𝝀∗\)−ϵn\]\\displaystyle\\leq\\\|\\mathrm\{vec\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\\|\_\{q\}\\bigl\[W\_\{p\}\(\\mathbb\{P\}^\{\*\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\-\\epsilon\_\{n\}\\bigr\]≤0\.\\displaystyle\\leq 0\.\(125\)The WDRO penaltyϵn​‖vec⁡\(𝐋^∗\)‖q\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\\|\_\{q\}thus exactly absorbs the Kantorovich transport cost fromℙ∗\\mathbb\{P\}^\{\*\}tob𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}, rendering term \(I\) non\-positive\.

For term \(II\), since theϵn​‖vec‖q\\epsilon\_\{n\}\\\|\\mathrm\{vec\}\\\|\_\{q\}contributions cancel inRϵn∗−R^ϵnR^\{\*\}\_\{\\epsilon\_\{n\}\}\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}:

Rϵn∗​\(𝐋^∗\)−R^ϵn​\(𝐋^∗\)\\displaystyle R^\{\*\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)=Rb𝝀∗​\(𝐋^∗\)−Rb^K∗​\(𝐋^∗\)\\displaystyle=R\_\{b^\{\*\}\_\{\\bm\{\\lambda\}\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-R\_\{\\hat\{b\}^\{\*\}\_\{K\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)=tr⁡\(\(Σ¯𝝀∗−Σ^𝝀\)​𝐋^∗\)\.\\displaystyle=\\mathrm\{tr\}\\\!\\bigl\(\(\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\-\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\\hat\{\\mathbf\{L\}\}^\{\*\}\\bigr\)\.\(126\)Applying Cauchy\-Schwarz with‖𝐋^∗‖F≤B\\\|\\hat\{\\mathbf\{L\}\}^\{\*\}\\\|\_\{F\}\\leq B:

tr⁡\(\(Σ¯𝝀∗−Σ^𝝀\)​𝐋^∗\)≤B​‖Σ¯𝝀∗−Σ^𝝀‖F\.\\displaystyle\\mathrm\{tr\}\\\!\\bigl\(\(\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\-\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\\hat\{\\mathbf\{L\}\}^\{\*\}\\bigr\)\\;\\leq\\;B\\\|\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\-\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}\.\(127\)Applying Assumption[III\.5](https://arxiv.org/html/2608.19914#S3.Thmassumption5)withΣ^m=nm−1​∑i𝚺m,i\\hat\{\\Sigma\}\_\{m\}=n\_\{m\}^\{\-1\}\\sum\_\{i\}\\bm\{\\Sigma\}\_\{m,i\}:

B​‖Σ¯𝝀∗−Σ^𝝀‖F≤B​Lbary​∑m=1Mλm​‖Σm−Σ^m‖F\.\\displaystyle B\\\|\\bar\{\\Sigma\}^\{\*\}\_\{\\bm\{\\lambda\}\}\-\\hat\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\\\|\_\{F\}\\;\\leq\\;BL\_\{\\mathrm\{bary\}\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}\.\(128\)Since‖Σm−Σ^m‖F=sup‖𝐌‖F≤1\|tr⁡\(\(Σm−Σ^m\)​𝐌\)\|\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}=\\sup\_\{\\\|\\mathbf\{M\}\\\|\_\{F\}\\leq 1\}\|\\mathrm\{tr\}\(\(\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\)\\mathbf\{M\}\)\|, the Rademacher symmetrization lemma applied to the i\.i\.d\. sample\{𝚺m,i\}i=1nm\\\{\\bm\{\\Sigma\}\_\{m,i\}\\\}\_\{i=1\}^\{n\_\{m\}\}yields

𝔼\[∥Σm−Σ^m∥F\]≤2𝔼𝝈m​\[1nm​‖∑i=1nmσm,i​𝚺m,i‖F\]⏟=:ℛnmF\.\\displaystyle\\mathbb\{E\}\\bigl\[\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}\\bigr\]\\;\\leq\\;2\\,\\underbrace\{\\mathbb\{E\}\_\{\\bm\{\\sigma\}\_\{m\}\}\\\!\\left\[\\frac\{1\}\{n\_\{m\}\}\\Bigl\\\|\\sum\_\{i=1\}^\{n\_\{m\}\}\\sigma\_\{m,i\}\\bm\{\\Sigma\}\_\{m,i\}\\Bigr\\\|\_\{F\}\\right\]\}\_\{=:\\;\\mathcal\{R\}\_\{n\_\{m\}\}^\{\\mathrm\{F\}\}\}\.\(129\)By definition \([70](https://arxiv.org/html/2608.19914#S3.E70)\),ℜn​\(ℒB,𝝀\)=B​∑mλm​ℛnmF\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\)=B\\sum\_\{m\}\\lambda\_\{m\}\\mathcal\{R\}\_\{n\_\{m\}\}^\{\\mathrm\{F\}\}, so multiplying \([129](https://arxiv.org/html/2608.19914#A2.E129)\) byB​Lbary​λmBL\_\{\\mathrm\{bary\}\}\\lambda\_\{m\}and summing gives

B​Lbary​∑m=1Mλm​𝔼​\[‖Σm−Σ^m‖F\]≤2​Lbary​ℜn​\(ℒB,𝝀\)\.\\displaystyle BL\_\{\\mathrm\{bary\}\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\,\\mathbb\{E\}\\bigl\[\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}\\bigr\]\\;\\leq\\;2L\_\{\\mathrm\{bary\}\}\\,\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\)\.\(130\)For concentration, replacing any single𝐳m,j\\mathbf\{z\}\_\{m,j\}changes‖Σm−Σ^m‖F\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}by at most2​D𝒵2/nm2D\_\{\\mathcal\{Z\}\}^\{2\}/n\_\{m\}\. McDiarmid’s inequality at confidenceδ/M\\delta/Mfor eachmm, combined with \([129](https://arxiv.org/html/2608.19914#A2.E129)\), gives with probability at least1−δ/M1\-\\delta/M:

‖Σm−Σ^m‖F≤2​ℛnmF\+D𝒵2​2​log⁡\(M/δ\)nm\.\\displaystyle\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}\\;\\leq\\;2\\mathcal\{R\}\_\{n\_\{m\}\}^\{\\mathrm\{F\}\}\+D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{\\frac\{2\\log\(M/\\delta\)\}\{n\_\{m\}\}\}\.\(131\)A union bound overm∈\[M\]m\\in\[M\]yields, with probability at least1−δ1\-\\delta:

B​Lbary​∑m=1Mλm​‖Σm−Σ^m‖F≤2​Lbary​ℜn​\(ℒB,𝝀\)\\displaystyle BL\_\{\\mathrm\{bary\}\}\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\\|\\Sigma\_\{m\}\-\\hat\{\\Sigma\}\_\{m\}\\\|\_\{F\}\\;\\leq\\;2L\_\{\\mathrm\{bary\}\}\\,\\mathfrak\{R\}\_\{n\}\(\\mathcal\{L\}\_\{B\};\\bm\{\\lambda\}\)\+B​Lbary​D𝒵2​2​log⁡\(M/δ\)nmin\.\\displaystyle\\quad\+\\frac\{BL\_\{\\mathrm\{bary\}\}D\_\{\\mathcal\{Z\}\}^\{2\}\\sqrt\{2\\log\(M/\\delta\)\}\}\{\\sqrt\{n\_\{\\min\}\}\}\.\(132\)Substituting \([132](https://arxiv.org/html/2608.19914#A2.E132)\) into \([128](https://arxiv.org/html/2608.19914#A2.E128)\) and \([127](https://arxiv.org/html/2608.19914#A2.E127)\) gives term \(II\)≤Δn​\(δ\)\\,\\leq\\Delta\_\{n\}\(\\delta\)with probability at least1−δ1\-\\delta\. Combining with \([125](https://arxiv.org/html/2608.19914#A2.E125)\) and \([122](https://arxiv.org/html/2608.19914#A2.E122)\):

Rℙ∗​\(𝐋^∗\)−R^ϵn​\(𝐋^∗\)≤0\+Δn​\(δ\)=Δn​\(δ\),\\displaystyle R\_\{\\mathbb\{P\}^\{\*\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\-\\hat\{R\}\_\{\\epsilon\_\{n\}\}\(\\hat\{\\mathbf\{L\}\}^\{\*\}\)\\;\\leq\\;0\+\\Delta\_\{n\}\(\\delta\)\\;=\\;\\Delta\_\{n\}\(\\delta\),\(133\)which establishes \([72](https://arxiv.org/html/2608.19914#S3.E72)\)\. Part \(iii\) is immediate from \([71](https://arxiv.org/html/2608.19914#S3.E71)\)\.□\\hfill\\square

## Appendix CProof of Lemma[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmlemma2)

Proof\.Throughout, writeF⁡\(ν\)=∑m=1Mλm​W22​\(ν,ℙm∗\)F\(\\nu\)=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}W\_\{2\}^\{2\}\(\\nu,\\mathbb\{P\}\_\{m\}^\{\*\}\)for the population barycenter functional andF^​\(ν\)=∑m=1Mλm​W22​\(ν,ℙ^m\)\\hat\{F\}\(\\nu\)=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\allowbreak W\_\{2\}^\{2\}\(\\nu,\\hat\{\\mathbb\{P\}\}\_\{m\}\)for its empirical counterpart, so thatb𝝀∗=arg⁡minν⁡F⁡\(ν\)b^\{\*\}\_\{\\bm\{\\lambda\}\}=\\arg\\min\_\{\\nu\}F\(\\nu\)andb^M∗=arg⁡minν​F^​\(ν\)\\hat\{b\}^\{\*\}\_\{M\}=\\arg\\min\_\{\\nu\}\\hat\{F\}\(\\nu\)\. Setδ=W2​\(b^M∗,b𝝀∗\)\\delta=W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\),ξm=W2​\(ℙ^m,ℙm∗\)\\xi\_\{m\}=W\_\{2\}\(\\hat\{\\mathbb\{P\}\}\_\{m\},\\mathbb\{P\}\_\{m\}^\{\*\}\)form∈\[M\]m\\in\[M\], andΔm=W2​\(b𝝀∗,ℙm∗\)\\Delta\_\{m\}=W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\), so thatℋ𝝀2=∑m=1Mλm​Δm2\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Delta\_\{m\}^\{2\}and, by definition,ξM=∑m=1Mλm​ξm2\\xi\_\{M\}=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}^\{2\}; since𝝀∈△M\\bm\{\\lambda\}\\in\\triangle^\{M\}, both\{λm​Δm2\}\\\{\\lambda\_\{m\}\\Delta\_\{m\}^\{2\}\\\}and\{λm​ξm2\}\\\{\\lambda\_\{m\}\\xi\_\{m\}^\{2\}\\\}are convex combinations, a fact used repeatedly below\. Our goal is to controlδ\\deltain terms ofξM\\xi\_\{M\}andℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\.

We first translate the sub\-optimality gapF⁡\(b^M∗\)−F⁡\(b𝝀∗\)F\(\\hat\{b\}^\{\*\}\_\{M\}\)\-F\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)into a bound onδ\\deltaitself\. Assumption[III\.4](https://arxiv.org/html/2608.19914#S3.Thmassumption4)postulates thatFFisκ\\kappa\-strongly convex along generalized geodesics on\(𝒫2​\(𝒵\),W2\)\(\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\),W\_\{2\}\); combined with the first\-order optimality ofb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}as the unique global minimizer ofFF, strong geodesic convexity yields, for everyν∈𝒫2​\(𝒵\)\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\),

F⁡\(ν\)−F⁡\(b𝝀∗\)≥κ​W22​\(ν,b𝝀∗\),\\displaystyle F\(\\nu\)\-F\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\;\\geq\\;\\kappa\\,W\_\{2\}^\{2\}\(\\nu,b^\{\*\}\_\{\\bm\{\\lambda\}\}\),\(134\)because the first\-order term in the geodesic convexity inequality is non\-negative at a minimizer and may be discarded without weakening the bound\. Instantiating \([134](https://arxiv.org/html/2608.19914#A3.E134)\) atν=b^M∗\\nu=\\hat\{b\}^\{\*\}\_\{M\}gives

κ​δ2≤F⁡\(b^M∗\)−F⁡\(b𝝀∗\)\.\\displaystyle\\kappa\\,\\delta^\{2\}\\;\\leq\\;F\\\!\\bigl\(\\hat\{b\}^\{\*\}\_\{M\}\\bigr\)\-F\\\!\\bigl\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)\.\(135\)It therefore suffices to bound the right\-hand side of \([135](https://arxiv.org/html/2608.19914#A3.E135)\)\.

Becauseb^M∗\\hat\{b\}^\{\*\}\_\{M\}minimizesF^\\hat\{F\}over𝒫2​\(𝒵\)\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\)whileb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}is merely a feasible, generally suboptimal, point ofF^\\hat\{F\}, we haveF^​\(b^M∗\)≤F^​\(b𝝀∗\)\\hat\{F\}\(\\hat\{b\}^\{\*\}\_\{M\}\)\\leq\\hat\{F\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\. Adding and subtractingF^​\(b^M∗\)\\hat\{F\}\(\\hat\{b\}^\{\*\}\_\{M\}\)andF^​\(b𝝀∗\)\\hat\{F\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)insideF⁡\(b^M∗\)−F⁡\(b𝝀∗\)F\(\\hat\{b\}^\{\*\}\_\{M\}\)\-F\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)and then discarding the non\-positive quantityF^​\(b^M∗\)−F^​\(b𝝀∗\)\\hat\{F\}\(\\hat\{b\}^\{\*\}\_\{M\}\)\-\\hat\{F\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)produces the basic inequality

F⁡\(b^M∗\)−F⁡\(b𝝀∗\)\\displaystyle F\\\!\\bigl\(\\hat\{b\}^\{\*\}\_\{M\}\\bigr\)\-F\\\!\\bigl\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\\bigr\)≤\[F⁡\(b^M∗\)−F^​\(b^M∗\)\]⏟=:T1\\displaystyle\\leq\\underbrace\{\\bigl\[F\(\\hat\{b\}^\{\*\}\_\{M\}\)\-\\hat\{F\}\(\\hat\{b\}^\{\*\}\_\{M\}\)\\bigr\]\}\_\{=:T\_\{1\}\}\+\[F^​\(b𝝀∗\)−F⁡\(b𝝀∗\)\]⏟=:T2\.\\displaystyle\\quad\+\\underbrace\{\\bigl\[\\hat\{F\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\-F\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\bigr\]\}\_\{=:T\_\{2\}\}\.\(136\)Expanding the definitions ofFFandF^\\hat\{F\}term by term,

T1\\displaystyle T\_\{1\}=∑m=1Mλm​\[W22​\(b^M∗,ℙm∗\)−W22​\(b^M∗,ℙ^m\)\],\\displaystyle=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\\!\\bigl\[W\_\{2\}^\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\-W\_\{2\}^\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\bigr\],T2\\displaystyle T\_\{2\}=∑m=1Mλm​\[W22​\(b𝝀∗,ℙ^m\)−W22​\(b𝝀∗,ℙm∗\)\]\.\\displaystyle=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\\!\\bigl\[W\_\{2\}^\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\-W\_\{2\}^\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\bigr\]\.\(137\)Both sums involve differences of squared Wasserstein distances from a fixed reference measure to two nearby targets, which we now bound uniformly\. For anyν∈𝒫2​\(𝒵\)\\nu\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\)and anyα,β∈𝒫2​\(𝒵\)\\alpha,\\beta\\in\\mathcal\{P\}\_\{2\}\(\\mathcal\{Z\}\), factoring the difference of squares and applying the reverse triangle inequality\|W2​\(ν,α\)−W2​\(ν,β\)\|≤W2​\(α,β\)\|W\_\{2\}\(\\nu,\\alpha\)\-W\_\{2\}\(\\nu,\\beta\)\|\\leq W\_\{2\}\(\\alpha,\\beta\)gives

\|W22​\(ν,α\)−W22​\(ν,β\)\|\\displaystyle\\bigl\|W\_\{2\}^\{2\}\(\\nu,\\alpha\)\-W\_\{2\}^\{2\}\(\\nu,\\beta\)\\bigr\|=\|W2​\(ν,α\)−W2​\(ν,β\)\|⋅\(W2​\(ν,α\)\+W2​\(ν,β\)\)\\displaystyle=\\bigl\|W\_\{2\}\(\\nu,\\alpha\)\-W\_\{2\}\(\\nu,\\beta\)\\bigr\|\\cdot\\bigl\(W\_\{2\}\(\\nu,\\alpha\)\+W\_\{2\}\(\\nu,\\beta\)\\bigr\)≤\[W2​\(ν,α\)\+W2​\(ν,β\)\]​W2​\(α,β\)\.\\displaystyle\\leq\\bigl\[W\_\{2\}\(\\nu,\\alpha\)\+W\_\{2\}\(\\nu,\\beta\)\\bigr\]\\,W\_\{2\}\(\\alpha,\\beta\)\.\(138\)We apply \([138](https://arxiv.org/html/2608.19914#A3.E138)\) to each summand ofT1T\_\{1\}withν=b^M∗\\nu=\\hat\{b\}^\{\*\}\_\{M\},α=ℙm∗\\alpha=\\mathbb\{P\}\_\{m\}^\{\*\},β=ℙ^m\\beta=\\hat\{\\mathbb\{P\}\}\_\{m\}, so thatW2​\(α,β\)=ξmW\_\{2\}\(\\alpha,\\beta\)=\\xi\_\{m\}; it remains to bound the sumW2​\(b^M∗,ℙm∗\)\+W2​\(b^M∗,ℙ^m\)W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\+W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\. The triangle inequality throughb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}givesW2​\(b^M∗,ℙm∗\)≤W2​\(b^M∗,b𝝀∗\)\+W2​\(b𝝀∗,ℙm∗\)=δ\+ΔmW\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\leq W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\+W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)=\\delta\+\\Delta\_\{m\}, and a further application of the triangle inequality throughℙm∗\\mathbb\{P\}\_\{m\}^\{\*\}givesW2​\(b^M∗,ℙ^m\)≤W2​\(b^M∗,ℙm∗\)\+W2​\(ℙm∗,ℙ^m\)≤δ\+Δm\+ξmW\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\leq W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\+W\_\{2\}\(\\mathbb\{P\}\_\{m\}^\{\*\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\leq\\delta\+\\Delta\_\{m\}\+\\xi\_\{m\}\. Summing these two bounds,W2​\(b^M∗,ℙm∗\)\+W2​\(b^M∗,ℙ^m\)≤2​δ\+2​Δm\+ξmW\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\+W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\leq 2\\delta\+2\\Delta\_\{m\}\+\\xi\_\{m\}, and substituting into \([138](https://arxiv.org/html/2608.19914#A3.E138)\) yields the termwise bound

W22​\(b^M∗,ℙm∗\)−W22​\(b^M∗,ℙ^m\)≤\(2​δ\+2​Δm\+ξm\)​ξm\.\\displaystyle W\_\{2\}^\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\-W\_\{2\}^\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\;\\leq\\;\(2\\delta\+2\\Delta\_\{m\}\+\\xi\_\{m\}\)\\,\\xi\_\{m\}\.\(139\)The same argument applied to each summand ofT2T\_\{2\}, now withν=b𝝀∗\\nu=b^\{\*\}\_\{\\bm\{\\lambda\}\},α=ℙ^m\\alpha=\\hat\{\\mathbb\{P\}\}\_\{m\},β=ℙm∗\\beta=\\mathbb\{P\}\_\{m\}^\{\*\}\(soW2​\(α,β\)=ξmW\_\{2\}\(\\alpha,\\beta\)=\\xi\_\{m\}again\), givesW2​\(b𝝀∗,ℙ^m\)≤W2​\(b𝝀∗,ℙm∗\)\+W2​\(ℙm∗,ℙ^m\)=Δm\+ξmW\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\\leq W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\+W\_\{2\}\(\\mathbb\{P\}\_\{m\}^\{\*\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)=\\Delta\_\{m\}\+\\xi\_\{m\}andW2​\(b𝝀∗,ℙm∗\)=ΔmW\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)=\\Delta\_\{m\}, so thatW2​\(b𝝀∗,ℙ^m\)\+W2​\(b𝝀∗,ℙm∗\)≤2​Δm\+ξmW\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\+W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\leq 2\\Delta\_\{m\}\+\\xi\_\{m\}and hence

W22​\(b𝝀∗,ℙ^m\)−W22​\(b𝝀∗,ℙm∗\)≤\(2​Δm\+ξm\)​ξm\.\\displaystyle W\_\{2\}^\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\hat\{\\mathbb\{P\}\}\_\{m\}\)\-W\_\{2\}^\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)\\;\\leq\\;\(2\\Delta\_\{m\}\+\\xi\_\{m\}\)\\,\\xi\_\{m\}\.\(140\)Weighting \([139](https://arxiv.org/html/2608.19914#A3.E139)\) and \([140](https://arxiv.org/html/2608.19914#A3.E140)\) byλm\\lambda\_\{m\}, summing overm∈\[M\]m\\in\[M\], and substituting into \([137](https://arxiv.org/html/2608.19914#A3.E137)\) gives

T1\+T2\\displaystyle T\_\{1\}\+T\_\{2\}≤2​δ​∑m=1Mλm​ξm\+4​∑m=1Mλm​Δm​ξm\\displaystyle\\leq 2\\delta\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}\+4\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Delta\_\{m\}\\xi\_\{m\}\+2∑m=1Mλmξm2,\\displaystyle\\quad\+2\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}^\{2\},\(141\)which, together with \([136](https://arxiv.org/html/2608.19914#A3.E136)\), boundsF⁡\(b^M∗\)−F⁡\(b𝝀∗\)F\(\\hat\{b\}^\{\*\}\_\{M\}\)\-F\(b^\{\*\}\_\{\\bm\{\\lambda\}\}\)by the right\-hand side of \([141](https://arxiv.org/html/2608.19914#A3.E141)\)\.

The three sums in \([141](https://arxiv.org/html/2608.19914#A3.E141)\) are controlled by the Cauchy–Schwarz inequality with respect to the probability weights𝝀\\bm\{\\lambda\}\. Since∑m=1Mλm=1\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}=1, treatingλm\\lambda\_\{m\}as a probability measure on\[M\]\[M\]and applying Cauchy–Schwarz to the pairs\(ξm,1\)\(\\xi\_\{m\},1\)and\(Δm,ξm\)\(\\Delta\_\{m\},\\xi\_\{m\}\)gives

∑m=1Mλm​ξm≤\(∑m=1Mλm​ξm2\)1/2​\(∑m=1Mλm\)1/2=ξM1/2,\\displaystyle\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}\\;\\leq\\;\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}^\{2\}\\Bigr\)^\{1/2\}\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Bigr\)^\{1/2\}=\\xi\_\{M\}^\{1/2\},\(142\)∑m=1Mλm​Δm​ξm\\displaystyle\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Delta\_\{m\}\\xi\_\{m\}≤\(∑m=1Mλm​Δm2\)1/2​\(∑m=1Mλm​ξm2\)1/2\\displaystyle\\leq\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Delta\_\{m\}^\{2\}\\Bigr\)^\{1/2\}\\Bigl\(\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\xi\_\{m\}^\{2\}\\Bigr\)^\{1/2\}=ℋ𝝀​ξM1/2,\\displaystyle=\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\,\\xi\_\{M\}^\{1/2\},\(143\)while the third sum equalsξM\\xi\_\{M\}exactly, by definition\. Substituting \([142](https://arxiv.org/html/2608.19914#A3.E142)\)–\([143](https://arxiv.org/html/2608.19914#A3.E143)\) into \([141](https://arxiv.org/html/2608.19914#A3.E141)\) and then into \([135](https://arxiv.org/html/2608.19914#A3.E135)\) yields the single quadratic inequality

κ​δ2≤2​δ​ξM1/2\+4​ℋ𝝀​ξM1/2\+2​ξM\.\\displaystyle\\kappa\\,\\delta^\{2\}\\;\\leq\\;2\\delta\\,\\xi\_\{M\}^\{1/2\}\+4\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\,\\xi\_\{M\}^\{1/2\}\+2\\xi\_\{M\}\.\(144\)
It remains to extract an explicit bound onδ\\deltafrom \([144](https://arxiv.org/html/2608.19914#A3.E144)\)\. Rearranging, \([144](https://arxiv.org/html/2608.19914#A3.E144)\) states thatδ≥0\\delta\\geq 0satisfiesg⁡\(δ\)≤0g\(\\delta\)\\leq 0for the univariate quadraticg⁡\(x\)=κ​x2−2​ξM1/2​x−Cg\(x\)=\\kappa x^\{2\}\-2\\xi\_\{M\}^\{1/2\}x\-CwithC:=4​ℋ𝝀​ξM1/2\+2​ξM≥0C:=4\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\xi\_\{M\}^\{1/2\}\+2\\xi\_\{M\}\\geq 0\. Sinceκ\>0\\kappa\>0,ggis a strictly convex parabola opening upward withg⁡\(0\)=−C≤0g\(0\)=\-C\\leq 0; consequentlyggadmits a unique non\-negative rootx∗≥0x^\{\\ast\}\\geq 0, andg⁡\(x\)≤0g\(x\)\\leq 0precisely on the interval\[x−,x∗\]\[x\_\{\-\},x^\{\\ast\}\]withx−≤0≤x∗x\_\{\-\}\\leq 0\\leq x^\{\\ast\}\. Asδ≥0\\delta\\geq 0, this forcesδ≤x∗\\delta\\leq x^\{\\ast\}, where the quadratic formula gives

x∗=2​ξM1/2\+4​ξM\+4​κ​C2​κ=ξM1/2\+ξM\+κ​Cκ\.\\displaystyle x^\{\\ast\}=\\frac\{2\\xi\_\{M\}^\{1/2\}\+\\sqrt\{4\\xi\_\{M\}\+4\\kappa C\}\}\{2\\kappa\}=\\frac\{\\xi\_\{M\}^\{1/2\}\+\\sqrt\{\\xi\_\{M\}\+\\kappa C\}\}\{\\kappa\}\.\(145\)Substitutingκ​C=4​κ​ℋ𝝀​ξM1/2\+2​κ​ξM\\kappa C=4\\kappa\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\xi\_\{M\}^\{1/2\}\+2\\kappa\\xi\_\{M\}and collecting theξM\\xi\_\{M\}terms gives

δ≤ξM1/2\+\(1\+2​κ\)​ξM\+4​κ​ℋ𝝀​ξM1/2κ\.\\displaystyle\\delta\\;\\leq\\;\\frac\{\\xi\_\{M\}^\{1/2\}\+\\sqrt\{\(1\+2\\kappa\)\\,\\xi\_\{M\}\+4\\kappa\\,\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\,\\xi\_\{M\}^\{1/2\}\}\}\{\\kappa\}\.\(146\)Applying the subadditivity of the square root,a\+b≤a\+b\\sqrt\{a\+b\}\\leq\\sqrt\{a\}\+\\sqrt\{b\}fora,b≥0a,b\\geq 0, to the two non\-negative terms under the radical in \([146](https://arxiv.org/html/2608.19914#A3.E146)\), witha=\(1\+2​κ\)​ξMa=\(1\+2\\kappa\)\\xi\_\{M\}andb=4​κ​ℋ𝝀​ξM1/2b=4\\kappa\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\xi\_\{M\}^\{1/2\}, gives

\(1\+2​κ\)​ξM\+4​κ​ℋ𝝀​ξM1/2\\displaystyle\\sqrt\{\(1\+2\\kappa\)\\xi\_\{M\}\+4\\kappa\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\xi\_\{M\}^\{1/2\}\}≤1\+2​κ​ξM1/2\\displaystyle\\leq\\sqrt\{1\+2\\kappa\}\\;\\xi\_\{M\}^\{1/2\}\+2​κ​ℋ𝝀1/2​ξM1/4,\\displaystyle\\quad\+2\\sqrt\{\\kappa\}\\;\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\,\\xi\_\{M\}^\{1/4\},\(147\)using\(ξM1/2\)2=ξM1/2\\sqrt\{\(\\xi\_\{M\}^\{1/2\}\)^\{2\}\}=\\xi\_\{M\}^\{1/2\}and4​κ​ℋ𝝀​ξM1/2=2​κ​ℋ𝝀​ξM1/4=2​κ​ℋ𝝀1/2​ξM1/4\\sqrt\{4\\kappa\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\\xi\_\{M\}^\{1/2\}\}=2\\sqrt\{\\kappa\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\}\\,\\xi\_\{M\}^\{1/4\}=2\\sqrt\{\\kappa\}\\,\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\xi\_\{M\}^\{1/4\}\. Combining \([146](https://arxiv.org/html/2608.19914#A3.E146)\) and \([147](https://arxiv.org/html/2608.19914#A3.E147)\),

δ≤\(1\+1\+2​κ\)​ξM1/2\+2​κ​ℋ𝝀1/2​ξM1/4κ\.\\displaystyle\\delta\\;\\leq\\;\\frac\{\\bigl\(1\+\\sqrt\{1\+2\\kappa\}\\,\\bigr\)\\,\\xi\_\{M\}^\{1/2\}\+2\\sqrt\{\\kappa\}\\,\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\xi\_\{M\}^\{1/4\}\}\{\\kappa\}\.\(148\)Finally, settingC0:=1\+1\+2​κ\+2​κC\_\{0\}:=1\+\\sqrt\{1\+2\\kappa\}\+2\\sqrt\{\\kappa\}, we have both1\+1\+2​κ≤C01\+\\sqrt\{1\+2\\kappa\}\\leq C\_\{0\}and2​κ≤C02\\sqrt\{\\kappa\}\\leq C\_\{0\}by construction, so that bounding each coefficient in \([148](https://arxiv.org/html/2608.19914#A3.E148)\) byC0C\_\{0\}gives

δ≤C0κ​\(ℋ𝝀1/2​ξM1/4\+ξM1/2\),\\displaystyle\\delta\\;\\leq\\;\\frac\{C\_\{0\}\}\{\\kappa\}\\Bigl\(\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{1/2\}\\xi\_\{M\}^\{1/4\}\+\\xi\_\{M\}^\{1/2\}\\Bigr\),\(149\)which is exactly \([83](https://arxiv.org/html/2608.19914#S4.E83)\), withC0C\_\{0\}a universal constant independent ofMM,𝝀\\bm\{\\lambda\}, and the source distributions\. This completes the proof\.□\\hfill\\square

## Appendix DProof of Theorem[IV\.2](https://arxiv.org/html/2608.19914#S4.Thmtheorem2)

Proof\.We first identify the barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}explicitly under the common\-covariance model\. SubstitutingΣm=Σ\\Sigma\_\{m\}=\\Sigmafor everym∈\[M\]m\\in\[M\]into the covariance fixed\-point equation of Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)\(\([23](https://arxiv.org/html/2608.19914#S3.E23)\)\) shows thatΣ¯=Σ\\bar\{\\Sigma\}=\\Sigmais a solution, sinceΣ1/2​Σ​Σ1/2=Σ2=Σ\\sqrt\{\\Sigma^\{1/2\}\\Sigma\\Sigma^\{1/2\}\}=\\sqrt\{\\Sigma^\{2\}\}=\\Sigmaand hence the right\-hand side collapses to∑m=1Mλm​Σ=Σ\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Sigma=\\Sigmaby∑mλm=1\\sum\_\{m\}\\lambda\_\{m\}=1; as Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)guarantees the fixed point is unique in𝕊\+⁣\+N\\mathbb\{S\}\_\{\+\+\}^\{N\}, this is*the*barycentric covariance, and together with the mean formulaμ¯𝝀=∑mλm​μm\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\mu\_\{m\}of Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)we obtain

b𝝀∗=𝒩⁡\(μ¯𝝀,Σ\)\.\\displaystyle b^\{\*\}\_\{\\bm\{\\lambda\}\}=\\mathcal\{N\}\(\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\},\\,\\Sigma\)\.\(150\)This identification also lets us relate the inter\-source heterogeneityℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}of \([82](https://arxiv.org/html/2608.19914#S4.E82)\) to the scatter matrixΔ𝝀\\Delta\_\{\\bm\{\\lambda\}\}: sinceb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}and eachℙm∗\\mathbb\{P\}\_\{m\}^\{\*\}are Gaussian with the identical covarianceΣ\\Sigma, the Gelbrich formula givesW2​\(b𝝀∗,ℙm∗\)2=‖μm−μ¯𝝀‖2\+B2​\(Σ,Σ\)=‖μm−μ¯𝝀‖2W\_\{2\}\(b^\{\*\}\_\{\\bm\{\\lambda\}\},\\mathbb\{P\}\_\{m\}^\{\*\}\)^\{2\}=\\\|\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\\\|^\{2\}\+B^\{2\}\(\\Sigma,\\Sigma\)=\\\|\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\\\|^\{2\}, becauseB⁡\(Σ,Σ\)=0B\(\\Sigma,\\Sigma\)=0; weighting byλm\\lambda\_\{m\}and summing overmmtherefore yieldsℋ𝝀2=∑mλm​‖μm−μ¯𝝀‖2=tr⁡\(∑mλm​\(μm−μ¯𝝀\)​\(μm−μ¯𝝀\)T\)=tr⁡\(Δ𝝀\)\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}=\\sum\_\{m\}\\lambda\_\{m\}\\\|\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\\\|^\{2\}=\\mathrm\{tr\}\\bigl\(\\sum\_\{m\}\\lambda\_\{m\}\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\\bigr\)=\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\), an identity used repeatedly below\.

We next compute the covariance of the pooled distributionℙ𝝀∗\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}by conditioning on source membership\. Letι∈\[M\]\\iota\\in\[M\]be a random index withPr⁡\(ι=m\)=λm\\Pr\(\\iota=m\)=\\lambda\_\{m\}, and letX∼ℙ𝝀∗X\\sim\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}be generated hierarchically asX=μι\+Σ1/2​ZX=\\mu\_\{\\iota\}\+\\Sigma^\{1/2\}ZwithZ∼𝒩⁡\(0,𝐈N\)Z\\sim\\mathcal\{N\}\(0,\\mathbf\{I\}\_\{N\}\)independent ofι\\iota; this construction reproducesℙ𝝀∗=∑mλm​𝒩​\(μm,Σ\)\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\mathcal\{N\}\(\\mu\_\{m\},\\Sigma\)by construction\. ThenX−μ¯𝝀=\(μι−μ¯𝝀\)\+Σ1/2​ZX\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}=\(\\mu\_\{\\iota\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\+\\Sigma^\{1/2\}Z, and expanding the outer product,

\(X−μ¯𝝀\)​\(X−μ¯𝝀\)T=\(μι−μ¯𝝀\)​\(μι−μ¯𝝀\)T\\displaystyle\(X\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(X\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}=\(\\mu\_\{\\iota\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(\\mu\_\{\\iota\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\+Σ1/2​Z​\(μι−μ¯𝝀\)T\+\(μι−μ¯𝝀\)​ZT​Σ1/2\\displaystyle\\quad\+\\Sigma^\{1/2\}Z\(\\mu\_\{\\iota\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\+\(\\mu\_\{\\iota\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)Z^\{T\}\\Sigma^\{1/2\}\+Σ1/2​Z​ZT​Σ1/2\.\\displaystyle\\quad\+\\Sigma^\{1/2\}ZZ^\{T\}\\Sigma^\{1/2\}\.\(151\)Taking the conditional expectation givenι\\iotaand using𝔼⁡\[Z\]=𝟎\\mathbb\{E\}\[Z\]=\\mathbf\{0\},𝔼⁡\[Z​ZT\]=𝐈N\\mathbb\{E\}\[ZZ^\{T\}\]=\\mathbf\{I\}\_\{N\}\(independence ofZZfromι\\iota\), the two cross terms vanish and the last term reduces toΣ\\Sigma, giving𝔼⁡\[\(X−μ¯𝝀\)​\(X−μ¯𝝀\)T∣ι=m\]=\(μm−μ¯𝝀\)​\(μm−μ¯𝝀\)T\+Σ\\mathbb\{E\}\\bigl\[\(X\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(X\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\\mid\\iota=m\\bigr\]=\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\+\\Sigma\. Averaging overι\\iotawith weightsλm\\lambda\_\{m\}then yields

Cov⁡\(ℙ𝝀∗\)\\displaystyle\\mathrm\{Cov\}\(\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}\)=∑m=1Mλm​\[Σ\+\(μm−μ¯𝝀\)​\(μm−μ¯𝝀\)T\]\\displaystyle=\\sum\_\{m=1\}^\{M\}\\lambda\_\{m\}\\Bigl\[\\Sigma\+\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)\(\\mu\_\{m\}\-\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}\)^\{T\}\\Bigr\]=Σ\+Δ𝝀,\\displaystyle=\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\},\(152\)using∑mλm=1\\sum\_\{m\}\\lambda\_\{m\}=1to isolateΣ\\Sigmaand the definition \([88](https://arxiv.org/html/2608.19914#S4.E88)\) for the second term\. SinceΔ𝝀⪰0\\Delta\_\{\\bm\{\\lambda\}\}\\succeq 0andΔ𝝀≠𝟎\\Delta\_\{\\bm\{\\lambda\}\}\\neq\\mathbf\{0\}precisely whenℋ𝝀=tr⁡\(Δ𝝀\)\>0\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}=\\sqrt\{\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)\}\>0, identity \([152](https://arxiv.org/html/2608.19914#A4.E152)\) shows that heterogeneous source means strictly inflate the covariance of the pooled distribution relative to that of the true barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}, whose covariance remainsΣ\\Sigmaby \([150](https://arxiv.org/html/2608.19914#A4.E150)\)\.

This covariance inflation is converted into a Wasserstein separation via the Gelbrich inequality\[[39](https://arxiv.org/html/2608.19914#bib.bib39)\], which states that for anyα,β∈𝒫2​\(ℝN\)\\alpha,\\beta\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{N\}\)with means𝔼α​\[X\]\\mathbb\{E\}\_\{\\alpha\}\[X\],𝔼β​\[Y\]\\mathbb\{E\}\_\{\\beta\}\[Y\]and covariancesCov⁡\(α\)\\mathrm\{Cov\}\(\\alpha\),Cov⁡\(β\)\\mathrm\{Cov\}\(\\beta\),

W22​\(α,β\)\\displaystyle W\_\{2\}^\{2\}\(\\alpha,\\beta\)≥‖𝔼α​\[X\]−𝔼β​\[Y\]‖2\\displaystyle\\geq\\bigl\\\|\\mathbb\{E\}\_\{\\alpha\}\[X\]\-\\mathbb\{E\}\_\{\\beta\}\[Y\]\\bigr\\\|^\{2\}\+B2​\(Cov⁡\(α\),Cov⁡\(β\)\),\\displaystyle\\quad\+B^\{2\}\\\!\\bigl\(\\mathrm\{Cov\}\(\\alpha\),\\mathrm\{Cov\}\(\\beta\)\\bigr\),\(153\)this inequality holding for arbitrary distributions with the stated moments, with equality when both are Gaussian\. Takingα=ℙ𝝀∗\\alpha=\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}, with meanμ¯𝝀\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}and covarianceΣ\+Δ𝝀\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}by \([152](https://arxiv.org/html/2608.19914#A4.E152)\), andβ=b𝝀∗\\beta=b^\{\*\}\_\{\\bm\{\\lambda\}\}, with meanμ¯𝝀\\bar\{\\mu\}\_\{\\bm\{\\lambda\}\}and covarianceΣ\\Sigmaby \([150](https://arxiv.org/html/2608.19914#A4.E150)\), the two means coincide and the mean term in \([153](https://arxiv.org/html/2608.19914#A4.E153)\) vanishes identically, leaving

W22​\(ℙ𝝀∗,b𝝀∗\)≥B2​\(Σ\+Δ𝝀,Σ\),\\displaystyle W\_\{2\}^\{2\}\(\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\;\\geq\\;B^\{2\}\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\},\\Sigma\),\(154\)which is the first inequality in \([89](https://arxiv.org/html/2608.19914#S4.E89)\)\. It remains to lower boundB2​\(Σ\+Δ𝝀,Σ\)B^\{2\}\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\},\\Sigma\)explicitly in terms ofℋ𝝀\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\.

WritingF⁡\(ρ,σ\)≜tr⁡\(\(ρ1/2​σ​ρ1/2\)1/2\)F\(\\rho,\\sigma\)\\triangleq\\mathrm\{tr\}\\bigl\(\(\\rho^\{1/2\}\\sigma\\rho^\{1/2\}\)^\{1/2\}\\bigr\)for the Uhlmann–Bures quantum fidelity, the Bures–Wasserstein distance admits the identity

B2​\(Σ,Σ\+Δ𝝀\)\\displaystyle B^\{2\}\(\\Sigma,\\,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)=tr⁡\(Σ\)\+tr⁡\(Σ\+Δ𝝀\)\\displaystyle=\\mathrm\{tr\}\(\\Sigma\)\+\\mathrm\{tr\}\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)−2​F​\(Σ,Σ\+Δ𝝀\),\\displaystyle\\quad\-2\\,F\(\\Sigma,\\,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\),\(155\)so that a lower bound onB2B^\{2\}follows from an*upper*bound on the fidelity termF⁡\(Σ,Σ\+Δ𝝀\)F\(\\Sigma,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)\. We obtain such a bound by symmetrizingΣ\\SigmaandΣ\+Δ𝝀\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}over the orthogonal group\. The fidelityFFis jointly concave on𝕊\+N×𝕊\+N\\mathbb\{S\}\_\{\+\}^\{N\}\\times\\mathbb\{S\}\_\{\+\}^\{N\}by the Lieb concavity theorem\[[45](https://arxiv.org/html/2608.19914#bib.bib45)\], and it is invariant under simultaneous conjugation by any orthogonal matrix,F⁡\(U​ρ​UT,U​σ​UT\)=F⁡\(ρ,σ\)F\(U\\rho U^\{T\},U\\sigma U^\{T\}\)=F\(\\rho,\\sigma\)for allU∈O⁡\(N\)U\\in\\mathrm\{O\}\(N\), since conjugation byUUmerely relabels the eigenbasis without altering the eigenvalues ofρ1/2​σ​ρ1/2\\rho^\{1/2\}\\sigma\\rho^\{1/2\}\. Letμ\\mudenote the Haar probability measure onO⁡\(N\)\\mathrm\{O\}\(N\)and regardU↦\(U​Σ​UT,U⁡\(Σ\+Δ𝝀\)​UT\)U\\mapsto\\bigl\(U\\Sigma U^\{T\},\\,U\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)U^\{T\}\\bigr\)as aμ\\mu\-random pair of matrices\. Jensen’s inequality applied to the jointly concave mapFFgives

F⁡\(∫U​Σ​UT​𝑑μ​\(U\),∫U⁡\(Σ\+Δ𝝀\)​UT​𝑑μ​\(U\)\)\\displaystyle F\\\!\\left\(\\int U\\Sigma U^\{T\}\\,d\\mu\(U\),\\;\\int U\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)U^\{T\}\\,d\\mu\(U\)\\right\)≥∫F⁡\(U​Σ​UT,U⁡\(Σ\+Δ𝝀\)​UT\)​𝑑μ​\(U\),\\displaystyle\\geq\\int F\\bigl\(U\\Sigma U^\{T\},U\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)U^\{T\}\\bigr\)\\,d\\mu\(U\),\(156\)and by unitary invariance the integrand on the right equals the constantF⁡\(Σ,Σ\+Δ𝝀\)F\(\\Sigma,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\), so that the right\-hand side of \([156](https://arxiv.org/html/2608.19914#A4.E156)\) evaluates toF⁡\(Σ,Σ\+Δ𝝀\)F\(\\Sigma,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)\. On the left\-hand side, the twirling identity∫O⁡\(N\)U​A​UT​𝑑μ​\(U\)=\(tr⁡\(A\)/N\)​𝐈\\int\_\{\\mathrm\{O\}\(N\)\}UAU^\{T\}\\,d\\mu\(U\)=\(\\mathrm\{tr\}\(A\)/N\)\\,\\mathbf\{I\}, valid for anyA∈𝕊NA\\in\\mathbb\{S\}^\{N\}\[[46](https://arxiv.org/html/2608.19914#bib.bib46)\], gives∫U​Σ​UT​𝑑μ=σ¯​𝐈\\int U\\Sigma U^\{T\}\\,d\\mu=\\bar\{\\sigma\}\\mathbf\{I\}and∫U⁡\(Σ\+Δ𝝀\)​UT​𝑑μ=\(σ¯\+d¯\)​𝐈\\int U\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)U^\{T\}\\,d\\mu=\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}, where we writeσ¯:=tr⁡\(Σ\)/N\\bar\{\\sigma\}:=\\mathrm\{tr\}\(\\Sigma\)/Nandd¯:=tr⁡\(Δ𝝀\)/N=ℋ𝝀2/N\\bar\{d\}:=\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)/N=\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}/N\(the trace identity established above\)\. Combining these two evaluations of \([156](https://arxiv.org/html/2608.19914#A4.E156)\) yields the scalar reduction

F⁡\(Σ,Σ\+Δ𝝀\)≤F⁡\(σ¯​𝐈,\(σ¯\+d¯\)​𝐈\)\.\\displaystyle F\(\\Sigma,\\,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)\\;\\leq\\;F\\bigl\(\\bar\{\\sigma\}\\mathbf\{I\},\\,\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}\\bigr\)\.\(157\)The right\-hand side of \([157](https://arxiv.org/html/2608.19914#A4.E157)\) is now an explicit scalar quantity: sinceσ¯​𝐈\\bar\{\\sigma\}\\mathbf\{I\}and\(σ¯\+d¯\)​𝐈\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}are proportional to the identity and therefore commute,\(σ¯​𝐈\)1/2​\(σ¯\+d¯\)​𝐈​\(σ¯​𝐈\)1/2=σ¯​\(σ¯\+d¯\)​𝐈\(\\bar\{\\sigma\}\\mathbf\{I\}\)^\{1/2\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}\(\\bar\{\\sigma\}\\mathbf\{I\}\)^\{1/2\}=\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}, whose square root isσ¯​\(σ¯\+d¯\)​𝐈\\sqrt\{\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}\\,\\mathbf\{I\}, giving

F⁡\(σ¯​𝐈,\(σ¯\+d¯\)​𝐈\)=N​σ¯​\(σ¯\+d¯\)=N​σ¯​1\+d¯/σ¯\.\\displaystyle F\\bigl\(\\bar\{\\sigma\}\\mathbf\{I\},\(\\bar\{\\sigma\}\+\\bar\{d\}\)\\mathbf\{I\}\\bigr\)=N\\sqrt\{\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}=N\\bar\{\\sigma\}\\sqrt\{1\+\\bar\{d\}/\\bar\{\\sigma\}\}\.\(158\)
Substituting \([157](https://arxiv.org/html/2608.19914#A4.E157)\) and \([158](https://arxiv.org/html/2608.19914#A4.E158)\) into \([155](https://arxiv.org/html/2608.19914#A4.E155)\), and usingtr⁡\(Σ\)\+tr⁡\(Σ\+Δ𝝀\)=N​σ¯\+N⁡\(σ¯\+d¯\)=N⁡\(2​σ¯\+d¯\)\\mathrm\{tr\}\(\\Sigma\)\+\\mathrm\{tr\}\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)=N\\bar\{\\sigma\}\+N\(\\bar\{\\sigma\}\+\\bar\{d\}\)=N\(2\\bar\{\\sigma\}\+\\bar\{d\}\), gives

B2​\(Σ,Σ\+Δ𝝀\)≥N⁡\(2​σ¯\+d¯\)−2​N​σ¯​1\+d¯/σ¯\.\\displaystyle B^\{2\}\(\\Sigma,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)\\;\\geq\\;N\(2\\bar\{\\sigma\}\+\\bar\{d\}\)\-2N\\bar\{\\sigma\}\\sqrt\{1\+\\bar\{d\}/\\bar\{\\sigma\}\}\.\(159\)The right\-hand side of \([159](https://arxiv.org/html/2608.19914#A4.E159)\) is a perfect square in disguise: expandingN​\(σ¯\+d¯−σ¯\)2=N⁡\[\(σ¯\+d¯\)−2​σ¯​\(σ¯\+d¯\)\+σ¯\]=N⁡\(2​σ¯\+d¯\)−2​N​σ¯​\(σ¯\+d¯\)N\\bigl\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\-\\sqrt\{\\bar\{\\sigma\}\}\\bigr\)^\{2\}=N\\bigl\[\(\\bar\{\\sigma\}\+\\bar\{d\}\)\-2\\sqrt\{\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}\+\\bar\{\\sigma\}\\bigr\]=N\(2\\bar\{\\sigma\}\+\\bar\{d\}\)\-2N\\sqrt\{\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}, and notingσ¯​\(σ¯\+d¯\)=σ¯​1\+d¯/σ¯\\sqrt\{\\bar\{\\sigma\}\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}=\\bar\{\\sigma\}\\sqrt\{1\+\\bar\{d\}/\\bar\{\\sigma\}\}, shows that the right\-hand side of \([159](https://arxiv.org/html/2608.19914#A4.E159)\) equals exactlyN​\(σ¯\+d¯−σ¯\)2N\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\-\\sqrt\{\\bar\{\\sigma\}\}\)^\{2\}\. Rationalizing this difference via\(σ¯\+d¯−σ¯\)​\(σ¯\+d¯\+σ¯\)=d¯\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\-\\sqrt\{\\bar\{\\sigma\}\}\)\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\+\\sqrt\{\\bar\{\\sigma\}\}\)=\\bar\{d\}gives the equivalent closed form

N​\(σ¯\+d¯−σ¯\)2=N​d¯2\(σ¯\+d¯\+σ¯\)2\.\\displaystyle N\\bigl\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\-\\sqrt\{\\bar\{\\sigma\}\}\\bigr\)^\{2\}=\\frac\{N\\bar\{d\}^\{2\}\}\{\\bigl\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\+\\sqrt\{\\bar\{\\sigma\}\}\\bigr\)^\{2\}\}\.\(160\)Sinced¯≥0\\bar\{d\}\\geq 0, we haveσ¯≤σ¯\+d¯\\sqrt\{\\bar\{\\sigma\}\}\\leq\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}, so the denominator in \([160](https://arxiv.org/html/2608.19914#A4.E160)\) satisfies\(σ¯\+d¯\+σ¯\)2≤\(2​σ¯\+d¯\)2=4​\(σ¯\+d¯\)\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\+\\sqrt\{\\bar\{\\sigma\}\}\)^\{2\}\\leq\\bigl\(2\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\\bigr\)^\{2\}=4\(\\bar\{\\sigma\}\+\\bar\{d\}\); replacing the denominator by this upper bound can only increase the fraction, so

N​d¯2\(σ¯\+d¯\+σ¯\)2≥N​d¯24​\(σ¯\+d¯\)\.\\displaystyle\\frac\{N\\bar\{d\}^\{2\}\}\{\(\\sqrt\{\\bar\{\\sigma\}\+\\bar\{d\}\}\+\\sqrt\{\\bar\{\\sigma\}\}\)^\{2\}\}\\;\\geq\\;\\frac\{N\\bar\{d\}^\{2\}\}\{4\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}\.\(161\)Substitutingσ¯=tr⁡\(Σ\)/N\\bar\{\\sigma\}=\\mathrm\{tr\}\(\\Sigma\)/Nandd¯=tr⁡\(Δ𝝀\)/N\\bar\{d\}=\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)/Ninto \([161](https://arxiv.org/html/2608.19914#A4.E161)\) and simplifying,

N​d¯24​\(σ¯\+d¯\)=N⋅\(tr⁡\(Δ𝝀\)/N\)24​\(tr⁡\(Σ\)/N\+tr⁡\(Δ𝝀\)/N\)\\displaystyle\\frac\{N\\bar\{d\}^\{2\}\}\{4\(\\bar\{\\sigma\}\+\\bar\{d\}\)\}=\\frac\{N\\cdot\\bigl\(\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)/N\\bigr\)^\{2\}\}\{4\\bigl\(\\mathrm\{tr\}\(\\Sigma\)/N\+\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)/N\\bigr\)\}=\(tr⁡\(Δ𝝀\)\)24​\(tr⁡\(Σ\)\+tr⁡\(Δ𝝀\)\)=ℋ𝝀44​\(tr⁡\(Σ\)\+ℋ𝝀2\),\\displaystyle=\\frac\{\\bigl\(\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)\\bigr\)^\{2\}\}\{4\\bigl\(\\mathrm\{tr\}\(\\Sigma\)\+\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)\\bigr\)\}=\\frac\{\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{4\}\}\{4\\bigl\(\\mathrm\{tr\}\(\\Sigma\)\+\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\\bigr\)\},\(162\)where the last equality usestr⁡\(Δ𝝀\)=ℋ𝝀2\\mathrm\{tr\}\(\\Delta\_\{\\bm\{\\lambda\}\}\)=\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\. Chaining \([159](https://arxiv.org/html/2608.19914#A4.E159)\)–\([162](https://arxiv.org/html/2608.19914#A4.E162)\) establishes

B2​\(Σ,Σ\+Δ𝝀\)≥ℋ𝝀44​\(tr⁡\(Σ\)\+ℋ𝝀2\)\.\\displaystyle B^\{2\}\(\\Sigma,\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\}\)\\;\\geq\\;\\frac\{\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{4\}\}\{4\\bigl\(\\mathrm\{tr\}\(\\Sigma\)\+\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\\bigr\)\}\.\(163\)Finally, sincetr⁡\(Σ\)≤N​σmax​\(Σ\)\\mathrm\{tr\}\(\\Sigma\)\\leq N\\sigma\_\{\\max\}\(\\Sigma\)\(the trace is the sum ofNNeigenvalues each bounded by the largest one\), enlarging the denominator in \([163](https://arxiv.org/html/2608.19914#A4.E163)\) fromtr⁡\(Σ\)\+ℋ𝝀2\\mathrm\{tr\}\(\\Sigma\)\+\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}toN​σmax​\(Σ\)\+ℋ𝝀2N\\sigma\_\{\\max\}\(\\Sigma\)\+\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}can only decrease the fraction, so

B2​\(Σ\+Δ𝝀,Σ\)≥ℋ𝝀44​N​σmax​\(Σ\)\+4​ℋ𝝀2,\\displaystyle B^\{2\}\(\\Sigma\+\\Delta\_\{\\bm\{\\lambda\}\},\\Sigma\)\\;\\geq\\;\\frac\{\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{4\}\}\{4N\\sigma\_\{\\max\}\(\\Sigma\)\+4\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}^\{2\}\},\(164\)which is the second inequality in \([89](https://arxiv.org/html/2608.19914#S4.E89)\)\. The right\-hand side is strictly positive wheneverℋ𝝀\>0\\mathcal\{H\}\_\{\\bm\{\\lambda\}\}\>0, since its numerator is then strictly positive while its denominator is finite, completing the proof of the quantitative lower bound \([89](https://arxiv.org/html/2608.19914#S4.E89)\)\.

The contrasting almost\-sure convergenceW2​\(b^M∗,b𝝀∗\)→0W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\to 0asnmin→∞n\_\{\\min\}\\to\\inftyfollows from the finite\-sample concentration inequality of Theorem[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmtheorem1): that theorem bounds, for everyϵ\>0\\epsilon\>0, the probability thatW2​\(b^M∗,b𝝀∗\)W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)exceedsϵ\\epsilonby a term decaying exponentially innminn\_\{\\min\}, which is summable overnmin∈ℕn\_\{\\min\}\\in\\mathbb\{N\}; the Borel–Cantelli lemma then implies that the event\{W2\(b^M∗,b𝝀∗\)\>ϵ\}\\\{W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\>\\epsilon\\\}occurs for only finitely manynminn\_\{\\min\}almost surely, and lettingϵ↓0\\epsilon\\downarrow 0along a countable sequence yields almost\-sure convergence ofW2​\(b^M∗,b𝝀∗\)W\_\{2\}\(\\hat\{b\}^\{\*\}\_\{M\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)to zero\. In particular, the barycentric estimator is asymptotically consistent for the true barycenterb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}, whereas by \([89](https://arxiv.org/html/2608.19914#S4.E89)\) the pooled distributionℙ𝝀∗\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}remains bounded away fromb𝝀∗b^\{\*\}\_\{\\bm\{\\lambda\}\}by a fixed, sample\-size\-independent margin whenever the sources are heterogeneous: the mixing bias incurred by naive pooling is therefore irreducible, in contrast to the vanishing estimation error of the barycentric construction\.□\\square

## Appendix EProof of Proposition[IV\.1](https://arxiv.org/html/2608.19914#S4.Thmproposition1)

Proof\.Barycenter covariance in closed form\.Substitute the ansatzΣ=Q​diag​\(𝝈\)​QT\\Sigma=Q\\,\\mathrm\{diag\}\(\\bm\{\\sigma\}\)\\,Q^\{T\}into the fixed\-point equation \([23](https://arxiv.org/html/2608.19914#S3.E23)\)\. Since everyΣm\\Sigma\_\{m\}is diagonal in the same basisQQ, so isΣ1/2​Σm​Σ1/2\\Sigma^\{1/2\}\\Sigma\_\{m\}\\Sigma^\{1/2\}, withkk\-th diagonal entryσk​σm,k\\sigma\_\{k\}\\sigma\_\{m,k\}, whose square root isσk​σm,k\\sqrt\{\\sigma\_\{k\}\\sigma\_\{m,k\}\}; equation \([23](https://arxiv.org/html/2608.19914#S3.E23)\) therefore decouples intoNNindependent scalar equationsσk=∑mλm​σk​σm,k\\sigma\_\{k\}=\\sum\_\{m\}\\lambda\_\{m\}\\sqrt\{\\sigma\_\{k\}\\sigma\_\{m,k\}\}, i\.e\.σk=∑mλm​σm,k\\sqrt\{\\sigma\_\{k\}\}=\\sum\_\{m\}\\lambda\_\{m\}\\sqrt\{\\sigma\_\{m,k\}\}, which is uniquely solved byσ¯𝝀,k\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}of \([90](https://arxiv.org/html/2608.19914#S4.E90)\); uniqueness of the ambient fixed point \(Proposition[III\.1](https://arxiv.org/html/2608.19914#S3.Thmproposition1)\) then identifiesΣ¯𝝀=Q​diag​\(𝝈¯𝝀\)​QT\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}=Q\\,\\mathrm\{diag\}\(\\bar\{\\bm\{\\sigma\}\}\_\{\\bm\{\\lambda\}\}\)\\,Q^\{T\}as*the*barycentric covariance, givingb𝝀∗=𝒩⁡\(μ,Σ¯𝝀\)b^\{\*\}\_\{\\bm\{\\lambda\}\}=\\mathcal\{N\}\(\\mu,\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\.

Pooled covariance\.RepresentX∼ℙ𝝀∗X\\sim\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}hierarchically asX=μ\+Σι1/2​ZX=\\mu\+\\Sigma\_\{\\iota\}^\{1/2\}ZwithPr⁡\(ι=m\)=λm\\Pr\(\\iota=m\)=\\lambda\_\{m\}andZ∼𝒩⁡\(𝟎,𝐈N\)Z\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{N\}\)independent ofι\\iota\. Since the mean is common,Cov⁡\(X\)=𝔼ι​\[Σι1/2​𝔼​\[Z​ZT\]​Σι1/2\]=𝔼ι​\[Σι\]=Σ¯arith\\mathrm\{Cov\}\(X\)=\\mathbb\{E\}\_\{\\iota\}\\bigl\[\\Sigma\_\{\\iota\}^\{1/2\}\\,\\mathbb\{E\}\[ZZ^\{T\}\]\\,\\Sigma\_\{\\iota\}^\{1/2\}\\bigr\]=\\mathbb\{E\}\_\{\\iota\}\[\\Sigma\_\{\\iota\}\]=\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}, withkk\-th diagonal entryσ¯arith,k\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}in theQQ\-basis\.

Gelbrich reduction\.Applying the Gelbrich inequality \([153](https://arxiv.org/html/2608.19914#A4.E153)\) withα=ℙ𝝀∗\\alpha=\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\}\(meanμ\\mu, covarianceΣ¯arith\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\) andβ=b𝝀∗\\beta=b^\{\*\}\_\{\\bm\{\\lambda\}\}\(meanμ\\mu, covarianceΣ¯𝝀\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\), the mean term vanishes identically and

W22​\(ℙ𝝀∗,b𝝀∗\)≥B2​\(Σ¯arith,Σ¯𝝀\)\.\\displaystyle W\_\{2\}^\{2\}\(\\mathbb\{P\}^\{\*\}\_\{\\bm\{\\lambda\}\},b^\{\*\}\_\{\\bm\{\\lambda\}\}\)\\;\\geq\\;B^\{2\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\},\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\.\(165\)BecauseΣ¯arith\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}andΣ¯𝝀\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}are simultaneously diagonal inQQ\(Steps 1–2\), no twirling argument is needed: the fidelity term is exactlyF⁡\(Σ¯arith,Σ¯𝝀\)=∑kσ¯arith,k​σ¯𝝀,kF\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\},\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)=\\sum\_\{k\}\\sqrt\{\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\\,\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\}, and

B2​\(Σ¯arith,Σ¯𝝀\)=∑k=1N\(σ¯arith,k−σ¯𝝀,k\)2,\\displaystyle B^\{2\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\},\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)=\\sum\_\{k=1\}^\{N\}\\Bigl\(\\sqrt\{\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\}\-\\sqrt\{\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\}\\Bigr\)^\{\\\!2\},\(166\)which is \([91](https://arxiv.org/html/2608.19914#S4.E91)\)\. By Jensen’s inequality applied to the strictly concave mapt↦tt\\mapsto\\sqrt\{t\},σ¯arith,k≥σ¯𝝀,k\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\\geq\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}for everykk, with equality iffσm,k\\sigma\_\{m,k\}is constant acrossmm\(for thosemmwithλm\>0\\lambda\_\{m\}\>0\); hence each summand, and thus the total, is strictly positive as soon asΣi≠Σj\\Sigma\_\{i\}\\neq\\Sigma\_\{j\}for some pair withλi,λj\>0\\lambda\_\{i\},\\lambda\_\{j\}\>0\.

Closed\-form floor\.Writeak=σ¯arith,k≥bk=σ¯𝝀,k≥0a\_\{k\}=\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\\geq b\_\{k\}=\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\\geq 0\. Rationalizing,\(ak−bk\)2=\(ak−bk\)2/\(ak\+bk\)2≥\(ak−bk\)2/\(4​ak\)≥\(ak−bk\)2/\(4​σmax​\(Σ¯arith\)\)\(\\sqrt\{a\_\{k\}\}\-\\sqrt\{b\_\{k\}\}\)^\{2\}=\(a\_\{k\}\-b\_\{k\}\)^\{2\}/\(\\sqrt\{a\_\{k\}\}\+\\sqrt\{b\_\{k\}\}\)^\{2\}\\geq\(a\_\{k\}\-b\_\{k\}\)^\{2\}/\(4a\_\{k\}\)\\geq\(a\_\{k\}\-b\_\{k\}\)^\{2\}/\\bigl\(4\\sigma\_\{\\max\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\)\\bigr\), usingbk≤ak\\sqrt\{b\_\{k\}\}\\leq\\sqrt\{a\_\{k\}\}andak≤σmax​\(Σ¯arith\)a\_\{k\}\\leq\\sigma\_\{\\max\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\)\. Summing overkkand applying Cauchy–Schwarz,∑k\(ak−bk\)2≥1N​\(∑k\(ak−bk\)\)2=1N​Δtr2\\sum\_\{k\}\(a\_\{k\}\-b\_\{k\}\)^\{2\}\\geq\\frac\{1\}\{N\}\\bigl\(\\sum\_\{k\}\(a\_\{k\}\-b\_\{k\}\)\\bigr\)^\{2\}=\\frac\{1\}\{N\}\\Delta\_\{\\mathrm\{tr\}\}^\{2\}, whereΔtr:=tr⁡\(Σ¯arith\)−tr⁡\(Σ¯𝝀\)\\Delta\_\{\\mathrm\{tr\}\}:=\\mathrm\{tr\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\)\-\\mathrm\{tr\}\(\\bar\{\\Sigma\}\_\{\\bm\{\\lambda\}\}\)\. Finally, for anyXXtaking valuexmx\_\{m\}with probabilityλm\\lambda\_\{m\}, the elementary pairwise\-variance identityVar𝝀​\(X\)=12​∑i,jλi​λj​\(xi−xj\)2≥λi​λj​\(xi−xj\)2\\mathrm\{Var\}\_\{\\bm\{\\lambda\}\}\(X\)=\\tfrac\{1\}\{2\}\\sum\_\{i,j\}\\lambda\_\{i\}\\lambda\_\{j\}\(x\_\{i\}\-x\_\{j\}\)^\{2\}\\geq\\lambda\_\{i\}\\lambda\_\{j\}\(x\_\{i\}\-x\_\{j\}\)^\{2\}\(for any fixed pairi≠ji\\neq j, dropping the remaining nonnegative terms\), applied coordinatewise withX=σ⋅,kX=\\sqrt\{\\sigma\_\{\\cdot,k\}\}and summed overkk, givesΔtr=∑kVar𝝀​\(σ⋅,k\)≥λi​λj​∑k\(σi,k−σj,k\)2=λi​λj​B2​\(Σi,Σj\)\\Delta\_\{\\mathrm\{tr\}\}=\\sum\_\{k\}\\mathrm\{Var\}\_\{\\bm\{\\lambda\}\}\(\\sqrt\{\\sigma\_\{\\cdot,k\}\}\)\\geq\\lambda\_\{i\}\\lambda\_\{j\}\\sum\_\{k\}\(\\sqrt\{\\sigma\_\{i,k\}\}\-\\sqrt\{\\sigma\_\{j,k\}\}\)^\{2\}=\\lambda\_\{i\}\\lambda\_\{j\}\\,B^\{2\}\(\\Sigma\_\{i\},\\Sigma\_\{j\}\), where the last equality is the same diagonal fidelity computation as in Step 3, applied to the commuting pair\(Σi,Σj\)\(\\Sigma\_\{i\},\\Sigma\_\{j\}\)\. Chaining these four inequalities and maximizing over pairsi≠ji\\neq jyields

∑k=1N\(σ¯arith,k−σ¯𝝀,k\)2≥maxi≠j⁡λi2​λj2​B4​\(Σi,Σj\)4​N​σmax​\(Σ¯arith\),\\displaystyle\\sum\_\{k=1\}^\{N\}\\Bigl\(\\sqrt\{\\bar\{\\sigma\}\_\{\\mathrm\{arith\},k\}\}\-\\sqrt\{\\bar\{\\sigma\}\_\{\\bm\{\\lambda\},k\}\}\\Bigr\)^\{\\\!2\}\\;\\geq\\;\\max\_\{i\\neq j\}\\,\\frac\{\\lambda\_\{i\}^\{2\}\\lambda\_\{j\}^\{2\}\\,B^\{4\}\(\\Sigma\_\{i\},\\Sigma\_\{j\}\)\}\{4N\\,\\sigma\_\{\\max\}\(\\bar\{\\Sigma\}\_\{\\mathrm\{arith\}\}\)\},\(167\)which is \([92](https://arxiv.org/html/2608.19914#S4.E92)\), strictly positive wheneverΣi≠Σj\\Sigma\_\{i\}\\neq\\Sigma\_\{j\}for the maximizing pair\.□\\square

## References

- \[1\]Xiao\-Fei Zhang, Le Ou\-Yang, Ting Yan, Xiaohua Tony Hu, and Hong Yan\.A joint graphical model for inferring gene networks across multiple subpopulations and data types\.IEEE Transactions on Cybernetics, 51\(2\):1043–1055, 2019\.
- \[2\]Huijun Gao, Dongxu Lei, and Songlin Zhuang\.Bayesian modeling of gene regulatory networks in colorectal cancer organoids\.IEEE Transactions on Cybernetics, 2025\.
- \[3\]Yuqing Qian, Yizheng Wang, Junkai Liu, Quan Zou, Yijie Ding, Xiaoyi Guo, and Weiping Ding\.A survey on multi\-view fusion for predicting links in biomedical bipartite networks: Methods and applications\.Information Fusion, 117:102894, 2025\.
- \[4\]Geethu Joseph, Buddhika Nettasinghe, Vikram Krishnamurthy, and Pramod K Varshney\.Controllability of network opinion in Erdos–Rényi graphs using sparse control inputs\.SIAM Journal on Control and Optimization, 59\(3\):2321–2345, 2021\.
- \[5\]Sang\-Woong Lee, Jawad Tanveer, Amir Masoud Rahmani, Hamid Alinejad\-Rokny, Parisa Khoshvaght, Gholamreza Zare, Pegah Malekpour Alamdari, and Mehdi Hosseinzadeh\.SFGCN: Synergetic fusion\-based graph convolutional networks approach for link prediction in social networks\.Information Fusion, 114:102684, 2025\.
- \[6\]Lijia Ma, Jianqiang Li, Qiuzhen Lin, Maoguo Gong, Carlos A Coello Coello, and Zhong Ming\.Reliable link inference for network data with community structures\.IEEE Transactions on Cybernetics, 49\(9\):3347–3361, 2018\.
- \[7\]David I Shuman, Sunil K Narang, Pascal Frossard, Antonio Ortega, and Pierre Vandergheynst\.The emerging field of signal processing on graphs: Extending high\-dimensional data analysis to networks and other irregular domains\.IEEE Signal Processing Magazine, 30\(3\):83–98, 2013\.
- \[8\]Antonio Ortega, Pascal Frossard, Jelena Kovačević, José MF Moura, and Pierre Vandergheynst\.Graph signal processing: Overview, challenges, and applications\.Proceedings of the IEEE, 106\(5\):808–828, 2018\.
- \[9\]Wenling Li, Xiaoyan Fu, Bin Zhang, and Yang Liu\.Unscented Kalman filter of graph signals\.Automatica, 148:110796, 2023\.
- \[10\]Xiaowen Dong, Dorina Thanou, Pascal Frossard, and Pierre Vandergheynst\.Learning Laplacian matrix in smooth graph signal representations\.IEEE Transactions on Signal Processing, 64\(23\):6160–6173, 2016\.
- \[11\]Vassilis Kalofolias\.How to learn a graph from smooth signals\.InProceedings of the International Conference on Artificial Intelligence and Statistics \(AISTATS\), pages 920–929\. PMLR, 2016\.
- \[12\]Hilmi E Egilmez, Eduardo Pavez, and Antonio Ortega\.Graph learning from data under Laplacian and structural constraints\.IEEE Journal of Selected Topics in Signal Processing, 11\(6\):825–841, 2017\.
- \[13\]Sandeep Kumar, Jiaxi Ying, José Vinícius de M Cardoso, and Daniel P Palomar\.A unified framework for structured graph learning via spectral constraints\.Journal of Machine Learning Research, 21\(22\):1–60, 2020\.
- \[14\]Stefania Sardellitti, Sergio Barbarossa, and Paolo Di Lorenzo\.Graph topology inference based on transform learning\.InProceedings of the IEEE Global Conference on Signal and Information Processing \(GlobalSIP\), pages 356–360\. IEEE, 2016\.
- \[15\]Seyed Saman Saboksayr and Gonzalo Mateos\.Accelerated graph learning from smooth signals\.IEEE Signal Processing Letters, 28:2192–2196, 2021\.
- \[16\]Antonio G Marques, Santiago Segarra, Geert Leus, and Alejandro Ribeiro\.Stationary graph processes and spectral estimation\.IEEE Transactions on Signal Processing, 65\(22\):5911–5926, 2017\.
- \[17\]Dorina Thanou, Xiaowen Dong, Daniel Kressner, and Pascal Frossard\.Learning heat diffusion graphs\.IEEE Transactions on Signal and Information Processing over Networks, 3\(3\):484–499, 2017\.
- \[18\]Santiago Segarra, Antonio G Marques, Gonzalo Mateos, and Alejandro Ribeiro\.Network topology inference from spectral templates\.IEEE Transactions on Signal and Information Processing over Networks, 3\(3\):467–483, 2017\.
- \[19\]Yanli Yuan, Kun Guo, Zehui Xiong, and Tony QS Quek\.Joint network topology inference via structural fusion regularization\.IEEE Transactions on Knowledge and Data Engineering, 35\(10\):10351–10364, 2023\.
- \[20\]Jerome Friedman, Trevor Hastie, and Robert Tibshirani\.Sparse inverse covariance estimation with the graphical LASSO\.Biostatistics, 9\(3\):432–441, 2008\.
- \[21\]John C Duchi and Hongseok Namkoong\.Learning models with uniform performance via distributionally robust optimization\.The Annals of Statistics, 49\(3\):1378–1406, 2021\.
- \[22\]Erick Delage and Yinyu Ye\.Distributionally robust optimization under moment uncertainty with application to data\-driven problems\.Operations Research, 58\(3\):595–612, 2010\.
- \[23\]Wolfram Wiesemann, Daniel Kuhn, and Melvyn Sim\.Distributionally robust convex optimization\.Operations Research, 62\(6\):1358–1376, 2014\.
- \[24\]Peyman Mohajerin Esfahani and Daniel Kuhn\.Data\-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations\.Mathematical Programming, 171\(1\):115–166, 2018\.
- \[25\]Jose Blanchet and Karthyek Murthy\.Quantifying distributional model risk via optimal transport\.Mathematics of Operations Research, 44\(2\):565–600, 2019\.
- \[26\]Rui Gao and Anton Kleywegt\.Distributionally robust stochastic optimization with Wasserstein distance\.Mathematics of Operations Research, 48\(2\):603–655, 2023\.
- \[27\]Soroosh Shafieezadeh Abadeh, Peyman M Mohajerin Esfahani, and Daniel Kuhn\.Distributionally robust logistic regression\.InProceedings of the Advances in Neural Information Processing Systems \(NeurIPS\), volume 28, 2015\.
- \[28\]Aman Sinha, Hongseok Namkoong, Riccardo Volpi, and John Duchi\.Certifying some distributional robustness with principled adversarial training\.InProceedings of the International Conference on Learning Representations \(ICLR\), 2018\.
- \[29\]Xiang Zhang, Yinfei Xu, Mingjie Shao, and Yonina C Eldar\.Wasserstein distributionally robust graph learning via algorithm unrolling\.IEEE Transactions on Signal Processing, 73:676–690, 2025\.
- \[30\]Adriana Di Martino, Chao\-Gan Yan, Qingyang Li, Erin Denio, Francisco X Castellanos, Kaat Alaerts, Jeffrey S Anderson, Michal Assaf, Susan Y Bookheimer, Mirella Dapretto, et al\.The autism brain imaging data exchange: Towards a large\-scale evaluation of the intrinsic brain architecture in autism\.Molecular Psychiatry, 19\(6\):659–667, 2014\.
- \[31\]Ed Bullmore and Olaf Sporns\.Complex brain networks: Graph theoretical analysis of structural and functional systems\.Nature Reviews Neuroscience, 10\(3\):186–198, 2009\.
- \[32\]Sinno Jialin Pan and Qiang Yang\.A survey on transfer learning\.IEEE Transactions on Knowledge and Data Engineering, 22\(10\):1345–1359, 2009\.
- \[33\]Shai Ben\-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan\.A theory of learning from different domains\.Machine Learning, 79\(1\):151–175, 2010\.
- \[34\]Mehryar Mohri, Gary Sivek, and Ananda Theertha Suresh\.Agnostic federated learning\.InProceedings of the International Conference on Machine Learning \(ICML\), pages 4615–4625\. PMLR, 2019\.
- \[35\]Cédric Villani et al\.Optimal transport: Old and new, volume 338\.Springer, 2009\.
- \[36\]Martial Agueh and Guillaume Carlier\.Barycenters in the Wasserstein space\.SIAM Journal on Mathematical Analysis, 43\(2\):904–924, 2011\.
- \[37\]Marco Cuturi and Arnaud Doucet\.Fast computation of wasserstein barycenters\.InProceedings of the International Conference on Machine Learning \(ICML\), pages 685–693\. PMLR, 2014\.
- \[38\]Pedro C Álvarez\-Esteban, E Del Barrio, JA Cuesta\-Albertos, and C Matrán\.A fixed\-point approach to barycenters in Wasserstein space\.Journal of Mathematical Analysis and Applications, 441\(2\):744–762, 2016\.
- \[39\]Matthias Gelbrich\.On a formula for the L2 Wasserstein metric between measures on Euclidean and Hilbert spaces\.Mathematische Nachrichten, 147\(1\):185–203, 1990\.
- \[40\]Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamonjy\.Optimal transport for domain adaptation\.IEEE Transactions on Pattern Analysis and Machine Intelligence, 39\(9\):1853–1865, 2016\.
- \[41\]Vishal Monga, Yuelong Li, and Yonina C Eldar\.Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing\.IEEE Signal Processing Magazine, 38\(2\):18–44, 2021\.
- \[42\]Yan Yang, Jian Sun, Huibin Li, and Zongben Xu\.Deep ADMM\-Net for compressive sensing MRI\.InProceedings of the Advances in Neural Information Processing Systems \(NeurIPS\), volume 29, 2016\.
- \[43\]Harsh Shrivastava, Xinshi Chen, Binghong Chen, Guanghui Lan, Srinivas Aluru, Han Liu, and Le Song\.GLAD: Learning sparse graph recovery\.InProceedings of the International Conference on Learning Representations \(ICLR\), 2020\.
- \[44\]Siheng Chen, Yonina C Eldar, and Lingxiao Zhao\.Graph unrolling networks: Interpretable neural networks for graph signal denoising\.IEEE Transactions on Signal Processing, 69:3699–3713, 2021\.
- \[45\]Elliott H Lieb\.Convex trace functions and the Wigner\-Yanase\-Dyson conjecture\.Les Rencontres Physiciens\-Mathématiciens de Strasbourg\-RCP25, 19:0–35, 1973\.
- \[46\]Antonio Anna Mele\.Introduction to Haar measure tools in quantum information: A beginner’s tutorial\.Quantum, 8:1340, 2024\.
- \[47\]Daniel Kuhn, Peyman Mohajerin Esfahani, Viet Anh Nguyen, and Soroosh Shafieezadeh\-Abadeh\.Wasserstein distributionally robust optimization: Theory and applications in machine learning\.InOperations Research & Management Science in the Age of Analytics, pages 130–166\. Informs, 2019\.
- \[48\]Gabriel Peyré and Marco Cuturi\.Computational optimal transport: With applications to data science\.Now Foundations and Trends, 2019\.
- \[49\]Sinho Chewi, Tyler Maunu, Philippe Rigollet, and Austin J Stromme\.Gradient descent algorithms for Bures\-Wasserstein barycenters\.InProceedings of the Conference on Learning Theory \(CoLT\), pages 1276–1304\. PMLR, 2020\.
- \[50\]Ralph Tyrell Rockafellar\.Convex Analysis\.Princeton University Press, 2015\.
- \[51\]Bingsheng He and Xiaoming Yuan\.On the O\(1/n\) convergence rate of the Douglas–Rachford alternating direction method\.SIAM Journal on Numerical Analysis, 50\(2\):700–709, 2012\.
- \[52\]Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein\.Distributed optimization and statistical learning via the alternating direction method of multipliers\.Foundations and Trends® in Machine Learning, 3\(1\):1–122, 2011\.
- \[53\]Licheng Zhao, Yiwei Wang, Sandeep Kumar, and Daniel P Palomar\.Optimization algorithms for graph Laplacian estimation via ADMM and MM\.IEEE Transactions on Signal Processing, 67\(16\):4231–4244, 2019\.
- \[54\]Eugene Belilovsky, Kyle Kastner, Gaël Varoquaux, and Matthew B Blaschko\.Learning to discover sparse graphical models\.InProceedings of the International Conference on Machine Learning \(ICML\), pages 440–448\. PMLR, 2017\.
- \[55\]Xingyue Pu, Tianyue Cao, Xiaoyun Zhang, Xiaowen Dong, and Siheng Chen\.Learning to learn graph topologies\.InProceedings of the Advances in Neural Information Processing Systems \(NeurIPS\), volume 34, pages 4249–4262, 2021\.
- \[56\]Xiaolu Wang, Yuen\-Man Pun, and Anthony Man\-Cho So\.Distributionally robust graph learning from smooth signals under moment uncertainty\.IEEE Transactions on Signal Processing, 70:6216–6231, 2023\.