BREAD: Baseline-Referenced Explanations for Anomaly Diagnosis

# Baseline-Referenced Explanations for Anomaly Diagnosis
Source: [https://arxiv.org/html/2608.10587](https://arxiv.org/html/2608.10587)
Jiaqi QiuaJiaqi Qiu is with Department of Business Analytics, Amsterdam Business School, University of Amsterdam, Amsterdam, The Netherlands \(e\-mail: j\.qiu@uva\.nl\)\.Jannis Kurtzaand Inez M\. Zwetsloota aDepartment of Business AnalyticsAmsterdam Business School University of AmsterdamAmsterdamThe Netherlands

###### Abstract

Artificial Intelligence \(AI\)\-based prospective anomaly detection methods are increasingly deployed in high\-dimensional and nonlinear settings\. Among these approaches, AI\-based statistical process monitoring \(SPM\) is widely used, providing a structured framework for prospective monitoring\. Once an anomaly is detected, a diagnosis method is needed to identify the features driving the flagged observation away from normal behaviour\. Traditional SPM diagnosis methods are typically designed for specific detection models and cannot be directly applied to AI\-based methods\. Model\-agnostic explainable AI \(XAI\) offers a general framework for feature relevance explanation\. However, existing methods suffer from scalability limitations or assign relevance to noise features, reducing diagnosis accuracy\. We propose a scalable, baseline\-referenced diagnosis method that uses both the anomalous observation and normal baseline information\. We provide mathematical guarantees that under a mean\-shift anomaly setting, the proposed method achieves higher faithfulness in detecting the features causing the anomaly compared to LIME\. Simulation studies and a real\-world case study validate the effectiveness of the proposed method and show that it generates more faithful and accurate diagnosis results for AI\-based prospective anomaly detection methods\.

## 1Introduction

Prospective anomaly detection in sequential data is essential for identifying abnormal process behaviour before it leads to system failures\. Statistical Process Monitoring \(SPM\) provides a framework for this problem using control charts\(Zwetslootet al\.,[2023](https://arxiv.org/html/2608.10587#bib.bib35)\)\. It has been widely used in industry\(Colosimoet al\.,[2024](https://arxiv.org/html/2608.10587#bib.bib76)\), services\(Tsunget al\.,[2008](https://arxiv.org/html/2608.10587#bib.bib163)\), and healthcare\(Woodall,[2006](https://arxiv.org/html/2608.10587#bib.bib162)\)\. More recently, AI\-based SPM methods have been developed for complex industrial systems\(Leeet al\.,[2019](https://arxiv.org/html/2608.10587#bib.bib75); Cacciarelli and Kulahci,[2022](https://arxiv.org/html/2608.10587#bib.bib70); Magedet al\.,[2024](https://arxiv.org/html/2608.10587#bib.bib74); Qiuet al\.,[2025](https://arxiv.org/html/2608.10587#bib.bib161)\)\. However, in practice, detecting an anomaly is only the first step\. Practitioners also need diagnosis methods that explain how much each feature contributes to the detected signal, so that appropriate actions can be taken\.

Explainable AI \(XAI\) provides tools for post\-hoc explanation of model behaviourAdadi and Berrada \([2018](https://arxiv.org/html/2608.10587#bib.bib151)\); Arrietaet al\.\([2020](https://arxiv.org/html/2608.10587#bib.bib85)\); Dwivediet al\.\([2023](https://arxiv.org/html/2608.10587#bib.bib146)\)\. In prospective anomaly detection, especially during online monitoring, the diagnosis task is to quantify how much each feature contributed to a detected signal\. This requires methods that are computationally efficient and local\. Moreover, explanations need to be model\-agnostic, as different black\-box models may be used for monitoring\. These requirements make LIMERibeiroet al\.\([2016](https://arxiv.org/html/2608.10587#bib.bib93)\)an appealing choice\.

However, existing LIME techniques are not optimised for signal diagnosis\. Prior work has noted general limitations of LIME that are essential to signal diagnosis, including instability and limited fidelityZafar and Khan \([2019](https://arxiv.org/html/2608.10587#bib.bib148)\); Zhanget al\.\([2019a](https://arxiv.org/html/2608.10587#bib.bib147)\); Knabet al\.\([2025](https://arxiv.org/html/2608.10587#bib.bib181)\)\. More fundamentally, for the signal diagnosis setting, these methods explain a model output in the local neighbourhood of the queried instance\. For anomalies lying far from the usual behaviour, local perturbations may remain largely within abnormal regions of the feature space\. Consequently, these explanations are less informative for signal diagnosis, as they only describe the model behaviour locally around the anomaly and explain abnormal variation\. In contrast, the goal of signal explanation is to explain how the variables of an identified signal drive the process away from normal operation and into an anomalous state\. Thus, signal diagnosis requires explaining a local instance from a global reference perspective, presenting the normal behaviour\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/x1.png)Figure 1:Real case study from escalator monitoring using an AI\-based prospective anomaly detection method\. \(a\) Minute\-wise escalator energy usage data from multiple escalators\. \(b\) Observed unexpected shutdowns\. \(c\) VAE\-LSTM\-basedT2T^\{2\}control chart\(Magedet al\.,[2024](https://arxiv.org/html/2608.10587#bib.bib74)\), where anomalies align with a period of observed unexpected escalator shutdowns\. \(d\) Ordered LIME diagnosis\. \(e\) Ordered diagnosis by BREAD\. \(f\) Comparison of results, showing that the diagnosis by BREAD matches the ground truth\.To highlight the limitations of LIME, we introduce a real monitoring scenario\. We consider an escalator monitoring system\(Zwetslootet al\.,[2025](https://arxiv.org/html/2608.10587#bib.bib32)\), as shown in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(a\)\. Monitoring the energy data in real time allows engineers to assess the operational status of the escalator system\. A possible anomaly, for instance, an unexpected shutdown period depicted in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(b\), is expected to be detected by the monitoring model\. Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(c\) shows that the applied monitoring method detects anomalies during the shutdown period\. However, only detection is not enough for practitioners, as they also need to identify the escalators that caused the whole system to become anomalous\. The ordered diagnosis results are illustrated in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(d–e\)\. As depicted in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(f\), LIME fails to assign high relevance to the observed shutdown escalators\. Such a misdiagnosis may direct practitioners to take inappropriate actions\. Thus, this paper addresses the challenge of identifying the variables that actually drive the system to abnormal conditions under real\-time computational constraints\.

To address this, we propose a scalable model\-agnostic explanation method, Baseline\-Referenced Explanations for Anomaly Diagnosis \(BREAD\), for AI\-based prospective anomaly detection that incorporates a reference point representing the in\-control baseline information\. In essence, BREAD modifies the LIME algorithm by introducing a novel sampling scheme and a sampling\-aligned weighting kernel\. The main idea is to fit a regression model on the weighted sampled data around both the anomalous observation and a reference normal point\. By introducing the baseline information into the surrogate construction, the resulting explanations become sparser, concentrating relevance on anomaly related variables while assigning near\-zero relevance to irrelevant features\. In prospective anomaly detection, anomalous events are often caused by a limited subset of variables\. This concentration can improve the faithfulness of the explanations because such explanations are better aligned with the variables that actually drive the anomalous signal\. Meanwhile, the computational cost of BREAD is measured by the number of synthetic samples and the cost of the black\-box model, and thus, BREAD remains computationally efficient\.

For our method, we provide a theoretical analysis showing that, under certain assumptions that are both realistic and compatible with common monitoring settings, the proposed approach achieves improved faithfulness for signal diagnosis relative to LIME\. The proposed method is evaluated both in simulation studies and a real case study\. The simulations show the faithfulness, robustness of BREAD, and stability against sampling noise\. The case study revisits the motivating example in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)from a deployment perspective, providing an empirical evaluation of diagnosis alignment with observed shutdown events and computational efficiency\.

Thus, we summarise our main contributions as follows:

- •We propose BREAD, a scalable, model\-agnostic explanation method that incorporates baseline normal information through a novel sampling scheme and a modified weighting kernel\.
- •We provide theoretical proofs and analysis showing that, given an identified anomaly, BREAD yields asymptotically sparse explanations, which in turn improve alignment with the ground truth under the stated assumptions\.
- •We validate the theoretical results through simulation studies and a real\-world case study, demonstrating the effectiveness of the proposed method for black\-box\-based prospective anomaly detection\.

The remainder of the paper is organized as follows\. Section 2 reviews related work\. Section 3 formalizes the problem, presents BREAD, and establishes its theoretical properties\. Section 4 evaluates BREAD through simulation studies and the real\-world case study\. Section 5 concludes the paper\.

## 2Related Work

A major family of methods for prospective anomaly detection in sequential process data is statistical process monitoring \(SPM\), which uses control charts to detect departures from normal process behaviourZwetslootet al\.\([2023](https://arxiv.org/html/2608.10587#bib.bib35)\)\. Early work focused on decomposing alarms from traditional control charts, especially Hotelling’sT2T^\{2\}chart under Gaussian assumptions, to identify the variables most responsible for an out\-of\-control \(OOC\) signalMasonet al\.\([1995](https://arxiv.org/html/2608.10587#bib.bib36)\); Liet al\.\([2008](https://arxiv.org/html/2608.10587#bib.bib51)\); Kimet al\.\([2016](https://arxiv.org/html/2608.10587#bib.bib52)\)\. Further diagnosis methods have been developed beyond the Gaussian setting, for example, multivariate binomial processesHou \([2023](https://arxiv.org/html/2608.10587#bib.bib180)\)\. These methods are effective in finding a subset of variables driving a signal, but are tailored to specific chart structures, monitoring statistics, and distributional assumptions\. By contrast, AI\-based SPM methods typically apply complex embeddings to capture underlying process behaviour and extract representations from a latent space\(Yuet al\.,[2019](https://arxiv.org/html/2608.10587#bib.bib73); Yu and Zhao,[2019](https://arxiv.org/html/2608.10587#bib.bib72); Zhanget al\.,[2019b](https://arxiv.org/html/2608.10587#bib.bib71); Leeet al\.,[2019](https://arxiv.org/html/2608.10587#bib.bib75); Cacciarelli and Kulahci,[2022](https://arxiv.org/html/2608.10587#bib.bib70); Magedet al\.,[2024](https://arxiv.org/html/2608.10587#bib.bib74)\)\. Such methods have been shown to capture latent behaviour for process monitoring betterCacciarelli and Kulahci \([2023](https://arxiv.org/html/2608.10587#bib.bib69)\)\. However, the latent\-space representations are often difficult to translate into actionable information in the original space\.

This motivates the use of explainable AI \(XAI\) methods that quantify feature relevance for black\-box models\. A large body of work has proposed feature\-relevance methods such as saliency mapsSimonyanet al\.\([2013](https://arxiv.org/html/2608.10587#bib.bib144)\), Layer\-wise Relevance Propagation \(LRP\)Bachet al\.\([2015](https://arxiv.org/html/2608.10587#bib.bib153)\), Grad\-CAMSelvarajuet al\.\([2016](https://arxiv.org/html/2608.10587#bib.bib152)\), SHAP\-based approachesLundberg and Lee \([2017](https://arxiv.org/html/2608.10587#bib.bib90)\), and Integrated Gradients \(IG\)Sundararajanet al\.\([2017](https://arxiv.org/html/2608.10587#bib.bib106)\)\. In SPM, prior work has applied feature\-relevance XAI methods, including LRPAgarwalet al\.\([2021](https://arxiv.org/html/2608.10587#bib.bib142)\), IGSipple \([2020](https://arxiv.org/html/2608.10587#bib.bib175)\); Cacciarelli and Kulahci \([2022](https://arxiv.org/html/2608.10587#bib.bib70)\); Bhakteet al\.\([2022](https://arxiv.org/html/2608.10587#bib.bib129)\), SHAPKimet al\.\([2021](https://arxiv.org/html/2608.10587#bib.bib173)\); Hwang and Lee \([2021](https://arxiv.org/html/2608.10587#bib.bib172)\); Choiet al\.\([2022](https://arxiv.org/html/2608.10587#bib.bib170)\); Baek and Kim \([2023](https://arxiv.org/html/2608.10587#bib.bib115)\); Janget al\.\([2023](https://arxiv.org/html/2608.10587#bib.bib114)\); Szelążeket al\.\([2024](https://arxiv.org/html/2608.10587#bib.bib143)\)\. However, structured performance evaluation of these methods in SPM remains limited\. In addition, these methods are either computationally expensive or model\-specific\. For model\-specific methods, assumptions about gradients, architecture, or internal model structure limit their applicabilityAdadi and Berrada \([2018](https://arxiv.org/html/2608.10587#bib.bib151)\)\. In SPM settings, where the monitoring model may vary, model\-agnostic explanation methods are therefore especially attractive\.

Thus, among these feature\-relevance methods, LIMERibeiroet al\.\([2016](https://arxiv.org/html/2608.10587#bib.bib93)\)is particularly appealing because it is local, model\-agnostic, and computationally efficientMishraet al\.\([2017](https://arxiv.org/html/2608.10587#bib.bib150)\); Raboldet al\.\([2018](https://arxiv.org/html/2608.10587#bib.bib149)\); Zafar and Khan \([2019](https://arxiv.org/html/2608.10587#bib.bib148)\); Ottoet al\.\([2025](https://arxiv.org/html/2608.10587#bib.bib145)\); Jiaet al\.\([2025](https://arxiv.org/html/2608.10587#bib.bib182)\), and has been applied to alarm explanation in SPM byBhakteet al\.\([2023](https://arxiv.org/html/2608.10587#bib.bib113)\)\. However, prior work has also highlighted important limitations of LIME, including limited fidelity, instability due to stochastic neighbourhood sampling, and difficulties in domain\-specific adaptationZafar and Khan \([2019](https://arxiv.org/html/2608.10587#bib.bib148)\); Zhanget al\.\([2019a](https://arxiv.org/html/2608.10587#bib.bib147)\); Zhouet al\.\([2021](https://arxiv.org/html/2608.10587#bib.bib183)\); Tanet al\.\([2023](https://arxiv.org/html/2608.10587#bib.bib184)\); Knabet al\.\([2025](https://arxiv.org/html/2608.10587#bib.bib181)\)\. These issues are particularly critical in online SPM, where explanations must be both reliable and computationally efficient\. Moreover, the diagnosis objective is not just to explain the model output at a single point, but to identify the variables that drive the process away from the in\-control region\.

## 3Proposed Method

### 3\.1Problem Description

Letf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\rightarrow\\mathbb\{R\}denote an AI\-based anomaly detection model that maps an observation𝒕∈ℝd\\bm\{t\}\\in\\mathbb\{R\}^\{d\}to an anomaly scores∈ℝs\\in\\mathbb\{R\}, i\.e\.,s=f​\(𝒕\)s=f\(\\bm\{t\}\)\. A decision functionθs:ℝ→\{0,1\}\\theta\_\{s\}:\\mathbb\{R\}\\rightarrow\\\{0,1\\\}maps the anomaly scoressto a binary label, where0denotes normal behaviour and11denotes anomalous behaviour\. The corresponding normal region is defined as𝒜=\{𝒕∈ℝd:θs​\(f​\(𝒕\)\)=0\}\.\\mathcal\{A\}=\\\{\\bm\{t\}\\in\\mathbb\{R\}^\{d\}:\\theta\_\{s\}\(f\(\\bm\{t\}\)\)=0\\\}\.For example, in the SPM context,ssis often assumed to follow a certain distribution and a control limit is then set as a guideline of the distribution\.

During online monitoring, new data are denoted by𝒕j∈ℝd,j\>N\\bm\{t\}\_\{j\}\\in\\mathbb\{R\}^\{d\},j\>N\. An observation at timej∗j^\{\*\}is identified as an anomaly if𝒕j∗∉𝒜\\bm\{t\}\_\{j^\{\*\}\}\\notin\\mathcal\{A\}\. When an anomaly is detected, the diagnosis method is expected to provide an explanation indicating how much each feature contributes to the signal\.

### 3\.2Framework

Recall that our proposed method BREAD approximates the behaviour offfwith respect to𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}and the reference normal point via a linear surrogate model whose coefficients𝜷\\bm\{\\beta\}quantify the feature contribution from the reference normal point to the anomalous observation\.

In practice, we estimate𝜷\\bm\{\\beta\}by fitting a weighted ridge regression surrogate to synthetic samples drawn around both𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}and𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}, where𝒕ref∈𝒜\\bm\{t\}\_\{\\mathrm\{ref\}\}\\in\\mathcal\{A\}denotes the reference normal point that represents the normal operating conditions\. It is necessary to verify if𝒕ref∈𝒜\\bm\{t\}\_\{\\mathrm\{ref\}\}\\in\\mathcal\{A\}when choosing the mean of the normal training data as𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}, as it is not guaranteed for nonlinear or non\-convex normal regions\. After estimating the coefficients𝜷^\\hat\{\\bm\{\\beta\}\}, we report the topKKfeatures according to\|β^i\|\|\\hat\{\{\\beta\}\}\_\{i\}\|as the explanation for the detected signal\.

Synthetic samples are generated by adding random perturbations from a distribution, e\.g\.,𝒩d​\(0,Σ\)\\mathcal\{N\}\_\{d\}\(0,\\Sigma\), to both𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}and𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}\. Through this generation procedure, we obtain the sampled set𝒵=\{𝒛i\}i=1n\\mathcal\{Z\}=\\\{\\bm\{z\}\_\{i\}\\\}^\{n\}\_\{i=1\}withnnsampled points and the corresponding matrixZ∈ℝn×dZ\\in\\mathbb\{R\}^\{n\\times d\}, whoseii\-th row is𝒛i⊺\\bm\{z\}\_\{i\}^\{\\intercal\}\. For notational convenience, assume thatnnis even, with the firstn2\\frac\{n\}\{2\}points sampled around𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}and the remainingn2\\frac\{n\}\{2\}points sampled around𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}\. We then obtain the corresponding outputsyi=f​\(𝒛i\)y\_\{i\}=f\(\\bm\{z\}\_\{i\}\), and define𝒚=\(y1,…,yn\)⊺∈ℝn\\bm\{y\}=\(y\_\{1\},\\ldots,y\_\{n\}\)^\{\\intercal\}\\in\\mathbb\{R\}^\{n\}\.

After sampling, each sampled point𝒛i\\bm\{z\}\_\{i\}is weighted by a Gaussian\-type kernel,

πi=π𝒕j∗,𝒕ref​\(𝒛i\)=exp⁡\(−d𝒕j∗,𝒕ref2​\(𝒛i\)σ2\),\\pi\_\{i\}=\\pi\_\{\\bm\{t\}\_\{j^\{\*\}\},\\bm\{t\}\_\{\\mathrm\{ref\}\}\}\(\\bm\{z\}\_\{i\}\)=\\exp\\left\(\-\\frac\{d^\{2\}\_\{\\bm\{t\}\_\{j^\{\*\}\},\\bm\{t\}\_\{\\mathrm\{ref\}\}\}\(\\bm\{z\}\_\{i\}\)\}\{\\sigma^\{2\}\}\\right\),\(1\)whereσ\\sigmais the kernel width controlling the weights, andd𝒕j∗,𝒕refd\_\{\\bm\{t\}\_\{j^\{\*\}\},\\bm\{t\}\_\{\\mathrm\{ref\}\}\}denotes the distance of the sampled data with respect to the anomalous observation𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}and the reference point𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}\. Specifically, the distance is defined as

d𝒕j∗,𝒕ref​\(𝒛i\)=‖𝒛i−𝒕j∗‖2⋅‖𝒛i−𝒕ref‖2\.d\_\{\\bm\{t\}\_\{j^\{\*\}\},\\bm\{t\}\_\{\\mathrm\{ref\}\}\}\(\\bm\{z\}\_\{i\}\)=\\sqrt\{\{\\\|\\bm\{z\}\_\{i\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\}\_\{2\}\\cdot\{\\\|\\bm\{z\}\_\{i\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\}\_\{2\}\}\.\(2\)This distance assigns large weights to samples that are close to either the detected anomaly or the reference normal point\. As a boundary case, when𝒕j∗=𝒕ref\\bm\{t\}\_\{j^\{\*\}\}=\\bm\{t\}\_\{\\mathrm\{ref\}\}, \([1](https://arxiv.org/html/2608.10587#S3.E1)\) and \([2](https://arxiv.org/html/2608.10587#S3.E2)\) imply that the kernel used in BREAD becomes a one\-point weighting, which is analogous to the weighting used in vanilla LIME\. Figure[2](https://arxiv.org/html/2608.10587#S3.F2)compares the weight distribution of both kernels\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/ellipse_decision_boundary_T2_heatmap.png)\(a\)
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/ellipse_decision_boundary_T2_ref_heatmap.png)\(b\)

Figure 2:Heat maps of weights under the same kernel widthσ\\sigma\. \(a\) Weighting using the anomalous point only\. Weights are higher when closer to𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}\. \(b\) Weighting using both the reference point and the anomalous point\. Weights are higher when either near𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}or𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}\.We estimate𝜷^\\hat\{\\bm\{\\beta\}\}by the following weighted loss function,

L​\(f,𝜷,β0,Π\)\\displaystyle L\(f,\\bm\{\\beta\},\\beta\_\{0\},\\Pi\)=∑i=1nπi​\(𝒛i\)​\(𝜷⊺​𝒛i\+β0−f​\(𝒛i\)\)2\\displaystyle=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i\}\)\\left\(\{\\bm\{\\beta\}\}^\{\\intercal\}\\bm\{z\}\_\{i\}\+\{\\beta\_\{0\}\}\-f\(\\bm\{z\}\_\{i\}\)\\right\)^\{2\}=\(Z​𝜷\+β0​𝟏−𝒚\)⊺​Π​\(Z​𝜷\+β0​𝟏−𝒚\),\\displaystyle=\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\\bm\{1\}\}\-\\bm\{y\}\)^\{\\intercal\}\\Pi\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\\bm\{1\}\}\-\\bm\{y\}\),whereΠ=diag​\(π1,…​πn\)\\Pi=\\text\{diag\}\(\\pi\_\{1\},\\ldots\\pi\_\{n\}\)denotes the diagonal matrix of sample weights for each𝒛i\\bm\{z\}\_\{i\},β0\\beta\_\{0\}is the intercept term,𝟏\\bm\{1\}is a vector of ones with lengthnn, and𝒚∈ℝn\\bm\{y\}\\in\\mathbb\{R\}^\{n\}stacks the monitoring model outputsyi=f​\(𝒛i\)y\_\{i\}=f\(\\bm\{z\}\_\{i\}\)\. Using the weighted sampled data, we fit a regression model by minimising the regularised loss,

min𝜷∈ℝd,β0∈ℝ⁡12​\(Z​𝜷\+β0​𝟏−𝒚\)⊺​Π​\(Z​𝜷\+β0​𝟏−𝒚\)\+λ2​𝜷⊺​𝑰d​𝜷,\\min\_\{\\bm\{\\beta\}\\in\\mathbb\{R\}^\{d\},\\,\\beta\_\{0\}\\in\\mathbb\{R\}\}\\,\\frac\{1\}\{2\}\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\\bm\{1\}\}\-\\bm\{y\}\)^\{\\intercal\}\\Pi\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\\bm\{1\}\}\-\\bm\{y\}\)\+\\frac\{\\lambda\}\{2\}\{\\bm\{\\beta\}\}^\{\\intercal\}\\bm\{I\}\_\{d\}\{\\bm\{\\beta\}\},\(3\)where12\\frac\{1\}\{2\}is included to simplify the derivatives,λ\\lambdais a positive penalty weight that shrinks the coefficients toward zero, and𝑰d∈ℝd×d\\bm\{I\}\_\{d\}\\in\\mathbb\{R\}^\{d\\times d\}is an identity matrix\. The weighted ridge surrogate linear model has the analytical solution \(see Supplementary Appendix B\)\.

𝜷^=\(A\+λ​𝑰d\)−1​𝒃,\\hat\{\\bm\{\\beta\}\}=\(A\+\\lambda\\bm\{I\}\_\{d\}\)^\{\-1\}\\bm\{b\},\(4\)whereA=\(Z⊺​Π​Z−Z⊺​Π​𝟏​\(Z⊺​Π​𝟏\)⊺∑i=1nπi\),A=\\left\(Z^\{\\intercal\}\\Pi Z\-\\frac\{Z^\{\\intercal\}\\Pi\\bm\{1\}\\left\(Z^\{\\intercal\}\\Pi\\bm\{1\}\\right\)^\{\\intercal\}\}\{\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\}\\right\),and𝒃=Z⊺​Π​𝒚−1∑i=1nπi​\(Z⊺​Π​𝟏\)​\(𝟏⊺​Π​𝒚\)\\bm\{b\}=Z^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\}\\left\(Z^\{\\intercal\}\\Pi\\bm\{1\}\\right\)\\left\(\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}\\right\)\. From \([3](https://arxiv.org/html/2608.10587#S3.E3)\), sinceλ\>0\\lambda\>0andπi\>0\\pi\_\{i\}\>0, the weighted ridge objective is strictly convex\. Therefore, the solution exists and is unique\. This property ensures that the proposed surrogate explanation is well defined\. The diagnosis for the identified anomaly𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}is then given by the indices of theKKlargest components of\|𝜷^\|\|\\hat\{\\bm\{\\beta\}\}\|\.

### 3\.3Analytical Properties

This section discusses the analytical properties of BREAD, focusing on \(i\) asymptotic sparsity when an anomaly occurs in the feature space, and \(ii\) stability to perturbations\.

We next analyse whether the surrogate coefficients concentrate on the truly shifted features as the anomaly becomes farther from the normal reference point\. Specifically, let the normal reference point be𝒕ref∈ℝd\\bm\{t\}\_\{\\mathrm\{ref\}\}\\in\\mathbb\{R\}^\{d\}, and suppose an anomaly𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}is detected\. Denote the shift magnitudeδ=‖𝒕j∗−𝒕ref‖2\>0\\delta=\\\|\\bm\{t\}\_\{j^\{\*\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\>0and the direction𝒖=\(𝒕j∗−𝒕ref\)/δ\\bm\{u\}=\(\\bm\{t\}\_\{j^\{\*\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\)/\\delta, yielding

𝒕j∗=𝒕ref\+δ​𝒖\.\\bm\{t\}\_\{j^\{\*\}\}=\\bm\{t\}\_\{\\mathrm\{ref\}\}\+\\delta\\bm\{u\}\.\(5\)Let𝒦=\{ℓ:uℓ≠0\}\\mathcal\{K\}=\\\{\\ell:u\_\{\\ell\}\\neq 0\\\}denote the set of shifted features andℛ=\{1,…,d\}∖𝒦\\mathcal\{R\}=\\\{1,\\dots,d\\\}\\setminus\\mathcal\{K\}, where\|𝒦\|<d\|\\mathcal\{K\}\|<d\.

As described above, our method generates𝒵\(δ\)=\{𝒛i\(δ\)\}i=1n\\mathcal\{Z\}^\{\(\\delta\)\}=\\\{\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\\\}\_\{i=1\}^\{n\}and index setsI0\(δ\)I\_\{0\}^\{\(\\delta\)\}andI1\(δ\)I\_\{1\}^\{\(\\delta\)\}with\|I0\(δ\)\|=\|I1\(δ\)\|=n/2\|I\_\{0\}^\{\(\\delta\)\}\|=\|I\_\{1\}^\{\(\\delta\)\}\|=n/2containing samples around𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}and𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}, respectively\. The following two assumptions formalise the sampling design and the output structure\.

###### Assumption 3\.1\(Geometric sampled data assumption\)\.

There exist constantsc,ε,ρ\>0c,\\varepsilon,\\rho\>0\(independent ofδ\\delta\) and core index setsJ0\(δ\)⊂I0\(δ\)J^\{\(\\delta\)\}\_\{0\}\\subset I^\{\(\\delta\)\}\_\{0\}andJ1\(δ\)⊂I1\(δ\)J^\{\(\\delta\)\}\_\{1\}\\subset I^\{\(\\delta\)\}\_\{1\}with\|J0\(δ\)\|=\|J1\(δ\)\|≥1\|J^\{\(\\delta\)\}\_\{0\}\|=\|J^\{\(\\delta\)\}\_\{1\}\|\\geq 1, such that‖𝒛i\(δ\)−𝒕ref‖2≤cδ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq\\frac\{c\}\{\\delta^\{2\}\}, fori∈J0\(δ\),i\\in J^\{\(\\delta\)\}\_\{0\},and‖𝒛i\(δ\)−𝒕j∗‖2≤cδ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\frac\{c\}\{\\delta^\{2\}\}, fori∈J1\(δ\)\.i\\in J^\{\(\\delta\)\}\_\{1\}\.Furthermore, for the non\-core points,ε≤‖𝒛i\(δ\)−𝒕ref‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq\\rho, fori∈I0\(δ\)∖J0\(δ\),i\\in I^\{\(\\delta\)\}\_\{0\}\\setminus J^\{\(\\delta\)\}\_\{0\},andε≤‖𝒛i\(δ\)−𝒕j∗‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\rho, fori∈I1\(δ\)∖J1\(δ\)\.i\\in I^\{\(\\delta\)\}\_\{1\}\\setminus J^\{\(\\delta\)\}\_\{1\}\.

Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)is consistent with a local perturbation scheme and guarantees at least one sampled point in each group has non\-trivial weights\. We also assume thatnnis fixed\.

Next, we impose a structural assumption on the corresponding outputs\. Write these outputs asyi\(δ\)=f​\(𝒛i\(δ\)\)y\_\{i\}^\{\(\\delta\)\}=f\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\), and definey¯s\(δ\)\\bar\{y\}\_\{s\}^\{\(\\delta\)\}as the group mean overIs\(δ\)I\_\{s\}^\{\(\\delta\)\}fors∈\{0,1\}s\\in\\\{0,1\\\}\. Because the signal is caused by the shift, we assume that the black\-box monitoring output has a persistent between group contrastΔ​\(δ\):=y¯1\(δ\)−y¯0\(δ\)\\Delta\(\\delta\):=\\bar\{y\}\_\{1\}^\{\(\\delta\)\}\-\\bar\{y\}\_\{0\}^\{\(\\delta\)\}, while within\-group fluctuations are bounded\. To ensure that the dominant output change is the between\-groupΔ​\(δ\)\\Delta\(\\delta\), we additionally assume that the black\-box monitoring output is nearly constant on the core points near each centre\.

###### Assumption 3\.2\(Output structure\)\.

Assume there exist constantsBy<∞B\_\{y\}<\\infty,cΔ,cy\>0c\_\{\\Delta\},c\_\{y\}\>0independent ofδ\\deltasuch that for eachs∈\{0,1\}s\\in\\\{0,1\\\}, eachi∈Is\(δ\)i\\in I\_\{s\}^\{\(\\delta\)\},\|yi\(δ\)−y¯s\(δ\)\|≤By,\\bigl\|\{y\}\_\{i\}^\{\(\\delta\)\}\-\\bar\{y\}\_\{s\}^\{\(\\delta\)\}\\bigr\|\\leq B\_\{y\},andΔ​\(δ\)≥cΔ\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\. Moreover, fors∈\{0,1\}s\\in\\\{0,1\\\}andi∈Js\(δ\)i\\in J\_\{s\}^\{\(\\delta\)\},\|yi\(δ\)−y¯s\(δ\)\|≤cyδ\.\|\{y\}\_\{i\}^\{\(\\delta\)\}\-\\bar\{y\}\_\{s\}^\{\(\\delta\)\}\|\\leq\\frac\{c\_\{y\}\}\{\\delta\}\.

Note that Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)is a monitoring\-diagnosis design condition, which is similar to local Lipschitz continuity, and formalises the local output structure after a meaningful monitoring signal has been detected\. Specifically, it requires the monitoring model to produce a persistent output contrast between the synthetic samples around the detected anomalous point and samples around the reference normal point\.

Assumptions[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)–[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)are both practically satisfied in the monitoring context\. Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)is typically satisfied by selecting an appropriate sample scheme\. For example, it can be achieved by including the two centre points𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}and𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}in the synthetic sample set\. Then their kernel weights are equal to one\. The remaining non\-core samples can be generated from a truncated normal distribution around their centres\. Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)is satisfied when a meaningful monitoring signal has been detected, as the anomalous and reference outputs show a persistent anomaly score difference\. Together, the two assumptions ensure that \(i\) each set contains at least one “core” point extremely close to its centre, and \(ii\) the anomaly scoref​\(𝒕j∗\)f\(\\bm\{t\}\_\{j^\{\*\}\}\)exceeds the threshold due to the shift in𝒦\\mathcal\{K\}, rather than the bias\.

Under the geometric design and output structure, we establish the following asymptotic sparsity guarantee for BREAD\.

###### Theorem 3\.3\(Asymptotic sparsity\)\.

Suppose the shift setup in \([5](https://arxiv.org/html/2608.10587#S3.E5)\) holds, the synthetic data matrixZ\(δ\)Z^\{\(\\delta\)\}satisfies Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), and the corresponding outputs,yi\(δ\)=f​\(𝐳i\(δ\)\)\{y\}\_\{i\}^\{\(\\delta\)\}=f\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\), satisfy Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)\. Let𝛃^\\hat\{\\bm\{\\beta\}\}denote the estimator in \([4](https://arxiv.org/html/2608.10587#S3.E4)\) and defineϵ=‖𝛃^ℛ‖2‖𝛃^𝒦‖2\\epsilon=\\dfrac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\. Then,

ϵ→0,as​δ→∞\.\{\\epsilon\}\\rightarrow 0,\\qquad\\text\{as\}\\,\\delta\\rightarrow\\infty\.

Theorem[3\.3](https://arxiv.org/html/2608.10587#S3.Thmtheorem3)establishes that, as the anomaly moves further from the normal reference, the sampling geometry and kernel weighting jointly decouple the shifted and unshifted feature blocks in the surrogate, driving relevance to concentrate on𝒦\\mathcal\{K\}and vanish onℛ\\mathcal\{R\}\. The proof is provided in Appendix C\.

In contrast, LIME does not enjoy this guarantee that the ratio of coefficients need not converge to zero, even asδ→∞\\delta\\rightarrow\\infty\. To illustrate this comparison, consider a simple two\-dimensional case with𝒕ref=\(0,0\)\\bm\{t\}\_\{\\mathrm\{ref\}\}=\(0,0\)and𝒕j∗=\(δ,0\)\\bm\{t\}\_\{j^\{\*\}\}=\(\\delta,0\)\. In this case, the anomaly is generated only by the first feature, so𝒦=\{1\}\\mathcal\{K\}=\\\{1\\\}andℛ=\{2\}\\mathcal\{R\}=\\\{2\\\}\. Suppose the black\-box model is locally quadratic in the neighbourhood of𝒕j∗\\bm\{t\}\_\{j^\{\*\}\},f​\(z1,z2\)=z12\+γ​z1​z2f\(z\_\{1\},z\_\{2\}\)=z\_\{1\}^\{2\}\+\\gamma z\_\{1\}z\_\{2\}withγ≠0\\gamma\\neq 0\. Then∇f​\(δ,0\)=\(2​δ,γ​δ\)⊺\\nabla f\(\\delta,0\)=\(2\\delta,\\gamma\\delta\)^\{\\intercal\}\. Thus, although the shift occurs only in the first feature, the local sensitivity offfaround𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}has a nonzero component in the second feature\. A LIME explanation fitted locally around𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}may assign a nonzero coefficient to the non\-shifted feature, and the coefficients ratio is approximatelyϵL≈\|γ\|/2\\epsilon^\{L\}\\approx\|\\gamma\|/2\. By contrast, Theorem[3\.3](https://arxiv.org/html/2608.10587#S3.Thmtheorem3)states that the proposed reference\-based surrogate yields‖𝜷^ℛ‖2/‖𝜷^𝒦‖2→0\|\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\|\|\_\{2\}/\|\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\|\|\_\{2\}\\rightarrow 0asδ→∞\\delta\\rightarrow\\infty\. This shows that LIME’s coefficient ratio is bounded away from zero by a constantϵL≈\|γ\|/2\\epsilon^\{L\}\\approx\|\\gamma\|/2, regardless of how largeδ\\deltabecomes, which is a fundamental limitation that BREAD overcomes\.

Finally, we study the stability of the estimator via its sensitivity to perturbations in\(A,𝒃\)\(A,\\bm\{b\}\)\.

###### Lemma 3\.4\(Sensitivity\)\.

Consider the linear system\(A\+λ​𝐈d\)​𝛃=𝐛\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\bm\{\\beta\}=\\bm\{b\}, whereAAand𝐛\\bm\{b\}are defined as in \([4](https://arxiv.org/html/2608.10587#S3.E4)\)\. If‖\(A\+λ​𝐈d\)−1‖p​‖Δ​\(A\+λ​𝐈d\)‖p<1\\\|\(A\+\\lambda\\bm\{I\}\_\{d\}\)^\{\-1\}\\\|\_\{p\}\\\|\\Delta\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}<1, then

‖Δ​𝜷‖p‖𝜷‖p≤\\displaystyle\\frac\{\\\|\\Delta\\bm\{\\beta\}\\\|\_\{p\}\}\{\\\|\\bm\{\\beta\}\\\|\_\{p\}\}\\leqκp​\(\(A\+λ​𝑰d\)\)1−κp​\(\(A\+λ​𝑰d\)\)​‖Δ​\(A\+λ​𝑰d\)‖p‖\(A\+λ​𝑰d\)‖p\\displaystyle\\frac\{\\kappa\_\{p\}\(\(A\+\\lambda\\bm\{I\}\_\{d\}\)\)\}\{1\-\\kappa\_\{p\}\(\(A\+\\lambda\\bm\{I\}\_\{d\}\)\)\\,\\dfrac\{\\\|\\Delta\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}\}\{\\\|\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}\}\}\(6\)\(‖Δ​\(A\+λ​𝑰d\)‖p‖\(A\+λ​𝑰d\)‖p\+‖Δ​𝒃‖p‖𝒃‖p\),\\displaystyle\\left\(\\frac\{\\\|\\Delta\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}\}\{\\\|\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}\}\+\\frac\{\\\|\\Delta\\bm\{b\}\\\|\_\{p\}\}\{\\\|\\bm\{b\}\\\|\_\{p\}\}\\right\),whereκp​\(\(A\+λ​𝐈d\)\)=‖\(A\+λ​𝐈d\)‖p​‖\(A\+λ​𝐈d\)−1‖p\\kappa\_\{p\}\(\(A\+\\lambda\\bm\{I\}\_\{d\}\)\)=\\\|\(A\+\\lambda\\bm\{I\}\_\{d\}\)\\\|\_\{p\}\\\|\(A\+\\lambda\\bm\{I\}\_\{d\}\)^\{\-1\}\\\|\_\{p\}is the \(pp\-\) condition number of the matrixA\+λ​𝐈dA\+\\lambda\\bm\{I\}\_\{d\}, whileΔ​𝛃,Δ​\(A\+λ​𝐈d\)\\Delta\\bm\{\\beta\},\\Delta\(A\+\\lambda\\bm\{I\}\_\{d\}\), andΔ​𝐛\\Delta\\bm\{b\}denote the perturbation in𝛃\\bm\{\\beta\},A\+λ​𝐈dA\+\\lambda\\bm\{I\}\_\{d\}, and𝐛\\bm\{b\}, respectively\.

Lemma[3\.4](https://arxiv.org/html/2608.10587#S3.Thmtheorem4)shows that the relative perturbation in the estimated coefficient vector is controlled by the condition number ofA\+λ​𝑰dA\+\\lambda\\bm\{I\}\_\{d\}and by the perturbations inAAand𝒃\\bm\{b\}, which follows from standard perturbation theory for linear systems\. Therefore, when the weighted ridge system is well conditioned, the generated explanation is stable with respect to small perturbations in the weighted sampled data, or model outputs\.

## 4Experiments

In this section, we evaluate BREAD using a synthetic simulation and a case study\.

### 4\.1Simulation Study

The simulation is designed to mimic high\-dimensional sequential data with temporal dependence, cross\-sectional correlation, and seasonal variation\. Although the monitoring model used in this study follows an SPM control chart framework, the evaluation focuses on the more general task of diagnosing detected anomalies\. Specifically, we evaluate the proposed approach along three dimensions: \(i\) faithfulness measures alignment of diagnosis with the ground truth; \(ii\) robustness shows sensitivity of the explanation to observation perturbations; \(iii\) stability shows the consistency across resampled datasets\.

#### 4\.1\.1Data Generation and Monitoring Model Setup

We first generate a baseline data set representing normal behaviour, drawn from a multivariate autoregressive model of order one \(AR\(1\)\) with block\-structured cross\-sectional correlation and an additive seasonal component\. Specifically, we simulated=500d=500features overn=2000n=2000time points\. The covariance structureΣ0\\Sigma\_\{0\}is constructed to exhibit block\-wise dependence \(i\.e\., correlation blocks\) to reflect clustered feature relationships\. Variables are ordered by correlation blocks, yielding a clear block\-diagonal pattern with strong within\-block dependence \(\>0\.8\>0\.8\) and comparatively weaker between\-block correlations \(lighter off\-diagonal regions,<0\.5<0\.5\)\. Two kinds of seasonality with periods of2424and168168are induced by first\-order Fourier components\. The data generation procedure is detailed in Appendix D\.2 of the Supplementary\.

To generate anomalous test data, we introduce feature shifts, defined as \([5](https://arxiv.org/html/2608.10587#S3.E5)\), fromt=1600t=1600\. These shifts define three anomaly scenarios, adapted fromCacciarelli and Kulahci \([2022](https://arxiv.org/html/2608.10587#bib.bib70)\), ranging from−5\-5to55stepped by11, yielding 33 experimental conditions in total\. The zero\-shift setting is used as the reference case\.

- •Scenario 1Anomaly within one block The means of all 53 features in the first correlation block are shifted by the given value\.
- •Scenario 2Anomaly across blocks From each block, three features are randomly selected and their means are shifted, for a total of 45 features\.
- •Scenario 3Combination of Scenarios 1&2 The means of all 53 features in the first block are shifted, and three features from each remaining block are also shifted, for a total of 95 features\.

For prospective anomaly detection, we use an LSTM\-basedT2T^\{2\}control chart model\. The anomaly scoressis defined as theT2T^\{2\}statistic computed on the LSTM residuals, while the thresholdθs\\theta\_\{s\}is set based on anFFdistribution\(Sullivan and Woodall,[1996](https://arxiv.org/html/2608.10587#bib.bib160)\)when given the Type I error rateα\\alpha\. Furthermore, the empirical anomaly detection performance of this method is shown in Appendix D\.3\.

#### 4\.1\.2Benchmark Methods

For faithfulness and robustness, we compare BREAD against vanilla LIME and a naive baseline that reduces KernelSHAP\(Lundberg and Lee,[2017](https://arxiv.org/html/2608.10587#bib.bib90)\)to leave\-one\-out coalitions\. Specifically, the vanilla LIME is the original LIME using the Gaussian kernel and the ridge regression as the surrogate model\. Sample size, kernel bandwidthσ\\sigmaand ridge penalty parameterλ\\lambdaare fixed across all experiments for both LIME and BREAD\. For the naive method, the reference point is selected in the same way as in BREAD\. Detailed hyperparameters are discussed in Appendix D\.4\. In addition, when comparing faithfulness, we introduce the expected faithfulness of detecting the features causing the error when randomly selectingKKfeatures fromdd\. For comparing sampling stability, we only consider sampling\-based methods, i\.e\., BREAD and LIME\.

#### 4\.1\.3Faithfulness

Faithfulness is assessed by comparing the top\-KKfeatures returned by each explanation method to the known ground\-truth affected features \(i\.e\., those in the shifted block\) for the first detected anomalous observation per run\. Specifically, we repeat the full simulation and detection pipeline across 1000 random seeds, extract the first outlier identified under each seed, and quantify agreement between the top\-KKexplanations and the ground truth by the cosine similaritySFS\_\{F\}\. Given the set𝒦\\mathcal\{K\}ofKKfeatures driving the process to abnormal, and a setDrD\_\{r\}of the selected top\-KKfeatures, the faithfulness at runrris defined asSF\(r\)​\(𝒦,Dr\)=\|𝒦∩Dr\|/\(\|𝒦\|⋅\|Dr\|\)S^\{\(r\)\}\_\{F\}\(\\mathcal\{K\},D\_\{r\}\)=\|\\mathcal\{K\}\\cap D\_\{r\}\|/\\sqrt\{\(\|\\mathcal\{K\}\|\\cdot\|D\_\{r\}\|\)\}\.

The expected cosine similarity of a randomKK\-feature selection isK/dK/d\(Random Guess in Figure[3](https://arxiv.org/html/2608.10587#S4.F3)\), following a hypergeometric distribution, which gives𝔼​\(SF\)=0\.106,0\.09,0\.190\\mathbb\{E\}\(S\_\{F\}\)=0\.106,0\.09,0\.190for Scenarios 1–3, respectively\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Faith_comparison_legend.png)

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Faith_comparison_S1.png)\(a\)Scenario 1
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Faith_comparison_S2.png)\(b\)Scenario 2
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Faith_comparison_S3.png)\(c\)Scenario 3

Figure 3:Explanation faithfulness under varying mean shifts\.Panels \(a\)–\(c\) depict explanation faithfulness against the mean shift with shaded±1\\pm 1standard deviation region for Scenarios 1–3, respectively\. Line styles distinguish the three methods and the random pick \(LIME: circle; BREAD: square; Naive: diamond; random guess: dashed\)\.Figure[3](https://arxiv.org/html/2608.10587#S4.F3)depicts faithfulness of the three methods across three scenarios with different shifts in mean\. For all methods, when there is no shift in mean, the top\-KKfeature importance is very close to the results of random pick\. Meanwhile, for all methods, as the magnitude of shift increases, the faithfulness score also increases\. Specifically, BREAD achieves the highest faithfulness score, while LIME remains relatively low and flat\. Moreover, the performance gap between these two methods increases rapidly as the shift becomes large, which is consistent with Theorem[3\.3](https://arxiv.org/html/2608.10587#S3.Thmtheorem3)\. In addition, the naive method improves with the size of shift as well, its faithfulness score generally falls between LIME and BREAD\.

In terms of computational cost, LIME and BREAD are comparable in runtime atd=500d=500, whereas the naive method is four times slower\. Results are detailed in Appendix D\.5\.

#### 4\.1\.4Robustness

The robustness test is designed to assess how perturbation in observation affects the diagnosis results\. It is evaluated as the consistency of the explanation method for the same anomaly type, i\.e\., with the same mean shift under the same scenario\. For each anomaly type, we perform 1000 runs\. Since all anomalous observations are generated from the same underlying shift, any variation in diagnosis reflects sensitivity to noise rather than a change in fault source\. The robustness score for runrris the average pairwise cosine similarity of the top\-KKfeature sets across allNrN\_\{r\}identified signals,SR\(r\)=1\(Nr2\)​∑i=1Nr∑j=i\+1Nr\|Dr,i∩Dr,j\|\(\|Dr,i\|⋅\|Dr,j\|\),S^\{\(r\)\}\_\{R\}=\\frac\{1\}\{\\binom\{N\_\{r\}\}\{2\}\}\\sum\_\{i=1\}^\{N\_\{r\}\}\\sum\_\{j=i\+1\}^\{N\_\{r\}\}\\frac\{\|D\_\{r,i\}\\cap D\_\{r,j\}\|\}\{\\sqrt\{\(\|D\_\{r,i\}\|\\cdot\|D\_\{r,j\}\|\)\}\},whereDr,iD\_\{r,i\}denotes the features selected in the top\-KKfor the signaliiin runrr\. The overall robustness score is the average pairwise similaritySR\(r\)S^\{\(r\)\}\_\{R\}across all 1000 runs\. Note that this metric only measures robustness, but not faithfulness\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Robustness_comparison_legend.png)

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Robustness_comparison_S1.png)\(a\)Scenario 1
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Robustness_comparison_S2.png)\(b\)Scenario 2
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Robustness_comparison_S3.png)\(c\)Scenario 3

Figure 4:Explanation robustness under varying mean shifts\.Panels \(a\)–\(c\) correspond to Scenarios 1–3, respectively\. In each panel, explanation robustness is plotted with variability bands against the magnitude of the mean shift for LIME \(circle\), BREAD \(square\), and Naive \(diamond\)\.The robustness results for the three methods are depicted in Figure[4](https://arxiv.org/html/2608.10587#S4.F4)\. Robustness improves with the mean shift size for all scenarios, where consistency is lowest near the zero mean shift\. This means all methods are likely to report random noise\. The key result is the rapid increase in robustness as the mean shift grows, indicating consistent diagnoses once the anomaly signal is larger than the noise level\. In addition, all methods show higher robustness under Scenario 3\. Across scenarios, BREAD achieves the highest robustness for moderate\-to\-large mean shifts\. LIME shows higher robustness despite low faithfulness near zero shifts, suggesting repeated potentially incorrect feature selection when the anomaly signal is weak\.

#### 4\.1\.5Stability

Stability is evaluated by repeatedly explaining the same outlier instance across 1000 random seeds at each sample size, and measuring the variability of the resulting feature attributions\. Specifically, the set cosine similaritySS​tS\_\{St\}between the top\-KKfeatures and the ground truth across repeated runs is used as the metric\. The outlier is fixed from one run under Scenario 1 with mean shift of22\. This criterion is particularly relevant for sampling\-based approaches such as LIME and its variants, which are sensitive to perturbation and sampling randomness\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Stability_Study_MultiSeeds_MeanAcc.png)\(a\)Mean
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/Stability_Study_MultiSeeds_StdAcc.png)\(b\)Standard deviation

Figure 5:Stability diagnosis results\.The horizontal axis shows the number of samples on a logarithmic scale\. \(a\) Mean faithfulness score, repeated 1000 runs with different random seeds at each sample size\. \(b\) Standard deviation of faithfulness score across the same repetitions\. Curves compare LIME and BREAD, highlighting how resampling size affects both the average faithfulness score and the stability\.Figure[5](https://arxiv.org/html/2608.10587#S4.F5)shows how the sample size affects average faithfulness score and stability across repetitions\. BREAD achieves a consistently high faithfulness score even at a relatively small sample size, and its standard deviation decreases rapidly as the number of samples increases\. In contrast, LIME shows lower mean faithfulness score, which improves gradually with the size of sampled data, while the deviation stays elevated and declines notably only at larger sample sizes\.

### 4\.2Case study

In this section, we revisit the case mentioned in the introduction and implement the proposed method in this real data set to show its applicability and performance in practice\.

#### 4\.2\.1Data Description and Monitoring

We select 16 escalators and monitor their daily operating status\. As a concrete fault type with known ground truth, we use the unexpected shutdowns as the fault type, which happened to Escalators 08, 13, and 14 \(see Appendix D\.8, Figure 3 of the Supplementary\)\.

The VAE\-LSTM\-basedT2T^\{2\}control charts proposed byMagedet al\.\([2024](https://arxiv.org/html/2608.10587#bib.bib74)\)are applied as the monitoring method\. The monitoring results are illustrated in aT2T^\{2\}control chart in Figure[1](https://arxiv.org/html/2608.10587#S1.F1)\(c\)\. Note that, since the model may raise a false alarm, we set the empirical false alarm probability \(FAP\) relatively low\.

#### 4\.2\.2Signal Diagnosis

We apply both LIME and BREAD to all OOC signals detected during the shutdown period\. For each signal, both methods generate a feature importance score for each of the 16 escalators, using the same number of synthetic samples \(n=2000n=2000\)\. The in\-control reference point used by the proposed method is estimated from the training data as the component\-wise mean of the in\-control observations\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/LIME_T2.png)\(a\)
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/BREAD_T2.png)\(b\)

Figure 6:Box plots ofl2l\_\{2\}\-regularized feature importance\.Each box plot summarizes the median, range, and outliers for feature importance\. \(a\) LIME distributes importance broadly across non\-shut\-down escalators \(e\.g\., 00–02, 03, 07, and 12\), failing to concentrate relevance on the ground\-truth Escalators 08, 13, 14\. \(b\) BREAD assigns the largest importance consistently to Escalators 08, 13, and 14, with near\-zero, low\-variance relevance for all remaining escalators, correctly recovering the fault source\.The diagnosis results from all the signals are illustrated in box plots, see Figure[6](https://arxiv.org/html/2608.10587#S4.F6)\. The results show a clear qualitative difference between the two methods\.

LIME assigns large importance values to Escalators 03 and 07 \(median positive importance≈0\.25\\approx 0\.25–0\.400\.40\) and large negative importance to Escalators 00, 01, and 02 \(median≈−0\.25\\approx\-0\.25–−0\.40\-0\.40\), while most remaining escalators receive near\-zero scores\. Critically, the three ground\-truth shut down escalators \(08, 13, 14\) are not consistently identified among the top\-ranked features, indicating that LIME’s diagnosis is misaligned with the actual fault source\.

The BREAD method produces markedly different and more faithful results\. Escalator 08 receives the largest negative importance \(median≈−0\.80\\approx\-0\.80\), and Escalators 13 and 14 receive consistently large negative importance \(median≈−0\.25\\approx\-0\.25–−0\.50\-0\.50\)\. The remaining 13 escalators are assigned near\-zero importance with low variance, reflecting the asymptotic sparsity property established in Section[3\.3](https://arxiv.org/html/2608.10587#S3.SS3)\. This concentration of relevance on the true fault variables is consistent across signals, demonstrating both faithfulness and stability\.

The proposed method correctly identifies the three shutdown escalators across all detected signals, whereas LIME consistently mis\-attributes importance to normal escalators\. This aligns with the theoretical sparsity result, that is, BREAD concentrates relevance on variables that drive the process away from normal operation\. Further, the low variance of importance scores for non\-faulty escalators \(Figure[6](https://arxiv.org/html/2608.10587#S4.F6)\(b\)\) indicates that BREAD is robust to input perturbations\. This is consistent with the simulation findings in Section[4\.1\.4](https://arxiv.org/html/2608.10587#S4.SS1.SSS4)\.

Note that, diagnosis quality is inherently coupled to the monitoring model quality\. In this case study, the VAE\-LSTM model is trained on a sufficiently long in\-control period and validated to achieve a low FAP before deployment\. In general, the proposed diagnosis method should be applied only after the monitoring model has been properly validated\.

## 5Conclusion and Limitation

In this paper, we propose the BREAD method for post\-hoc diagnosis in AI\-based prospective anomaly detection\. By incorporating in\-control reference information into local surrogate construction, BREAD generates explanations that are more aligned with the variables driving out\-of\-control signals while preserving the model\-agnosticism needed for applicability and computational efficiency needed for online deployment\. Theoretical analysis and empirical results on simulations and a real monitoring case study demonstrate that this global baseline\-aware explanation is effective for practical diagnosis in black\-box SPM control chart frameworks\.

Several important directions for future work remain to be explored\. First, the current method provides point estimates of feature relevance but does not quantify explanation uncertainty\. Future work can estimate the uncertainty of signal explanation for sampled local surrogates\. Second, the proposed model uses a single in\-control reference\. It would be valuable to investigate principled strategies for selection and extend to multi\-reference variants when the baseline operation is multi\-modal\. Third, although the method guarantees sparse relevance patterns, the subgroup of features remains unselected\. Future work could therefore investigate feature\-selection mechanisms on top of the proposed method to improve actionability in practice\.

## References

- A\. Adadi and M\. Berrada \(2018\)Peeking inside the black\-box: a survey on explainable artificial intelligence \(xai\)\.6,pp\. 52138–52160\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p2.1),[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- P\. Agarwal, M\. Tamer, and H\. Budman \(2021\)Explainability: relevance based dynamic deep learning algorithm for fault detection and diagnosis in chemical processes\.154,pp\. 107467\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- A\. B\. Arrieta, N\. Díaz\-Rodríguez, J\. Del Ser, A\. Bennetot, S\. Tabik, A\. Barbado, S\. García, S\. Gil\-López, D\. Molina, R\. Benjamins,et al\.\(2020\)Explainable artificial intelligence \(xai\): concepts, taxonomies, opportunities and challenges toward responsible ai\.Information fusion58,pp\. 82–115\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p2.1)\.
- S\. Bach, A\. Binder, G\. Montavon, F\. Klauschen, K\. Müller, and W\. Samek \(2015\)On pixel\-wise explanations for non\-linear classifier decisions by layer\-wise relevance propagation\.10\(7\),pp\. e0130140\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- M\. Baek and S\. B\. Kim \(2023\)Failure detection and primary cause identification of multivariate time series data in semiconductor equipment\.IEEE Access11,pp\. 54363–54372\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- A\. Bhakte, M\. Chakane, and R\. Srinivasan \(2023\)Alarm\-based explanations of process monitoring results from deep neural networks\.Computers & Chemical Engineering179,pp\. 108442\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- A\. Bhakte, V\. Pakkiriswamy, and R\. Srinivasan \(2022\)An explainable artificial intelligence based approach for interpretation of fault classification results from deep neural networks\.Chemical Engineering Science250,pp\. 117373\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- D\. Cacciarelli and M\. Kulahci \(2022\)A novel fault detection and diagnosis approach based on orthogonal autoencoders\.Computers & Chemical Engineering163,pp\. 107853\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1),[§2](https://arxiv.org/html/2608.10587#S2.p1.1),[§2](https://arxiv.org/html/2608.10587#S2.p2.1),[§4\.1\.1](https://arxiv.org/html/2608.10587#S4.SS1.SSS1.p2.4)\.
- D\. Cacciarelli and M\. Kulahci \(2023\)Hidden dimensions of the data: pca vs autoencoders\.Quality Engineering35\(4\),pp\. 741–750\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- H\. Choi, D\. Kim, J\. Kim, J\. Kim, and P\. Kang \(2022\)Explainable anomaly detection framework for predictive maintenance in manufacturing systems\.125,pp\. 109147\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- B\. M\. Colosimo, L\. A\. Jones\-Farmer, F\. M\. Megahed, K\. Paynabar, C\. Ranjan, and W\. H\. Woodall \(2024\)Statistical process monitoring from industry 2\.0 to industry 4\.0: insights into research and practice\.Technometrics,pp\. 1–24\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1)\.
- R\. Dwivedi, D\. Dave, H\. Naik, S\. Singhal, R\. Omer, P\. Patel, B\. Qian, Z\. Wen, T\. Shah, G\. Morgan,et al\.\(2023\)Explainable ai \(xai\): core ideas, techniques, and solutions\.55\(9\),pp\. 1–33\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p2.1)\.
- C\. Hou \(2023\)A simple test to determine the contributors of fraction nonconforming shifts in a multivariate binomial process\.35\(2\),pp\. 279–289\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- C\. Hwang and T\. Lee \(2021\)E\-sfd: explainable sensor fault detection in the ics anomaly detection system\.9,pp\. 140470–140486\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- K\. Jang, K\. E\. S\. Pilario, N\. Lee, I\. Moon, and J\. Na \(2023\)Explainable artificial intelligence for fault diagnosis of industrial processes\.IEEE Transactions on Industrial Informatics\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- X\. Jia, J\. Li, Y\. Lu, and W\. Li \(2025\)LIMEFLDL: a local interpretable model\-agnostic explanations approach for label distribution learning\.InForty\-second International Conference on Machine Learning,Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- D\. Kim, G\. Antariksa, M\. P\. Handayani, S\. Lee, and J\. Lee \(2021\)Explainable anomaly detection framework for maritime main engine sensor data\.21\(15\),pp\. 5200\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- J\. Kim, M\. K\. Jeong, E\. A\. Elsayed, K\. Al\-Khalifa, and A\. Hamouda \(2016\)An adaptive step\-down procedure for fault variable identification\.International Journal of Production Research54\(11\),pp\. 3187–3200\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- P\. Knab, S\. Marton, U\. Schlegel, and C\. Bartelt \(2025\)Which lime should i trust? concepts, challenges, and solutions\.InWorld Conference on Explainable Artificial Intelligence,pp\. 28–52\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p3.1),[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- S\. Lee, M\. Kwak, K\. Tsui, and S\. B\. Kim \(2019\)Process monitoring using variational autoencoder for high\-dimensional nonlinear processes\.Engineering Applications of Artificial Intelligence83,pp\. 13–27\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1),[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- J\. Li, J\. Jin, and J\. Shi \(2008\)Causation\-based t 2 decomposition for multivariate process monitoring and diagnosis\.Journal of Quality Technology40\(1\),pp\. 46–58\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- S\. M\. Lundberg and S\. Lee \(2017\)A unified approach to interpreting model predictions\.Advances in neural information processing systems30\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1),[§4\.1\.2](https://arxiv.org/html/2608.10587#S4.SS1.SSS2.p1.4)\.
- A\. Maged, C\. F\. Lui, S\. Haridy, and M\. Xie \(2024\)Variational autoencoders\-lstm based fault detection of time\-dependent high dimensional processes\.International Journal of Production Research62\(4\),pp\. 1092–1107\.Cited by:[Figure 1](https://arxiv.org/html/2608.10587#S1.F1),[Figure 1](https://arxiv.org/html/2608.10587#S1.F1.2.1),[§1](https://arxiv.org/html/2608.10587#S1.p1.1),[§2](https://arxiv.org/html/2608.10587#S2.p1.1),[§4\.2\.1](https://arxiv.org/html/2608.10587#S4.SS2.SSS1.p2.2)\.
- R\. L\. Mason, N\. D\. Tracy, and J\. C\. Young \(1995\)Decomposition of t 2 for multivariate control chart interpretation\.Journal of quality technology27\(2\),pp\. 99–108\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- S\. Mishra, B\. L\. Sturm, and S\. Dixon \(2017\)Local interpretable model\-agnostic explanations for music content analysis\.\.InISMIR,Vol\.53,pp\. 537–543\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- D\. Otto, J\. Kurtz, and S\. I\. Birbil \(2025\)Coherent local explanations for mathematical optimization\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- J\. Qiu, Y\. Lin, and I\. M\. Zwetsloot \(2025\)An lstm\-based predictive monitoring method for data with time\-varying variability\.63\(7\),pp\. 2622–2637\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1)\.
- J\. Rabold, M\. Siebers, and U\. Schmid \(2018\)Explaining black\-box classifiers with ilp–empowering lime with aleph to approximate non\-linear decisions with relational rules\.InInternational conference on inductive logic programming,pp\. 105–117\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- M\. T\. Ribeiro, S\. Singh, and C\. Guestrin \(2016\)“Why should i trust you?” explaining the predictions of any classifier\.InProceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining,pp\. 1135–1144\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p2.1),[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- R\. R\. Selvaraju, A\. Das, R\. Vedantam, M\. Cogswell, D\. Parikh, and D\. Batra \(2016\)Grad\-cam: why did you say that?\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- K\. Simonyan, A\. Vedaldi, and A\. Zisserman \(2013\)Deep inside convolutional networks: visualising image classification models and saliency maps\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- J\. Sipple \(2020\)Interpretable, multidimensional, multimodal anomaly detection with negative sampling for detection of device failure\.InInternational Conference on Machine Learning,pp\. 9016–9025\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- J\. H\. Sullivan and W\. H\. Woodall \(1996\)A comparison of multivariate control charts for individual observations\.28\(4\),pp\. 398–408\.Cited by:[§4\.1\.1](https://arxiv.org/html/2608.10587#S4.SS1.SSS1.p4.6)\.
- M\. Sundararajan, A\. Taly, and Q\. Yan \(2017\)Axiomatic attribution for deep networks\.InInternational conference on machine learning,pp\. 3319–3328\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- M\. Szelążek, S\. Bobek, and G\. J\. Nalepa \(2024\)Semantic data mining\-based decision support for quality assessment in steel industry\.41\(2\),pp\. e13319\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p2.1)\.
- Z\. Tan, Y\. Tian, and J\. Li \(2023\)GLIME: general, stable and local lime explanation\.36,pp\. 36250–36277\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- F\. Tsung, Y\. Li, and M\. Jin \(2008\)Statistical process control for multistage manufacturing and service operations: a review and some extensions\.3\(2\),pp\. 191–204\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1)\.
- W\. H\. Woodall \(2006\)The use of control charts in health\-care and public\-health surveillance\.38\(2\),pp\. 89–104\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1)\.
- J\. Yu, X\. Zheng, and S\. Wang \(2019\)A deep autoencoder feature learning method for process pattern recognition\.Journal of Process Control79,pp\. 1–15\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- W\. Yu and C\. Zhao \(2019\)Robust monitoring and fault isolation of nonlinear industrial processes using denoising autoencoder and elastic net\.IEEE Transactions on Control Systems Technology28\(3\),pp\. 1083–1091\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- M\. R\. Zafar and N\. M\. Khan \(2019\)DLIME: a deterministic local interpretable model\-agnostic explanations approach for computer\-aided diagnosis systems\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p3.1),[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- Y\. Zhang, K\. Song, Y\. Sun, S\. Tan, and M\. Udell \(2019a\)" Why should you trust my explanation?" understanding uncertainty in lime explanations\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p3.1),[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- Z\. Zhang, T\. Jiang, C\. Zhan, and Y\. Yang \(2019b\)Gaussian feature learning based on variational autoencoder for improving nonlinear process monitoring\.Journal of Process Control75,pp\. 136–155\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- Z\. Zhou, G\. Hooker, and F\. Wang \(2021\)S\-lime: stabilized\-lime for model explanation\.InProceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining,pp\. 2429–2438\.Cited by:[§2](https://arxiv.org/html/2608.10587#S2.p3.1)\.
- I\. Zwetsloot, L\. Jones\-Farmer, and W\. Woodall \(2023\)Monitoring univariate processes using control charts: some practical issues and advice\.Quality Engineering,pp\. 1–13\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p1.1),[§2](https://arxiv.org/html/2608.10587#S2.p1.1)\.
- I\. M\. Zwetsloot, Y\. Lin, J\. Qiu, L\. Li, W\. K\. F\. Lee, E\. Y\. S\. Yeung, C\. Y\. W\. Yeung, and C\. C\. L\. Wong \(2025\)Remaining useful life modelling with an escalator health condition analytic system\.Quality and Reliability Engineering International41\(8\),pp\. 3746–3758\.Cited by:[§1](https://arxiv.org/html/2608.10587#S1.p4.1)\.

BREAD: Baseline\-Referenced Explanations for Anomaly Diagnosis Supplementary Material

Jiaqi Qiu, Rob Goedhart, Jannis Kurtz and Inez M\. Zwetsloot

## Appendix APseudocode

The overall flow of our algorithm is outlined in Algorithm[1](https://arxiv.org/html/2608.10587#alg1)\.

Algorithm 1BREADInput:AI\-based black\-box monitoring model

ff, identified OOC signal

𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}, pre\-defined baseline reference point

𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}, number of samples

NN, kernel

πi\\pi\_\{i\}\.

Output:A set of the coefficients from a linear model

ggfor the anomaly score

f​\(𝒕j∗\)f\(\\bm\{t\}\_\{j^\{\*\}\}\)\.

𝒵←\[\]\\mathcal\{Z\}\\leftarrow\[\\,\];

𝐲←\[\]\\mathbf\{y\}\\leftarrow\[\\,\];

𝚷←\[\]\\mathbf\{\\Pi\}\\leftarrow\[\\,\];

for

i=1i=1to

N/2N/2do

𝒛i←SampleAround​𝒕j∗\\bm\{z\}\_\{i\}\\leftarrow\\textsc\{SampleAround\}\\bm\{t\}\_\{j^\{\*\}\};

yi←f​\(𝒛i\)y\_\{i\}\\leftarrow f\(\\bm\{z\}\_\{i\}\);

πi←πi​\(𝒛i\)\\pi\_\{i\}\\leftarrow\\pi\_\{i\}\(\\bm\{z\}\_\{i\}\);

𝒵←𝒵∪\{𝒛i\}\\mathcal\{Z\}\\leftarrow\\mathcal\{Z\}\\cup\\\{\\bm\{z\}\_\{i\}\\\};

𝐲←𝐲∪\{yi\}\\mathbf\{y\}\\leftarrow\\mathbf\{y\}\\cup\\\{y\_\{i\}\\\};

𝚷←𝚷∪\{πi\}\\mathbf\{\\Pi\}\\leftarrow\\mathbf\{\\Pi\}\\cup\\\{\\pi\_\{i\}\\\};

endfor

for

i=N/2\+1i=N/2\+1to

NNdo

𝒛i←SampleAround​𝒕ref\\bm\{z\}\_\{i\}\\leftarrow\\textsc\{SampleAround\}\\bm\{t\}\_\{\\mathrm\{ref\}\};

yi←f​\(𝒛i\)y\_\{i\}\\leftarrow f\(\\bm\{z\}\_\{i\}\);

πi←πi​\(𝒛i\)\\pi\_\{i\}\\leftarrow\\pi\_\{i\}\(\\bm\{z\}\_\{i\}\);

𝒵←𝒵∪\{𝒛i\}\\mathcal\{Z\}\\leftarrow\\mathcal\{Z\}\\cup\\\{\\bm\{z\}\_\{i\}\\\};

𝐲←𝐲∪\{yi\}\\mathbf\{y\}\\leftarrow\\mathbf\{y\}\\cup\\\{y\_\{i\}\\\};

𝚷←𝚷∪\{πi\}\\mathbf\{\\Pi\}\\leftarrow\\mathbf\{\\Pi\}\\cup\\\{\\pi\_\{i\}\\\};

endfor

g∗←arg⁡min​∑z∈𝒵12​πi​\(g​\(z\)−𝒚\)2\+Ω​\(g\)g^\{\*\}\\leftarrow\\displaystyle\\arg\\min\\,\\sum\_\{z\\in\\mathcal\{Z\}\}\\frac\{1\}\{2\}\\pi\_\{i\}\(g\(z\)\-\\bm\{y\}\)^\{2\}\+\\Omega\(g\),

where

Ω​\(g\)\\Omega\(g\)is the ridge term\.

Return

ExtractFeatureImportance​\(g∗\)\\textsc\{ExtractFeatureImportance\}\(g^\{\*\}\)\.

## Appendix BAnalytical Results of the Proposed Model

Denote the objective function12​\(Z​𝜷\+β0​𝟏−𝒚\)⊺​Π​\(Z​𝜷\+β0​𝟏−𝒚\)\+λ2​𝜷⊺​𝑰d​𝜷\\frac\{1\}\{2\}\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\}\\bm\{1\}\-\\bm\{y\}\)^\{\\intercal\}\\Pi\(Z\{\\bm\{\\beta\}\}\+\{\\beta\_\{0\}\}\\bm\{1\}\-\\bm\{y\}\)\+\\frac\{\\lambda\}\{2\}\{\\bm\{\\beta\}\}^\{\\intercal\}\\bm\{I\}\_\{d\}\{\\bm\{\\beta\}\}asJJ\.

Differentiate the objective function𝒥​\(𝜷′\)\\mathcal\{J\}\(\\bm\{\\beta\}^\{\\prime\}\)w\.r\.t𝜷′\\bm\{\\beta\}^\{\\prime\}, we obtain

∇𝜷^′𝒥​\(𝜷^′\)=Z′⁣⊺​Π​\(Z′​𝜷^′−𝒚\)\+λ​𝑰d\+1​𝜷^′\.\\nabla\_\{\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\}\\mathcal\{J\}\(\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\)=Z^\{\\prime\\intercal\}\\Pi\(Z^\{\\prime\}\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\-\\bm\{y\}\)\+\\lambda\\bm\{I\}\_\{d\+1\}\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\.Set∇𝜷^′𝒥​\(𝜷^′\)=0\\nabla\_\{\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\}\\mathcal\{J\}\(\\hat\{\\bm\{\\beta\}\}^\{\\prime\}\)=0, we get

\(Z′⁣⊺​Π​Z′\+λ​𝑰d\+1\)​𝜷^′=Z′⁣⊺​Π​𝒚\.\(Z^\{\\prime\\intercal\}\\Pi Z^\{\\prime\}\+\\lambda\\bm\{I\}\_\{d\+1\}\)\\hat\{\\bm\{\\beta\}\}^\{\\prime\}=Z^\{\\prime\\intercal\}\\Pi\\bm\{y\}\.\(7\)i\.e\.,

\(Z⊺​Π​Z\+λ​𝑰dZ⊺​Π​𝟏𝟏⊺​Π​Z𝟏⊺​Π​𝟏\)​\(𝜷^β^0\)=\(Z⊺​Π​𝒚𝟏⊺​Π​𝒚\)\\left\(\\begin\{matrix\}Z^\{\\intercal\}\\Pi Z\+\\lambda\\bm\{I\}\_\{d\}&Z^\{\\intercal\}\\Pi\\bm\{1\}\\\\ \\bm\{1\}^\{\\intercal\}\\Pi Z&\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{1\}\\end\{matrix\}\\right\)\\left\(\\begin\{matrix\}\\hat\{\\bm\{\\beta\}\}\\\\ \\hat\{\\beta\}\_\{0\}\\end\{matrix\}\\right\)=\\left\(\\begin\{matrix\}Z^\{\\intercal\}\\Pi\\bm\{y\}\\\\ \\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}\\end\{matrix\}\\right\)\(8\)
Eliminating the intercept termβ^0\\hat\{\\beta\}\_\{0\}from this linear system,

\(Z⊺​Π​Z\+λ​𝑰d−Z⊺​Π​𝟏​Z⊺​Π​𝟏⊺∑inπi\)​𝜷^=Z⊺​Π​𝒚−1∑inπi​Z⊺​Π​𝟏𝟏⊺​Π​𝒚\.\(Z^\{\\intercal\}\\Pi Z\+\\lambda\\bm\{I\}\_\{d\}\-\\frac\{Z^\{\\intercal\}\\Pi\\bm\{1\}\{Z^\{\\intercal\}\\Pi\\bm\{1\}\}^\{\\intercal\}\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\)\\hat\{\\bm\{\\beta\}\}=Z^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\{Z^\{\\intercal\}\\Pi\\bm\{1\}\}\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}\.\(9\)
Thus, we get

𝜷^=\(A\+λ​𝑰d\)−1​𝒃,\\hat\{\\bm\{\\beta\}\}=\(A\+\\lambda\\bm\{I\}\_\{d\}\)^\{\-1\}\\bm\{b\},\(10\)whereA=\(Z⊺​Π​Z−Z⊺​Π​𝟏​\(Z⊺​Π​𝟏\)⊺∑i=1nπi\),A=\\left\(Z^\{\\intercal\}\\Pi Z\-\\frac\{Z^\{\\intercal\}\\Pi\\bm\{1\}\\left\(Z^\{\\intercal\}\\Pi\\bm\{1\}\\right\)^\{\\intercal\}\}\{\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\}\\right\),and𝒃=Z⊺​Π​𝒚−1∑i=1nπi​\(Z⊺​Π​𝟏\)​\(𝟏⊺​Π​𝒚\)\\bm\{b\}=Z^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\}\\left\(Z^\{\\intercal\}\\Pi\\bm\{1\}\\right\)\\left\(\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}\\right\)\.

## Appendix CFaithfulness Proof

### C\.1Restatement of Assumptions and Theorem

For convenience, we restate the assumptions and theorem\.

###### Assumption C\.1\(Sampled data geometric assumption\)\.

There exist constantsc,ε,ρ\>0c,\\varepsilon,\\rho\>0\(independent ofδ\\delta\) and core index setsJ0\(δ\)⊂I0\(δ\)J^\{\(\\delta\)\}\_\{0\}\\subset I^\{\(\\delta\)\}\_\{0\}andJ1\(δ\)⊂I1\(δ\)J^\{\(\\delta\)\}\_\{1\}\\subset I^\{\(\\delta\)\}\_\{1\}with\|J0\(δ\)\|=\|J1\(δ\)\|≥1\|J^\{\(\\delta\)\}\_\{0\}\|=\|J^\{\(\\delta\)\}\_\{1\}\|\\geq 1, such that‖𝒛i\(δ\)−𝒕ref‖2≤cδ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq\\frac\{c\}\{\\delta^\{2\}\}, fori∈J0\(δ\),i\\in J^\{\(\\delta\)\}\_\{0\},and‖𝒛i\(δ\)−𝒕j∗‖2≤cδ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\frac\{c\}\{\\delta^\{2\}\}, fori∈J1\(δ\)\.i\\in J^\{\(\\delta\)\}\_\{1\}\.Furthermore, for the non\-core points,ε≤‖𝒛i\(δ\)−𝒕ref‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq\\rho, fori∈I0\(δ\)∖J0\(δ\),i\\in I^\{\(\\delta\)\}\_\{0\}\\setminus J^\{\(\\delta\)\}\_\{0\},andε≤‖𝒛i\(δ\)−𝒕j∗‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\rho, fori∈I1\(δ\)∖J1\(δ\)\.i\\in I^\{\(\\delta\)\}\_\{1\}\\setminus J^\{\(\\delta\)\}\_\{1\}\.

###### Assumption C\.2\(Output structure\)\.

Assume there exist constantsBy<∞B\_\{y\}<\\inftyandcΔ\>0c\_\{\\Delta\}\>0independent ofδ\\deltasuch that for eachs∈\{0,1\}s\\in\\\{0,1\\\},\|yi\(δ\)−y¯s\(δ\)\|≤By,\\bigl\|y\_\{i\}^\{\(\\delta\)\}\-\\bar\{y\}\_\{s\}^\{\(\\delta\)\}\\bigr\|\\leq B\_\{y\},andΔ​\(δ\)≥cΔ\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\. Moreover, fors∈\{0,1\}s\\in\\\{0,1\\\}andi∈Js\(δ\)i\\in J\_\{s\}^\{\(\\delta\)\},\|yi\(δ\)−y¯s\(δ\)\|=O​\(δ−1\)\.\|y\_\{i\}^\{\(\\delta\)\}\-\\bar\{y\}\_\{s\}^\{\(\\delta\)\}\|=O\(\\delta^\{\-1\}\)\.

###### Theorem C\.3\(Theorem 3\.3, restated\)\.

Suppose the shift setup in \(5\) holds, the synthetic data matrixZ\(δ\)Z^\{\(\\delta\)\}satisfies Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), and the corresponding outputs,yi\(δ\)=f​\(𝐳i\(δ\)\)\{y\}\_\{i\}^\{\(\\delta\)\}=f\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\), satisfy Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)\. Let𝛃^\\hat\{\\bm\{\\beta\}\}denote the estimator in \(4\) and defineϵ=‖𝛃^ℛ‖2‖𝛃^𝒦‖2\\epsilon=\\dfrac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\. Then,

ϵ→0,as​δ→∞\.\{\\epsilon\}\\rightarrow 0,\\qquad\\text\{as\}\\,\\delta\\rightarrow\\infty\.

### C\.2Notation and Setup

Throughout the proof, all asymptotic notation is with respect toδ→∞\\delta\\to\\inftywithnnfixed\. The notationO​\(⋅\)O\(\\cdot\),o​\(⋅\)o\(\\cdot\), andΩ​\(⋅\)\\Omega\(\\cdot\)has its standard meaning with respect to this limit\. That is, forf​\(δ\)f\(\\delta\)andg​\(δ\)\>0g\(\\delta\)\>0

- •f=O​\(g\)f=O\(g\): there existsC\>0C\>0andδ0\>0\\delta\_\{0\}\>0such that\|f​\(δ\)\|≤C​g​\(δ\)\|f\(\\delta\)\|\\leq Cg\(\\delta\)for allδ≥δ0\\delta\\geq\\delta\_\{0\};
- •f=o​\(g\)f=o\(g\):f​\(δ\)/g​\(δ\)→0f\(\\delta\)/g\(\\delta\)\\to 0asδ→∞\\delta\\to\\infty;
- •f=Ω​\(g\)f=\\Omega\(g\): there existsc\>0c\>0andδ0\>0\\delta\_\{0\}\>0such thatf​\(δ\)≥c​g​\(δ\)f\(\\delta\)\\geq cg\(\\delta\)for allδ≥δ0\\delta\\geq\\delta\_\{0\}\.

We denote the synthetic sample set as𝒵\(δ\)=\{𝒛i\(δ\)\}i=1n\\mathcal\{Z\}^\{\(\\delta\)\}=\\\{\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\\\}\_\{i=1\}^\{n\}and the corresponding matrixZZ, and index setsI0\(δ\)I\_\{0\}^\{\(\\delta\)\}andI1\(δ\)I\_\{1\}^\{\(\\delta\)\}with\|I0\(δ\)\|=\|I1\(δ\)\|=n/2\|I\_\{0\}^\{\(\\delta\)\}\|=\|I\_\{1\}^\{\(\\delta\)\}\|=n/2containing samples around𝒕ref\\bm\{t\}\_\{\\mathrm\{ref\}\}and𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}, respectively\. Under Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), we denote core index setsJ0\(δ\)⊂I0\(δ\)J^\{\(\\delta\)\}\_\{0\}\\subset I^\{\(\\delta\)\}\_\{0\}andJ1\(δ\)⊂I1\(δ\)J^\{\(\\delta\)\}\_\{1\}\\subset I^\{\(\\delta\)\}\_\{1\}with\|J0\(δ\)\|=\|J1\(δ\)\|≥1\|J^\{\(\\delta\)\}\_\{0\}\|=\|J^\{\(\\delta\)\}\_\{1\}\|\\geq 1\. For the non\-core points, we define the non\-core index set as

T\(δ\)=\(I0\(δ\)∖J0\(δ\)\)∪\(I1\(δ\)∖J1\(δ\)\)\.T^\{\(\\delta\)\}=\\bigl\(I\_\{0\}^\{\(\\delta\)\}\\setminus J\_\{0\}^\{\(\\delta\)\}\\bigr\)\\cup\\bigl\(I\_\{1\}^\{\(\\delta\)\}\\setminus J\_\{1\}^\{\(\\delta\)\}\\bigr\)\.We consider a setting that a signal is triggered by a shift, denoted by𝒕j∗=𝒕ref\+δ​𝒖\\bm\{t\}\_\{j^\{\*\}\}=\\bm\{t\}\_\{\\mathrm\{ref\}\}\+\\delta\\bm\{u\}, where𝒖\\bm\{u\}is a unit vector andδ:=‖𝒕j∗−𝒕ref‖2\\delta:=\\\|\\bm\{t\}\_\{j^\{\*\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\. Let𝒦=\{ℓ:uℓ≠0\}\\mathcal\{K\}=\\\{\\ell:\{u\}\_\{\\ell\}\\neq 0\\\}denote the set of shifted features with\|𝒦\|=k\|\\mathcal\{K\}\|=k, andℛ=\{1,…,d\}∖𝒦\\mathcal\{R\}=\\\{1,\\dots,d\\\}\\setminus\\mathcal\{K\}, we split the feature space

ℝd=ℝ𝒦⊕ℝℛ,\\mathbb\{R\}^\{d\}=\\mathbb\{R\}^\{\\mathcal\{K\}\}\\oplus\\mathbb\{R\}^\{\\mathcal\{R\}\},whereℝ𝒦\\mathbb\{R\}^\{\\mathcal\{K\}\}is the subspace of features leading𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}to a signal andℝℛ\\mathbb\{R\}^\{\\mathcal\{R\}\}is its unshifted complement of𝒕j∗\\bm\{t\}\_\{j^\{\*\}\}\. Thus, for a vector𝒃∈ℝd\\bm\{b\}\\in\\mathbb\{R\}^\{d\}, we write𝒃𝒦\\bm\{b\}\_\{\\mathcal\{K\}\}and𝒃ℛ\\bm\{b\}\_\{\\mathcal\{R\}\}for its sub\-vectors indexed by𝒦\\mathcal\{K\}andℛ\\mathcal\{R\}, respectively\.

Similarly, for a matrixM∈ℝd×dM\\in\\mathbb\{R\}^\{d\\times d\}, we writeM𝒦​𝒦,M𝒦​ℛ,Mℛ​𝒦M\_\{\\mathcal\{KK\}\},M\_\{\\mathcal\{KR\}\},M\_\{\\mathcal\{RK\}\}, andMℛ​ℛM\_\{\\mathcal\{RR\}\}for the corresponding block sub\-matrices\. Adapting the split, we write the linear system \([9](https://arxiv.org/html/2608.10587#A2.E9)\) as

M​𝜷^\\displaystyle M\\hat\{\\bm\{\\beta\}\}=𝒃,\\displaystyle=\\bm\{b\},\[M𝒦​𝒦M𝒦​ℛMℛ​𝒦Mℛ​ℛ\]​\[𝜷^𝒦𝜷^ℛ\]\\displaystyle\\begin\{bmatrix\}M\_\{\\mathcal\{KK\}\}&M\_\{\\mathcal\{KR\}\}\\\\ M\_\{\\mathcal\{RK\}\}&M\_\{\\mathcal\{RR\}\}\\end\{bmatrix\}\\begin\{bmatrix\}\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\\ \\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}=\[𝒃𝒦𝒃ℛ\],\\displaystyle=\\begin\{bmatrix\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\\\ \\bm\{b\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\},whereM:=A\+λ​𝑰=\(Z⊺​Π​Z−Z⊺​Π​𝟏​Z⊺​Π​𝟏⊺∑inπi\)\+λ​𝑰M:=A\+\\lambda\\bm\{I\}=\(Z^\{\\intercal\}\\Pi Z\-\\frac\{Z^\{\\intercal\}\\Pi\\bm\{1\}\{Z^\{\\intercal\}\\Pi\\bm\{1\}\}^\{\\intercal\}\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\)\+\\lambda\\bm\{I\},M𝒦​𝒦∈ℝk×kM\_\{\\mathcal\{KK\}\}\\in\\mathbb\{R\}^\{k\\times k\},Mℛ​ℛ∈ℝ\(d−k\)×\(d−k\)M\_\{\\mathcal\{RR\}\}\\in\\mathbb\{R\}^\{\(d\-k\)\\times\(d\-k\)\},M𝒦​ℛ∈ℝk×\(d−k\)M\_\{\\mathcal\{KR\}\}\\in\\mathbb\{R\}^\{k\\times\(d\-k\)\}, and𝒃=Z⊺​Π​𝒚−1∑inπi​Z⊺​Π​𝟏𝟏⊺​Π​𝒚\\bm\{b\}=Z^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\{Z^\{\\intercal\}\\Pi\\bm\{1\}\}\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}\. Therefore, we obtain

ϵ=‖𝜷^ℛ‖2‖𝜷^𝒦‖2,and​\[𝜷^𝒦𝜷^ℛ\]=\[M𝒦​𝒦M𝒦​ℛMℛ​𝒦Mℛ​ℛ\]−1​\[𝒃𝒦𝒃ℛ\]\\epsilon=\\frac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\},\\;\\text\{and\}\\;\\begin\{aligned\} \\begin\{bmatrix\}\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\\ \\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}&=\\begin\{bmatrix\}M\_\{\\mathcal\{KK\}\}&M\_\{\\mathcal\{KR\}\}\\\\ M\_\{\\mathcal\{RK\}\}&M\_\{\\mathcal\{RR\}\}\\end\{bmatrix\}^\{\-1\}\\begin\{bmatrix\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\\\ \\bm\{b\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}\\end\{aligned\}
Letsπ:=∑i=1nπis\_\{\\pi\}:=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}denote the sum of the weights\. We write

s0:=∑i∈I0\(δ\)πi,s1:=∑i∈I1\(δ\)πi\.s\_\{0\}:=\\sum\_\{i\\in I\_\{0\}^\{\(\\delta\)\}\}\\pi\_\{i\},\\qquad s\_\{1\}:=\\sum\_\{i\\in I\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\.
The weighted feature mean𝝁\\bm\{\\mu\}and weighted output meanμy\\mu\_\{y\}are defined as

𝝁=\(Z\(δ\)\)⊺​Π​𝟏sπ,μy=∑i=1nπi​yi\(δ\)sπ\.\\bm\{\\mu\}=\\frac\{\(Z^\{\(\\delta\)\}\)^\{\\intercal\}\\Pi\\bm\{1\}\}\{s\_\{\\pi\}\},\\qquad\\mu\_\{y\}=\\frac\{\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}y\_\{i\}^\{\(\\delta\)\}\}\{s\_\{\\pi\}\}\.
We can write

𝒃=\(Z\(δ\)\)⊺​Π​𝒚−1∑inπi​\(Z\(δ\)\)⊺​Π​𝟏𝟏⊺​Π​𝒚=∑i=1nπi​\(𝒛i\(δ\)−𝝁\)​\(yi\(δ\)−μy\)\.\\bm\{b\}=\{\(Z^\{\(\\delta\)\}\)\}^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\{\{\(Z^\{\(\\delta\)\}\)\}^\{\\intercal\}\\Pi\\bm\{1\}\}\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\)\(\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\)\.
For the norm in the proof,∥⋅∥2\\\|\\cdot\\\|\_\{2\}denotes two different objects depending on context\. For a vector𝒙∈ℝp\\bm\{x\}\\in\\mathbb\{R\}^\{p\},‖𝒙‖2:=\(∑j=1pxj2\)1/2\\\|\\bm\{x\}\\\|\_\{2\}:=\(\\sum\_\{j=1\}^\{p\}x\_\{j\}^\{2\}\)^\{1/2\}denotes the Euclidean norm\. For a matrixA∈ℝp×qA\\in\\mathbb\{R\}^\{p\\times q\},‖A‖2:=sup‖𝒙‖2=1‖A​𝒙‖2\\\|A\\\|\_\{2\}:=\\sup\_\{\\\|\\bm\{x\}\\\|\_\{2\}=1\}\\\|A\\bm\{x\}\\\|\_\{2\}denotes the spectral norm \(largest singular value\)\. For a symmetric positive semidefinite matrixAA,‖A‖2\\\|A\\\|\_\{2\}equals its largest eigenvalue\. We writeA⪰BA\\succeq Bto meanA−BA\-Bis positive semidefinite\.

### C\.3Boundingϵ\\epsilonbyδ\\delta

#### C\.3\.1Schur\-Complement Reduction

To prove Theorem[3\.3](https://arxiv.org/html/2608.10587#S3.Thmtheorem3), we first need to get the expressions of𝜷^𝒦\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}and𝜷^ℛ\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}by the Schur complement\. Specifically, given a symmetric SPD matrixMM,

M=\[PQQ⊺R\],M=\\begin\{bmatrix\}P&Q\\\\ Q^\{\\intercal\}&R\\end\{bmatrix\},we can write the inversion as

M−1=\[\(P−Q​R−1​Q⊺\)−1−\(P−Q​R−1​Q⊺\)−1​Q​R−1−R−1​Q⊺​\(P−Q​R−1​Q⊺\)−1R−1\+R−1​Q⊺​\(P−Q​R−1​Q⊺\)−1​Q​R−1\]\.M^\{\-1\}=\\begin\{bmatrix\}\(P\-QR^\{\-1\}Q^\{\\intercal\}\)^\{\-1\}&\-\(P\-QR^\{\-1\}Q^\{\\intercal\}\)^\{\-1\}QR^\{\-1\}\\\\ \-R^\{\-1\}Q^\{\\intercal\}\(P\-QR^\{\-1\}Q^\{\\intercal\}\)^\{\-1\}&R^\{\-1\}\+R^\{\-1\}Q^\{\\intercal\}\(P\-QR^\{\-1\}Q^\{\\intercal\}\)^\{\-1\}QR^\{\-1\}\\end\{bmatrix\}\.Then, we can write the estimator

\[𝜷^𝒦𝜷^ℛ\]\\displaystyle\\begin\{bmatrix\}\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\\ \\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}=\[M𝒦​𝒦M𝒦​ℛMℛ​𝒦Mℛ​ℛ\]−1​\[𝒃𝒦𝒃ℛ\]\\displaystyle=\\begin\{bmatrix\}M\_\{\\mathcal\{KK\}\}&M\_\{\\mathcal\{KR\}\}\\\\ M\_\{\\mathcal\{RK\}\}&M\_\{\\mathcal\{RR\}\}\\end\{bmatrix\}^\{\-1\}\\begin\{bmatrix\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\\\ \\bm\{b\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}=\[S−1−S−1​M𝒦​ℛ​Mℛ​ℛ−1−Mℛ​ℛ−1​M𝒦​ℛ⊺​S−1Mℛ​ℛ−1\+Mℛ​ℛ−1​M𝒦​ℛ⊺​S−1​M𝒦​ℛ​Mℛ​ℛ−1\]​\[𝒃𝒦𝒃ℛ\]\\displaystyle=\\begin\{bmatrix\}S^\{\-1\}&\-S^\{\-1\}M\_\{\\mathcal\{KR\}\}M\_\{\\mathcal\{RR\}\}^\{\-1\}\\\\ \-M\_\{\\mathcal\{RR\}\}^\{\-1\}M\_\{\\mathcal\{KR\}\}^\{\\intercal\}S^\{\-1\}&M\_\{\\mathcal\{RR\}\}^\{\-1\}\+M\_\{\\mathcal\{RR\}\}^\{\-1\}M\_\{\\mathcal\{KR\}\}^\{\\intercal\}S^\{\-1\}M\_\{\\mathcal\{KR\}\}M\_\{\\mathcal\{RR\}\}^\{\-1\}\\end\{bmatrix\}\\begin\{bmatrix\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\\\ \\bm\{b\}\_\{\\mathcal\{R\}\}\\end\{bmatrix\}whereS=M𝒦​𝒦−M𝒦​ℛ​Mℛ​ℛ−1​M𝒦​ℛ⊺S=M\_\{\\mathcal\{KK\}\}\-M\_\{\\mathcal\{KR\}\}M\_\{\\mathcal\{RR\}\}^\{\-1\}M\_\{\\mathcal\{KR\}\}^\{\\intercal\}is the Schur complement ofMℛ​ℛM\_\{\\mathcal\{RR\}\}inMM\. Thus,

𝜷^𝒦=S−1​𝒃𝒦−S−1​M𝒦​ℛ​Mℛ​ℛ−1​𝒃ℛ=S−1​\(𝒃𝒦−M𝒦​ℛ​Mℛ​ℛ−1​𝒃ℛ\)\\displaystyle\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}=S^\{\-1\}\\bm\{b\}\_\{\\mathcal\{K\}\}\-S^\{\-1\}M\_\{\\mathcal\{KR\}\}M^\{\-1\}\_\{\\mathcal\{RR\}\}\\bm\{b\}\_\{\\mathcal\{R\}\}=S^\{\-1\}\\left\(\\bm\{b\}\_\{\\mathcal\{K\}\}\-M\_\{\\mathcal\{KR\}\}M^\{\-1\}\_\{\\mathcal\{RR\}\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)𝜷^ℛ=−Mℛ​ℛ−1​M𝒦​ℛ⊺​S−1​𝒃𝒦\+\(Mℛ​ℛ−1\+Mℛ​ℛ−1​M𝒦​ℛ⊺​S−1​M𝒦​ℛ​Mℛ​ℛ−1\)​𝒃ℛ=Mℛ​ℛ−1​\(𝒃ℛ−M𝒦​ℛ⊺​𝜷^𝒦\)\\displaystyle\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}=\-M\_\{\\mathcal\{RR\}\}^\{\-1\}M\_\{\\mathcal\{KR\}\}^\{\\intercal\}S^\{\-1\}\\bm\{b\}\_\{\\mathcal\{K\}\}\+\(M\_\{\\mathcal\{RR\}\}^\{\-1\}\+M\_\{\\mathcal\{RR\}\}^\{\-1\}M\_\{\\mathcal\{KR\}\}^\{\\intercal\}S^\{\-1\}M\_\{\\mathcal\{KR\}\}M\_\{\\mathcal\{RR\}\}^\{\-1\}\)\\bm\{b\}\_\{\\mathcal\{R\}\}=M\_\{\\mathcal\{RR\}\}^\{\-1\}\\left\(\\bm\{b\}\_\{\\mathcal\{R\}\}\-M\_\{\\mathcal\{KR\}\}^\{\\intercal\}\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\right\)
Thus, we can writeϵ\\epsilonas,

ϵ\\displaystyle\\epsilon=‖𝜷^ℛ‖2‖𝜷^𝒦‖2\\displaystyle=\\frac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}=‖\(Mℛ​ℛ\)−1​\(𝒃ℛ−\(M𝒦​ℛ\)⊺​𝜷^𝒦\)‖2‖𝜷^𝒦‖2\\displaystyle=\\frac\{\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\left\(\\bm\{b\}\_\{\\mathcal\{R\}\}\-\(M\_\{\\mathcal\{KR\}\}\)^\{\\intercal\}\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\right\)\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}≤‖\(Mℛ​ℛ\)−1‖2​\(‖𝒃ℛ‖2\+‖M𝒦​ℛ‖2​‖𝜷^𝒦‖2\)‖𝜷^𝒦‖2\\displaystyle\\leq\\frac\{\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\\|\_\{2\}\(\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\+\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\)\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}=‖\(Mℛ​ℛ\)−1‖2​\(‖𝒃ℛ‖2‖𝜷^𝒦‖2\+‖M𝒦​ℛ‖2\)\.\\displaystyle=\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\\|\_\{2\}\\left\(\\frac\{\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\+\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\right\)\.SinceMℛ​ℛ⪰λ​𝑰M\_\{\\mathcal\{RR\}\}\\succeq\\lambda\\bm\{I\}, it follows that,

ϵ≤1λ​\(‖𝒃ℛ‖2‖𝜷^𝒦‖2\+‖M𝒦​ℛ‖2\)\.\\epsilon\\leq\\frac\{1\}\{\\lambda\}\\left\(\\frac\{\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\+\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\right\)\.Accordingly, the proof is divided into three parts\. First, we show that‖𝒃ℛ‖2\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}is small under the BREAD weighting\. Second, we show‖M𝒦​ℛ‖2\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}is asymptotically small\. Third, we show the lower bound on‖𝜷^𝒦‖2\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\. These imply thatϵ→0,\\epsilon\\rightarrow 0,asδ→∞\\delta\\rightarrow\\infty\.

#### C\.3\.2Preliminary Bounds

To boundϵ\\epsilonbyδ\\delta, we need the following lemmas that repeatedly used throughout\.

###### Lemma C\.4\(BREAD weight bounds\)\.

Under Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), there exist constantscB\>0c\_\{B\}\>0,KB<∞K\_\{B\}<\\infty, andC0\>0C\_\{0\}\>0, independent ofδ\\delta, such that for all sufficiently largeδ\\delta:

1. 1\.for everyi∈T\(δ\):=\(I0\(δ\)∖J0\(δ\)\)∪\(I1\(δ\)∖J1\(δ\)\)i\\in T^\{\(\\delta\)\}:=\(I\_\{0\}^\{\(\\delta\)\}\\setminus J\_\{0\}^\{\(\\delta\)\}\)\\cup\(I\_\{1\}^\{\(\\delta\)\}\\setminus J\_\{1\}^\{\(\\delta\)\}\), πi≤KB​e−cB​δ\.\\pi\_\{i\}\\leq K\_\{B\}e^\{\-c\_\{B\}\\delta\}\.
2. 2\.for everyi∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, e−C0/δ≤πi≤1;e^\{\-C\_\{0\}/\\delta\}\\leq\\pi\_\{i\}\\leq 1;

Then

sπ=∑i∈J0\(δ\)∪J1\(δ\)πi\+O​\(e−cB​δ\)=\|J0\(δ\)\|\+\|J1\(δ\)\|\+O​\(δ−1\)\+O​\(e−cB​δ\),s\_\{\\pi\}=\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\+O\(e^\{\-c\_\{B\}\\delta\}\)=\|J\_\{0\}^\{\(\\delta\)\}\|\+\|J\_\{1\}^\{\(\\delta\)\}\|\+O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\),and1/sπ=O​\(1\)1/s\_\{\\pi\}=O\(1\)\.

###### Proof\.

Recall the BREAD weights

πi=exp⁡\(−‖𝒛i\(δ\)−𝒕j∗‖2​‖𝒛i\(δ\)−𝒕ref‖2σ2\)\.\\pi\_\{i\}=\\exp\\\!\\left\(\-\\frac\{\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\}\{\\sigma^\{2\}\}\\right\)\.
\(i\) Fori∈I1\(δ\)∖J1\(δ\)i\\in I\_\{1\}^\{\(\\delta\)\}\\setminus J\_\{1\}^\{\(\\delta\)\}, Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)gives

ε≤‖𝒛i\(δ\)−𝒕j∗‖2≤ρ\.\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\rho\.Since𝒕j∗=𝒕ref\+δ​𝒖\\bm\{t\}\_\{j^\{\*\}\}=\\bm\{t\}\_\{\\mathrm\{ref\}\}\+\\delta\\bm\{u\},

𝒛i\(δ\)−𝒕ref=δ​𝒖\+\(𝒛i\(δ\)−𝒕j∗\)\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}=\\delta\\bm\{u\}\+\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\)Hence, by triangle inequality,

‖𝒛i\(δ\)−𝒕ref‖2=‖δ​𝒖\+\(𝒛i\(δ\)−𝒕j∗\)‖2≥δ−‖𝒛i\(δ\)−𝒕j∗‖2≥δ−ρ\.\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}=\\\|\\delta\\bm\{u\}\+\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\)\\\|\_\{2\}\\geq\\delta\-\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\geq\\delta\-\\rho\.
Hence, for allδ\>ρ\\delta\>\\rho,

πi≤exp⁡\(−ε​\(δ−ρ\)σ2\)\.\\pi\_\{i\}\\leq\\exp\\\!\\left\(\-\\frac\{\\varepsilon\(\\delta\-\\rho\)\}\{\\sigma^\{2\}\}\\right\)\.
Fori∈I0\(δ\)∖J0\(δ\)i\\in I\_\{0\}^\{\(\\delta\)\}\\setminus J\_\{0\}^\{\(\\delta\)\}, similarly,

‖𝒛i\(δ\)−𝒕ref‖2≥ε,‖𝒛i\(δ\)−𝒕j∗‖2≥δ−ρ\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\geq\\varepsilon,\\qquad\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\geq\\delta\-\\rho
Thus, the same bound holds\. Hence there existKB<∞K\_\{B\}<\\inftyandcB\>0c\_\{B\}\>0, such that

πi≤KB​e−cB​δ∀i∈T\(δ\)\.\\pi\_\{i\}\\leq K\_\{B\}e^\{\-c\_\{B\}\\delta\}\\qquad\\forall i\\in T^\{\(\\delta\)\}\.
\(ii\) Fori∈J0\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}, then‖𝒛i\(δ\)−𝒕ref‖2≤c/δ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq c/\\delta^\{2\}, it holds‖𝒛i\(δ\)−𝒕j∗‖2=‖δ​𝒖−\(𝒛i\(δ\)−𝒕ref\)‖2≤δ\+cδ2\.\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}=\\\|\\delta\\bm\{u\}\-\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\)\\\|\_\{2\}\\leq\\delta\+\\frac\{c\}\{\\delta^\{2\}\}\.Therefore

‖𝒛i\(δ\)−𝒕ref‖2​‖𝒛i\(δ\)−𝒕j∗‖2≤cδ\+c2δ4,\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\frac\{c\}\{\\delta\}\+\\frac\{c^\{2\}\}\{\\delta^\{4\}\},Then,

πi≥exp⁡\(−c/δ\+c2/δ4σ2\),i∈J0\(δ\)\.\\pi\_\{i\}\\geq\\exp\\left\(\-\\frac\{c/\\delta\+c^\{2\}/\\delta^\{4\}\}\{\\sigma^\{2\}\}\\right\),\\qquad i\\in J\_\{0\}^\{\(\\delta\)\}\.Hence,

1−πi=O​\(δ−1\)1\-\\pi\_\{i\}=O\(\\delta^\{\-1\}\)The same estimate holds fori∈J1\(δ\)i\\in J\_\{1\}^\{\(\\delta\)\}\. In particular, core weights are bounded away from0uniformly for largeδ\\delta\. Sincesπ=∑i=1nπis\_\{\\pi\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\},sπ=∑i∈J0\(δ\)∪J1\(δ\)πi\+O​\(e−cB​δ\)s\_\{\\pi\}=\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\+O\(e^\{\-c\_\{B\}\\delta\}\)\. Sinceπi=1\+O​\(δ−1\)\\pi\_\{i\}=1\+O\(\\delta^\{\-1\}\)andnnis fixed, we obtain

sπ=\|J0\(δ\)\|\+\|J1\(δ\)\|\+O​\(δ−1\)\+O​\(e−cB​δ\)\.s\_\{\\pi\}=\|J\_\{0\}^\{\(\\delta\)\}\|\+\|J\_\{1\}^\{\(\\delta\)\}\|\+O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\)\.Since\|J0\(δ\)\|\+\|J1\(δ\)\|≥2\|J\_\{0\}^\{\(\\delta\)\}\|\+\|J\_\{1\}^\{\(\\delta\)\}\|\\geq 2, we havesπ≥2s\_\{\\pi\}\\geq 2for largeδ\\delta, so1/sπ=O​\(1\)1/s\_\{\\pi\}=O\(1\)\. ∎

###### Lemma C\.5\(Bounds of weighted sum ratio\)\.

Recall

s0=∑i∈I0\(δ\)πi,s1=∑i∈I1\(δ\)πi,sπ=s0\+s1\.s\_\{0\}=\\sum\_\{i\\in I\_\{0\}^\{\(\\delta\)\}\}\\pi\_\{i\},\\qquad s\_\{1\}=\\sum\_\{i\\in I\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\},\\qquad s\_\{\\pi\}=s\_\{0\}\+s\_\{1\}\.In particular, there exists a constantc0\>0c\_\{0\}\>0, independent ofδ\\delta, such that

s0​s1\(sπ\)2≥c0\.\\frac\{s\_\{0\}s\_\{1\}\}\{\(s\_\{\\pi\}\)^\{2\}\}\\geq c\_\{0\}\.

###### Proof\.

By Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(ii\), each core set contains at least one point and each core weight is bounded below bye−C0/δe^\{\-C\_\{0\}/\\delta\}\. Hence, for all sufficiently largeδ\\delta,

s0≥e−C0/δ,s1≥e−C0/δ,sπ≤n\.s\_\{0\}\\geq\{e^\{\-C\_\{0\}/\\delta\}\},\\qquad s\_\{1\}\\geq e^\{\-C\_\{0\}/\\delta\},\\qquad s\_\{\\pi\}\\leq n\.Hence,

s0​s1\(sπ\)2≥e−2​C0/δn2→1n2≥0,\\frac\{s\_\{0\}s\_\{1\}\}\{\(s\_\{\\pi\}\)^\{2\}\}\\geq\\frac\{e^\{\-2C\_\{0\}/\\delta\}\}\{n^\{2\}\}\\rightarrow\\frac\{1\}\{n^\{2\}\}\\geq 0,Takingc0c\_\{0\}for all largeδ\\deltagives the results\. ∎

###### Lemma C\.6\(Pointwise bounds\)\.

Under Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), for all sufficiently largeδ\\delta,

1. \(i\)ℛ\\mathcal\{R\}\-block, core points\. Fori∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, ‖𝒛i,ℛ\(δ\)−𝒕ref,ℛ‖2=O​\(δ−2\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\\|\_\{2\}=O\(\\delta^\{\-2\}\)\.
2. \(ii\)ℛ\\mathcal\{R\}\-block, tail points\. Fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, ‖𝒛i,ℛ\(δ\)−𝒕ref,ℛ‖2=O​\(1\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\\|\_\{2\}=O\(1\)\.

###### Proof\.

1. \(i\)Fori∈J0\(δ\)∪J1\(δ\),i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\},since𝒖ℛ=0\\bm\{u\}\_\{\\mathcal\{R\}\}=0,𝒕j∗,ℛ=𝒕ref,ℛ\\bm\{t\}\_\{j^\{\*\},\\mathcal\{R\}\}=\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\. Fori∈J0\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}, Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)gives‖𝒛i\(δ\)−𝒕ref‖2≤c/δ2\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq c/\\delta^\{2\}\. Fori∈J1\(δ\)i\\in J\_\{1\}^\{\(\\delta\)\}, the same follows\. Thus, ‖𝒛i,ℛ\(δ\)−𝒕ref,ℛ‖2=O​\(δ−2\)\.\\left\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\right\\\|\_\{2\}=O\(\\delta^\{\-2\}\)\.
2. \(ii\)Fori∈T\(δ\),i\\in T^\{\(\\delta\)\},Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)givesε≤‖𝒛i\(δ\)−𝒕ref‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\}\}\\\|\_\{2\}\\leq\\rho, fori∈I0\(δ\)∖J0\(δ\),i\\in I^\{\(\\delta\)\}\_\{0\}\\setminus J^\{\(\\delta\)\}\_\{0\},andε≤‖𝒛i\(δ\)−𝒕j∗‖2≤ρ\\varepsilon\\leq\\\|\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{j^\{\*\}\}\\\|\_\{2\}\\leq\\rho, fori∈I1\(δ\)∖J1\(δ\)\.i\\in I^\{\(\\delta\)\}\_\{1\}\\setminus J^\{\(\\delta\)\}\_\{1\}\. Thus, theℛ\\mathcal\{R\}\-block is bounded byρ=O​\(1\)\\rho=O\(1\)\. ∎

###### Lemma C\.7\(Bounds for pointwise weighted mean\)\.

Under Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1),

1. \(i\)𝒦\\mathcal\{K\}\-block, all points\. For alli∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\}, ‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\(\\delta\)\.
2. \(ii\)ℛ\\mathcal\{R\}\-block, all points\. 𝝁ℛ=𝒕ref,ℛ\+O​\(δ−2\)\+O​\(KB​e−cB​δ\)\.\\bm\{\\mu\}\_\{\\mathcal\{R\}\}=\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\+O\(\\delta^\{\-2\}\)\+O\(K\_\{B\}e^\{\-c\_\{B\}\\delta\}\)\.Furthermore, fori∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, ‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2=O​\(δ−2\)\+O​\(e−cB​δ\),\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\(\\delta^\{\-2\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\),and fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, ‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2=O​\(1\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\(1\)\.

###### Proof\.

1. \(i\)For𝒛i,𝒦\(δ\)\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}, Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)gives either𝒛i,𝒦\(δ\)=𝒕ref,𝒦\+O​\(1\)\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+O\(1\)or𝒛i,𝒦\(δ\)=𝒕ref,𝒦\+δ​𝒖𝒦\+O​\(1\)\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+\\delta\\bm\{u\}\_\{\\mathcal\{K\}\}\+O\(1\)\. Thus𝒛i,𝒦\(δ\)\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}lies at distanceO​\(δ\)O\(\\delta\)from𝒕ref,𝒦\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\. Since𝝁𝒦\\bm\{\\mu\}\_\{\\mathcal\{K\}\}is a weighted average of𝒛i,𝒦\(δ\)\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}’s, it also satisfies𝝁𝒦=𝒕ref,𝒦\+O​\(δ\)\\bm\{\\mu\}\_\{\\mathcal\{K\}\}=\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+O\(\\delta\)\. Hence, ‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\(\\delta\)\.
2. \(ii\)Write 𝝁ℛ−𝒕ref,ℛ=1sπ​∑inπi​\(𝒛i,ℛ\(δ\)−𝒕ref,ℛ\)\.\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}=\\frac\{1\}\{s\_\{\\pi\}\}\\sum\_\{i\}^\{n\}\\pi\_\{i\}\\left\(\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\right\)\.Fori∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, by Lemmas[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)and[C\.6](https://arxiv.org/html/2608.10587#A3.Thmtheorem6)\(i\) ‖𝝁ℛ−𝒕ref,ℛ‖2=‖1sπ​∑inπi​\(𝒛i,ℛ\(δ\)−𝒕ref,ℛ\)‖2=O​\(δ−2\)​∑i∈J0\(δ\)∪J1\(δ\)πi=O​\(δ−2\)\.\\left\\\|\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\right\\\|\_\{2\}=\\left\\\|\\frac\{1\}\{s\_\{\\pi\}\}\\sum\_\{i\}^\{n\}\\pi\_\{i\}\\left\(\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\right\)\\right\\\|\_\{2\}=O\(\\delta^\{\-2\}\)\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}=O\(\\delta^\{\-2\}\)\.Similarly, fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, Lemmas[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)and[C\.6](https://arxiv.org/html/2608.10587#A3.Thmtheorem6)\(ii\) give ‖𝝁ℛ−𝒕ref,ℛ‖2≤ρsπ​∑i∈T\(δ\)πi=O​\(e−cB​δ\)\.∎\\left\\\|\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\-\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{R\}\}\\right\\\|\_\{2\}\\leq\\frac\{\\rho\}\{s\_\{\\pi\}\}\\sum\_\{i\\in T^\{\(\\delta\)\}\}\\pi\_\{i\}=O\(e^\{\-c\_\{B\}\\delta\}\)\.\\qed

###### Lemma C\.8\(Bounds for weighted outputs\)\.

Under Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2),

1. \(i\)maxi⁡\|yi\(δ\)−μy\|≤Δ​\(δ\)\+2​By=O​\(Δ​\(δ\)\+1\)\.\\max\_\{i\}\|y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\leq\\Delta\(\\delta\)\+2B\_\{y\}=O\(\\Delta\(\\delta\)\+1\)\.
2. \(ii\)Moreover, μy=y¯0\(δ\)\+s1sπ​Δ​\(δ\)\+O​\(δ−1\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\.\\mu\_\{y\}=\\bar\{y\}\_\{0\}^\{\(\\delta\)\}\+\\frac\{s\_\{1\}\}\{s\_\{\\pi\}\}\\Delta\(\\delta\)\+O\(\\delta^\{\-1\}\)\+O\\bigl\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\bigr\)\.

###### Proof\.

1. \(i\)Under Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2), fors∈\{0,1\}s\\in\\\{0,1\\\}, \|yi\(δ\)−y¯s\(δ\)\|≤By,i∈Is\(δ\),\|y^\{\(\\delta\)\}\_\{i\}\-\\bar\{y\}^\{\(\\delta\)\}\_\{s\}\|\\leq B\_\{y\},\\qquad i\\in I^\{\(\\delta\)\}\_\{s\},Hence, everyyi\(δ\)y^\{\(\\delta\)\}\_\{i\}lies in \[y¯0\(δ\)−By,y¯1\(δ\)\+By\],\[\\bar\{y\}^\{\(\\delta\)\}\_\{0\}\-B\_\{y\},\\bar\{y\}^\{\(\\delta\)\}\_\{1\}\+B\_\{y\}\],the end points are up to swapping if needed\. Sinceμy\\mu\_\{y\}is a convex combination of theyi\(δ\)y^\{\(\\delta\)\}\_\{i\}’s, it also lies in this interval\. Thus, \|yi\(δ\)−μy\|≤\|y¯0\(δ\)−y¯1\(δ\)\|\+2​By=Δ​\(δ\)\+2​By,\|y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\leq\|\\bar\{y\}^\{\(\\delta\)\}\_\{0\}\-\\bar\{y\}^\{\(\\delta\)\}\_\{1\}\|\+2B\_\{y\}=\\Delta\(\\delta\)\+2B\_\{y\},and max⁡\|yi\(δ\)−μy\|≤Δ​\(δ\)\+2​By\.\\max\|y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\leq\\Delta\(\\delta\)\+2B\_\{y\}\.
2. \(ii\)Fori∈J0\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}, Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)givesyi\(δ\)=y¯0\(δ\)\+O​\(δ−1\)y^\{\(\\delta\)\}\_\{i\}=\\bar\{y\}\_\{0\}^\{\(\\delta\)\}\+O\(\\delta^\{\-1\}\), and fori∈J1\(δ\)i\\in J\_\{1\}^\{\(\\delta\)\},yi\(δ\)=y¯1\(δ\)\+O​\(δ−1\)=y¯0\(δ\)\+Δ​\(δ\)\+O​\(δ−1\)y^\{\(\\delta\)\}\_\{i\}=\\bar\{y\}\_\{1\}^\{\(\\delta\)\}\+O\(\\delta^\{\-1\}\)=\\bar\{y\}\_\{0\}^\{\(\\delta\)\}\+\\Delta\(\\delta\)\+O\(\\delta^\{\-1\}\)\. Fori∈T\(δ\)i\\in T^\{\(\\delta\)\},\|yi\(δ\)−y¯s\(δ\)\|≤By\|y^\{\(\\delta\)\}\_\{i\}\-\\bar\{y\}\_\{s\}^\{\(\\delta\)\}\|\\leq B\_\{y\}andπi=O​\(e−cB​δ\)\\pi\_\{i\}=O\(e^\{\-c\_\{B\}\\delta\}\)by Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(i\)\. Splitting the weighted sum and using1/sπ=O​\(1\)1/s\_\{\\pi\}=O\(1\), μy=y¯0\(δ\)\+s1sπ​Δ​\(δ\)\+O​\(δ−1\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\.∎\\mu\_\{y\}=\\bar\{y\}\_\{0\}^\{\(\\delta\)\}\+\\frac\{s\_\{1\}\}\{s\_\{\\pi\}\}\\Delta\(\\delta\)\+O\(\\delta^\{\-1\}\)\+O\\bigl\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\bigr\)\.\\qed

#### C\.3\.3Boundingbℛ\{b\}\_\{\\mathcal\{R\}\}

We show that theℛ\\mathcal\{R\}\-block of𝒃\\bm\{b\}is asymptotically negligible\. Intuitively, this follows because theℛ\\mathcal\{R\}\-features of the core points fluctuate only at orderO​\(δ−2\)O\(\\delta^\{\-2\}\), while the non\-core points are exponentially small weighted\.

###### Lemma C\.9\(Upper bound on𝒃ℛ\\bm\{b\}\_\{\\mathcal\{R\}\}\)\.

Under Assumptions[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)and[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2),

‖𝒃ℛ‖2=O​\(\(Δ​\(δ\)\+1\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\),δ→∞\.\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\\big\(\(\\Delta\(\\delta\)\+1\)\\delta^\{\-2\}\\big\)\+O\\big\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\big\),\\qquad\\delta\\to\\infty\.

###### Proof\.

Since

𝒃=\(Z\(δ\)\)⊺​Π​𝒚−1∑inπi​\(Z\(δ\)\)⊺​Π​𝟏𝟏⊺​Π​𝒚=∑i=1nπi​\(𝒛i\(δ\)−𝝁\)​\(yi\(δ\)−μy\),\\bm\{b\}=\{\(Z^\{\(\\delta\)\}\)\}^\{\\intercal\}\\Pi\\bm\{y\}\-\\frac\{1\}\{\\sum\_\{i\}^\{n\}\\pi\_\{i\}\}\{\{\(Z^\{\(\\delta\)\}\)\}^\{\\intercal\}\\Pi\\bm\{1\}\}\\bm\{1\}^\{\\intercal\}\\Pi\\bm\{y\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\)\(\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\),we have,

‖𝒃ℛ‖2≤max⁡\|yi\(δ\)−μy\|​∑i=1nπi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2\.\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\leq\\max\|\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\.We bound‖𝒃ℛ‖2\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}by splitting the sum into the tail setT\(δ\)T^\{\(\\delta\)\}and the two core setsJ0\(δ\)∪J1\(δ\)J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\. Lemma[C\.7](https://arxiv.org/html/2608.10587#A3.Thmtheorem7)\(ii\) and triangle inequality give,

1. \(i\)fori∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, max⁡\|yi\(δ\)−μy\|​∑i∈J0\(δ\)∪J1\(δ\)πi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2≤max⁡\|yi\(δ\)−μy\|​∑i∈J0\(δ\)∪J1\(δ\)πi​\(O​\(δ−2\)\+O​\(e−cB​δ\)\)\.\\max\|\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\leq\\max\|\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\(O\(\\delta^\{\-2\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\)\)\.
2. \(ii\)fori∈T\(δ\)i\\in T^\{\(\\delta\)\},max⁡\|yi\(δ\)−μy\|​∑i∈T\(δ\)πi=O​\(max⁡\|yi\(δ\)−μy\|​e−cB​δ\)\\max\|\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|\\sum\_\{i\\in T^\{\(\\delta\)\}\}\\pi\_\{i\}=O\(\\max\|\{y\}^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|e^\{\-c\_\{B\}\\delta\}\)

By Lemma[C\.8](https://arxiv.org/html/2608.10587#A3.Thmtheorem8),maxi⁡\|yi\(δ\)−μy\|=O​\(Δ​\(δ\)\+1\)\.\\max\_\{i\}\|y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\|=O\(\\Delta\(\\delta\)\+1\)\.Therefore,

‖𝒃ℛ‖2≤O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\+O​\(\(Δ​\(δ\)\+1\)​δ−2\)∎\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\leq O\\big\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\big\)\+O\\big\(\(\\Delta\(\\delta\)\+1\)\\delta^\{\-2\}\\big\)\\qed

The preceding lemmas show that, under the BREAD kernel, the unshifted block𝒃ℛ\\bm\{b\}\_\{\\mathcal\{R\}\}is asymptotically negligible with order ofδ−2\\delta^\{\-2\}and up to exponentially small tail terms\.

#### C\.3\.4BoundingM𝒦​ℛM\_\{\\mathcal\{KR\}\}

This section bounds the cross\-block couplingM𝒦​ℛM\_\{\\mathcal\{KR\}\}\. This shows that, asymptotically, the BREAD linear system becomes nearly block diagonal\.

###### Lemma C\.10\(Upper bound onM𝒦​ℛM\_\{\\mathcal\{KR\}\}\)\.

Under Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1),

‖M𝒦​ℛ‖2=‖Mℛ​𝒦‖2=O​\(δ−1\)\+O​\(e−cB​δ\),δ→∞\.\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}=\\\|M\_\{\\mathcal\{RK\}\}\\\|\_\{2\}=O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\),\\qquad\\delta\\to\\infty\.

###### Proof\.

RecallM=A\+λ​𝑰M=A\+\\lambda\\bm\{I\}andMℛ​𝒦=Aℛ​𝒦=∑i=1nπi​\(𝒛i,ℛ\(δ\)−𝝁ℛ\)​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)⊺\.M\_\{\\mathcal\{R\}\\mathcal\{K\}\}=A\_\{\\mathcal\{R\}\\mathcal\{K\}\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\)\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)^\{\\intercal\}\.

‖M𝒦​ℛ‖2=‖Mℛ​𝒦‖2=‖∑i=1nπi​\(𝒛i,ℛ\(δ\)−𝝁ℛ\)​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)⊺‖2≤∑i=1nπi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2​‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}=\\\|M\_\{\\mathcal\{RK\}\}\\\|\_\{2\}=\\left\\\|\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\)\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)^\{\\intercal\}\\right\\\|\_\{2\}\\leq\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}
By Lemma[C\.7](https://arxiv.org/html/2608.10587#A3.Thmtheorem7)\(i\), for alli∈\{1,…,n\}i\\in\\\{1,\\dots,n\\\},‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ\)\.\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\(\\delta\)\.

Then, we split the sum into tail and core\.

Fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, by Lemmas[C\.7](https://arxiv.org/html/2608.10587#A3.Thmtheorem7)\(i\),‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2=O​\(1\)\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\(1\)and by[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(i\),πi≤KB​e−cB​δ\\pi\_\{i\}\\leq K\_\{B\}e^\{\-c\_\{B\}\\delta\}\. Thus,

∑i∈T\(δ\)πi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2​‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ​e−cB​δ\)=O​\(e−cB′​δ\),\\sum\_\{i\\in T^\{\(\\delta\)\}\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\(\\delta e^\{\-c\_\{B\}\\delta\}\)=O\(e^\{\-c^\{\\prime\}\_\{B\}\\delta\}\),for some0<cB′<cB\.0<c^\{\\prime\}\_\{B\}<c\_\{B\}\.

Fori∈J0\(δ\)∪J1\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}, by Lemmas[C\.7](https://arxiv.org/html/2608.10587#A3.Thmtheorem7)\(ii\),‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2=O​\(δ−2\)\+O​\(e−cB​δ\),\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\(\\delta^\{\-2\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\),and by[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(ii\),πi≤1\\pi\_\{i\}\\leq 1\. Combined with‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ\)\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\(\\delta\), it holds

∑i∈J0\(δ\)∪J1\(δ\)πi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2​‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2=O​\(δ−1\)\+O​\(δ​e−cB′​δ\)\.\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O\\left\(\\delta^\{\-1\}\\right\)\+O\\bigl\(\\delta e^\{\-c^\{\\prime\}\_\{B\}\\delta\}\\bigr\)\.
Hence,

‖M𝒦​ℛ‖2=‖Mℛ​𝒦‖2=‖∑i=1nπi​\(𝒛i,ℛ\(δ\)−𝝁ℛ\)​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)⊺‖2\\displaystyle\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}=\\\|M\_\{\\mathcal\{RK\}\}\\\|\_\{2\}=\\left\\\|\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\)\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)^\{\\intercal\}\\right\\\|\_\{2\}≤∑i=1nπi​‖𝒛i,ℛ\(δ\)−𝝁ℛ‖2​‖𝒛i,𝒦\(δ\)−𝝁𝒦‖2\\displaystyle\\leq\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{R\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=O​\(δ−1\)\+O​\(δ​e−cB′​δ\)∎\\displaystyle=O\\left\(\\delta^\{\-1\}\\right\)\+O\\bigl\(\\delta e^\{\-c^\{\\prime\}\_\{B\}\\delta\}\\bigr\)\\qed

Thus, the BREAD matrixMMis asymptotically block diagonal\.

#### C\.3\.5Lower bound on𝜷^𝒦\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}

It remains to show that‖𝜷^𝒦‖2\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}does not vanish too fast\. Write𝜷^𝒦=\(S\)−1​\(𝒃𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​𝒃ℛ\)\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}=\(S\)^\{\-1\}\\left\(\\bm\{b\}\_\{\\mathcal\{K\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)\. The preceding Lemmas[C\.9](https://arxiv.org/html/2608.10587#A3.Thmtheorem9), and[C\.10](https://arxiv.org/html/2608.10587#A3.Thmtheorem10)imply that term\(M𝒦​ℛ​\(Mℛ​ℛ\)−1​𝒃ℛ\)\\left\(M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)is negligible\. Thus, we first prove that the𝒃𝒦\\bm\{b\}\_\{\\mathcal\{K\}\}has the lower bound with orderδ​Δ​\(δ\)\\delta\\Delta\(\\delta\), then combine this with the upper bound on‖S‖2\\\|S\\\|\_\{2\}\.

###### Lemma C\.11\(Lower bound on‖𝒃𝒦‖2\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\)\.

Under Assumptions[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)and[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2),

‖𝒃𝒦‖2=Ω​\(δ​Δ​\(δ\)\),δ→∞\.\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\(\\delta\\Delta\(\\delta\)\),\\qquad\\delta\\to\\infty\.SinceΔ​\(δ\)≥cΔ\>0\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\>0, this implies

‖𝒃𝒦‖2=Ω​\(δ\)\.\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\(\\delta\)\.

###### Proof\.

Recall𝒃𝒦=∑i=1nπi​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)​\(yi\(δ\)−μy\)\.\\bm\{b\}\_\{\\mathcal\{K\}\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\)\.

To find the lower bound on‖𝒃𝒦‖2\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}, since‖𝒖𝒦‖2=1\\\|\\bm\{u\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=1, we consider

‖𝒃𝒦‖2≥\|𝒖𝒦⊺​𝒃𝒦\|=\|∑i=1nπi​\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)\|,\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\\geq\|\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{b\}\_\{\\mathcal\{K\}\}\|=\\left\|\\sum^\{n\}\_\{i=1\}\\pi\_\{i\}\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\mu\_\{\\bm\{u\}\}\)\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\)\\right\|,whereμ𝒖=1sπ​∑i=1nπi​𝒖𝒦⊺​𝒛i,𝒦\(δ\)\.\\mu\_\{\\bm\{u\}\}=\\frac\{1\}\{s\_\{\\pi\}\}\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\.

We first estimateμ𝒖\{\\mu\}\_\{\\bm\{u\}\}\. Consideri∈J0\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\},

𝒖𝒦⊺​𝒛i,𝒦\(δ\)=𝒖𝒦⊺​𝒕ref,𝒦\+O​\(δ−2\),\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+O\(\\delta^\{\-2\}\),while fori∈J1\(δ\)i\\in J\_\{1\}^\{\(\\delta\)\}, since𝒕j∗=𝒕ref\+δ​𝒖\\bm\{t\}\_\{j^\{\*\}\}=\\bm\{t\}\_\{\\mathrm\{ref\}\}\+\\delta\\bm\{u\}and‖𝒖𝒦‖2=1\\\|\\bm\{u\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=1,

𝒖𝒦⊺​𝒛i,𝒦\(δ\)=𝒖𝒦⊺​𝒕ref,𝒦\+δ\+O​\(δ−2\)\.\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+\\delta\+O\(\\delta^\{\-2\}\)\.Fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)implies𝒖𝒦⊺​𝒛i,𝒦\(δ\)=O​\(δ\)\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=O\(\\delta\), and Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)givesπi=O​\(e−cB​δ\)\\pi\_\{i\}=O\(e^\{\-c\_\{B\}\\delta\}\)\.

Combining all,

μ𝒖=𝒖𝒦⊺​𝒕ref,𝒦\+s1sπ​δ\+O​\(δ−2\)\+O​\(δ​e−cB​δ\)\.\{\\mu\}\_\{\\bm\{u\}\}=\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{t\}\_\{\\mathrm\{ref\},\\mathcal\{K\}\}\+\\frac\{s\_\{1\}\}\{s\_\{\\pi\}\}\\delta\+O\(\\delta^\{\-2\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\)\.
Then, we consider the‖𝒃𝒦‖2\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}for core points\.

Fori∈J0\(δ\)i\\in J\_\{0\}^\{\(\\delta\)\},

𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖=−s1sπ​δ\+O​\(δ−2\)\+O​\(δ​e−cB​δ\),\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}=\-\\frac\{s\_\{1\}\}\{s\_\{\\pi\}\}\\delta\+O\(\\delta^\{\-2\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\),and by Lemma[C\.8](https://arxiv.org/html/2608.10587#A3.Thmtheorem8)\(ii\),

yi\(δ\)−μy=−s1sπ​Δ​\(δ\)\+O​\(δ−1\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\.y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}=\-\\frac\{s\_\{1\}\}\{s\_\{\\pi\}\}\\Delta\(\\delta\)\+O\(\\delta^\{\-1\}\)\+O\\bigl\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\bigr\)\.Hence,

\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)\\displaystyle\\bigl\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}\\bigr\)\\bigl\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\\bigr\)=s12sπ2​δ​Δ​\(δ\)\+O​\(1\)\+O​\(Δ​\(δ\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​δ​e−cB​δ\)\.\\displaystyle=\\frac\{s\_\{1\}^\{2\}\}\{s\_\{\\pi\}^\{2\}\}\\delta\\Delta\(\\delta\)\+O\(1\)\+O\\\!\\bigl\(\\Delta\(\\delta\)\\delta^\{\-2\}\\bigr\)\+O\\\!\\bigl\(\(\\Delta\(\\delta\)\+1\)\\delta e^\{\-c\_\{B\}\\delta\}\\bigr\)\.
Fori∈J1\(δ\)i\\in J\_\{1\}^\{\(\\delta\)\}, similarly,

𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖=s0sπ​δ\+O​\(δ−2\)\+O​\(δ​e−cB​δ\),\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}=\\frac\{s\_\{0\}\}\{s\_\{\\pi\}\}\\delta\+O\(\\delta^\{\-2\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\),yi\(δ\)−μy=s0sπ​Δ​\(δ\)\+O​\(δ−1\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\),y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}=\\frac\{s\_\{0\}\}\{s\_\{\\pi\}\}\\Delta\(\\delta\)\+O\(\\delta^\{\-1\}\)\+O\\bigl\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\bigr\),and hence

\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)\\displaystyle\\bigl\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}\\bigr\)\\bigl\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\\bigr\)=s02sπ2​δ​Δ​\(δ\)\+O​\(1\)\+O​\(Δ​\(δ\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​δ​e−cB​δ\)\.\\displaystyle=\\frac\{s\_\{0\}^\{2\}\}\{s\_\{\\pi\}^\{2\}\}\\delta\\Delta\(\\delta\)\+O\(1\)\+O\\\!\\bigl\(\\Delta\(\\delta\)\\delta^\{\-2\}\\bigr\)\+O\\\!\\bigl\(\(\\Delta\(\\delta\)\+1\)\\delta e^\{\-c\_\{B\}\\delta\}\\bigr\)\.
Since Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(ii\) givesπi=1\+O​\(δ−1\)\\pi\_\{i\}=1\+O\(\\delta^\{\-1\}\)on the core sets, summing overJ0\(δ\)∪J1\(δ\)J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}yields

∑i∈J0\(δ\)∪J1\(δ\)πi​\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)\\displaystyle\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\\bigl\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}\\bigr\)\\bigl\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\\bigr\)=\[s12sπ2​∑i∈J0\(δ\)πi\+s02sπ2​∑i∈J1\(δ\)πi\]​δ​Δ​\(δ\)\\displaystyle=\\left\[\\frac\{s\_\{1\}^\{2\}\}\{s\_\{\\pi\}^\{2\}\}\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\}\\pi\_\{i\}\+\\frac\{s\_\{0\}^\{2\}\}\{s\_\{\\pi\}^\{2\}\}\\sum\_\{i\\in J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\\right\]\\delta\\Delta\(\\delta\)\+O​\(1\)\+O​\(Δ​\(δ\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​δ​e−cB​δ\)\.\\displaystyle\\qquad\\quad\+O\(1\)\+O\\\!\\bigl\(\\Delta\(\\delta\)\\delta^\{\-2\}\\bigr\)\+O\\\!\\bigl\(\(\\Delta\(\\delta\)\+1\)\\delta e^\{\-c\_\{B\}\\delta\}\\bigr\)\.
By Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4)\(ii\) and[C\.5](https://arxiv.org/html/2608.10587#A3.Thmtheorem5), since bothJ0J\_\{0\}andJ1J\_\{1\}contribute terms of the same sign inδ​Δ​\(δ\)\\delta\\Delta\(\\delta\), the sum is bounded below by a positive multiple ofδ​Δ​\(δ\)\\delta\\Delta\(\\delta\)\.

Therefore,

∑i∈J0\(δ\)∪J1\(δ\)πi​\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)=Ω​\(δ​Δ​\(δ\)\)\.\\displaystyle\\sum\_\{i\\in J\_\{0\}^\{\(\\delta\)\}\\cup J\_\{1\}^\{\(\\delta\)\}\}\\pi\_\{i\}\\bigl\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}\\bigr\)\\bigl\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\\bigr\)=\\Omega\\bigl\(\\delta\\Delta\(\\delta\)\\bigr\)\.\(11\)
Finally, we bound the tail part\.

Fori∈T\(δ\)i\\in T^\{\(\\delta\)\}, by Lemma[C\.4](https://arxiv.org/html/2608.10587#A3.Thmtheorem4),πi=O​\(e−cB​δ\)\\pi\_\{i\}=O\(e^\{\-c\_\{B\}\\delta\}\), by Lemma[C\.8](https://arxiv.org/html/2608.10587#A3.Thmtheorem8),yi\(δ\)−μy=O​\(Δ​\(δ\)\+1\)y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}=O\(\\Delta\(\\delta\)\+1\), and Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)implies𝒖𝒦⊺​𝒛i,𝒦\(δ\)=O​\(δ\)\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}=O\(\\delta\), and the bounds above yield𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖=O​\(δ\)\.\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}=O\(\\delta\)\.Hence

∑i∈T\(δ\)πi​\|\(𝒖𝒦⊺​𝒛i,𝒦\(δ\)−μ𝒖\)​\(yi\(δ\)−μy\)\|=O​\(\(Δ​\(δ\)\+1\)​δ​e−cB​δ\)=o​\(δ​Δ​\(δ\)\),\\sum\_\{i\\in T^\{\(\\delta\)\}\}\\pi\_\{i\}\\left\|\\bigl\(\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\{\\mu\}\_\{\\bm\{u\}\}\\bigr\)\\bigl\(y^\{\(\\delta\)\}\_\{i\}\-\\mu\_\{y\}\\bigr\)\\right\|=O\\bigl\(\(\\Delta\(\\delta\)\+1\)\\delta e^\{\-c\_\{B\}\\delta\}\\bigr\)=o\\bigl\(\\delta\\Delta\(\\delta\)\\bigr\),becauseΔ​\(δ\)≥cΔ\>0\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\>0\.

Combining this with \([11](https://arxiv.org/html/2608.10587#A3.E11)\), we obtain

\|𝒖𝒦⊺​𝒃𝒦\|=Ω​\(δ​Δ​\(δ\)\)\.\\bigl\|\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\bigr\|=\\Omega\\\!\\bigl\(\\delta\\Delta\(\\delta\)\\bigr\)\.Therefore,

‖𝒃𝒦‖2≥\|𝒖𝒦⊺​𝒃𝒦\|=Ω​\(δ​Δ​\(δ\)\)\.\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\\geq\\bigl\|\\bm\{u\}\_\{\\mathcal\{K\}\}^\{\\intercal\}\\bm\{b\}\_\{\\mathcal\{K\}\}\\bigr\|=\\Omega\\\!\\bigl\(\\delta\\Delta\(\\delta\)\\bigr\)\.SinceΔ​\(δ\)≥cΔ\>0\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\>0, it follows that

‖𝒃𝒦‖2=Ω​\(δ\)\.∎\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\(\\delta\)\.\\qed

We now control the scale of the𝒦\\mathcal\{K\}block, reflecting the fact that the dominant variation inAAis along the shift direction\.

###### Lemma C\.12\(Upper bound on‖S‖2\\\|S\\\|\_\{2\}\)\.

Write the Schur complementS=M𝒦​𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​\(M𝒦​ℛ\)⊺S=M\_\{\\mathcal\{KK\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\(M\_\{\{\\mathcal\{KR\}\}\}\)^\{\\intercal\},

‖S‖2=O​\(δ2\)\.\\\|S\\\|\_\{2\}=O\(\\delta^\{2\}\)\.

###### Proof\.

SinceMMis symmetric positive definite, its Schur complementS=M𝒦​𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​Mℛ​𝒦S=M\_\{\\mathcal\{KK\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}M\_\{\\mathcal\{RK\}\}is also symmetric positive definite\. Moreover,M𝒦​ℛ​\(Mℛ​ℛ\)−1​Mℛ​𝒦⪰0\.M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}M\_\{\\mathcal\{RK\}\}\\succeq 0\.Hence,

0⪯S⪯M𝒦​𝒦\.0\\preceq S\\preceq M\_\{\\mathcal\{KK\}\}\.Therefore,

‖S‖2≤‖M𝒦​𝒦‖2\.\\\|S\\\|\_\{2\}\\leq\\\|M\_\{\\mathcal\{KK\}\}\\\|\_\{2\}\.
SinceM𝒦​𝒦=λ​𝑰𝒦​𝒦\+A𝒦​𝒦,M\_\{\\mathcal\{KK\}\}=\\lambda\\bm\{I\}\_\{\\mathcal\{KK\}\}\+A\_\{\\mathcal\{KK\}\},whereA𝒦​𝒦=∑i=1nπi​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)​\(𝒛i,𝒦\(δ\)−𝝁𝒦\)⊺\.A\_\{\\mathcal\{KK\}\}=\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)\(\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\)^\{\\intercal\}\.By the triangle inequality,

‖A𝒦​𝒦‖2≤∑i=1nπi​‖𝒛i,𝒦\(δ\)−𝝁𝒦‖22\.\\\|A\_\{\\mathcal\{KK\}\}\\\|\_\{2\}\\leq\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}\\\|\\bm\{z\}\_\{i,\\mathcal\{K\}\}^\{\(\\delta\)\}\-\\bm\{\\mu\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}^\{2\}\.By Lemma[C\.7](https://arxiv.org/html/2608.10587#A3.Thmtheorem7)\(i\),

‖A𝒦​𝒦‖2≤∑i=1nπi​O​\(δ2\)\.\\\|A\_\{\\mathcal\{KK\}\}\\\|\_\{2\}\\leq\\sum\_\{i=1\}^\{n\}\\pi\_\{i\}O\(\\delta^\{2\}\)\.Combined withπi≤1\\pi\_\{i\}\\leq 1and fixednn,

‖S‖2≤‖M𝒦​𝒦‖2=O​\(δ2\)\.∎\\\|S\\\|\_\{2\}\\leq\\\|M\_\{\\mathcal\{KK\}\}\\\|\_\{2\}=O\(\\delta^\{2\}\)\.\\qed

###### Lemma C\.13\(Lower bound on the BREAD coefficients on𝒦\\mathcal\{K\}\)\.

Under Assumptions[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1)and[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2),

‖𝜷^𝒦‖2=Ω​\(Δ​\(δ\)δ\),δ→∞\.\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\\\!\\left\(\\frac\{\\Delta\(\\delta\)\}\{\\delta\}\\right\),\\qquad\\delta\\to\\infty\.In particular,

‖𝜷^𝒦‖2=Ω​\(δ−1\)\.\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\(\\delta^\{\-1\}\)\.

###### Proof\.

Recall

𝜷^𝒦=\(S\)−1​\(𝒃𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​𝒃ℛ\)\.\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}=\(S\)^\{\-1\}\\left\(\\bm\{b\}\_\{\\mathcal\{K\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)\.
Thus,

S​𝜷^𝒦=\(𝒃𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​𝒃ℛ\)\.S\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}=\\left\(\\bm\{b\}\_\{\\mathcal\{K\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)\.Then, we have,

‖\(𝒃𝒦−M𝒦​ℛ​\(Mℛ​ℛ\)−1​𝒃ℛ\)‖2≤‖S‖2​‖𝜷^𝒦‖2\.\\\|\\left\(\\bm\{b\}\_\{\\mathcal\{K\}\}\-M\_\{\\mathcal\{KR\}\}\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\bm\{b\}\_\{\\mathcal\{R\}\}\\right\)\\\|\_\{2\}\\leq\\\|S\\\|\_\{2\}\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\.Hence,

‖𝜷^𝒦‖2≥1‖S‖2​\(‖𝒃𝒦‖2−‖M𝒦​ℛ‖2​‖\(Mℛ​ℛ\)−1‖2​‖𝒃ℛ‖2\)\.\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\\geq\\frac\{1\}\{\\\|S\\\|\_\{2\}\}\\Big\(\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\-\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\\|\_\{2\}\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\Big\)\.SinceA⪰0A\\succeq 0andλ\>0\\lambda\>0, we haveM⪰λ​𝑰M\\succeq\\lambda\\bm\{I\}, hence

‖\(Mℛ​ℛ\)−1‖2≤1λ\.\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\\|\_\{2\}\\leq\\frac\{1\}\{\\lambda\}\.\(12\)
Using Lemmas[C\.9](https://arxiv.org/html/2608.10587#A3.Thmtheorem9), and[C\.10](https://arxiv.org/html/2608.10587#A3.Thmtheorem10),‖M𝒦​ℛ‖2=O​\(δ−1\)\+O​\(e−cB​δ\)\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}=O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\), and‖𝒃ℛ‖2=O​\(\(Δ​\(δ\)\+1\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\.\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\\big\(\(\\Delta\(\\delta\)\+1\)\\delta^\{\-2\}\\big\)\+O\\big\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\big\)\.Thus, the second term in parentheses isO​\(\(Δ​\(δ\)\+1\)​δ−3\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)O\\big\(\(\\Delta\(\\delta\)\+1\)\\delta^\{\-3\}\\big\)\+O\\big\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\big\), hence negligible\.

Hence, by Lemmas[C\.11](https://arxiv.org/html/2608.10587#A3.Thmtheorem11)and[C\.12](https://arxiv.org/html/2608.10587#A3.Thmtheorem12),

‖𝜷^𝒦‖2≥1‖S‖2​\(‖𝒃𝒦‖2−‖M𝒦​ℛ‖2​‖\(Mℛ​ℛ\)−1‖2​‖𝒃ℛ‖2\)≥1O​\(δ2\)​\(Ω​\(δ​Δ​\(δ\)\)\)=Ω​\(Δ​\(δ\)δ\),\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\\geq\\frac\{1\}\{\\\|S\\\|\_\{2\}\}\\Big\(\\\|\\bm\{b\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\-\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\\|\(M\_\{\\mathcal\{RR\}\}\)^\{\-1\}\\\|\_\{2\}\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\\Big\)\\geq\\frac\{1\}\{O\(\\delta^\{2\}\)\}\\big\(\\Omega\(\\delta\\Delta\(\\delta\)\)\\big\)=\\Omega\(\\frac\{\\Delta\(\\delta\)\}\{\\delta\}\),
Since,Δ​\(δ\)≥cΔ\>0\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\>0, in particular‖𝜷^𝒦‖2=Ω​\(δ−1\)\.\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\(\\delta^\{\-1\}\)\.∎

#### C\.3\.6Proof of Theorem[3\.3](https://arxiv.org/html/2608.10587#S3.Thmtheorem3)

We now combine the previous bounds to obtain the asymptotic sparsity of𝜷^\\hat\{\\bm\{\\beta\}\}\.

###### Theorem C\.14\(Theorem 3\.3, restated\)\.

Suppose the shift setup in \(5\) holds, the synthetic data matrixZ\(δ\)Z^\{\(\\delta\)\}satisfies Assumption[3\.1](https://arxiv.org/html/2608.10587#S3.Thmtheorem1), and the corresponding outputs,yi\(δ\)=f​\(𝐳i\(δ\)\)\{y\}\_\{i\}^\{\(\\delta\)\}=f\(\\bm\{z\}\_\{i\}^\{\(\\delta\)\}\), satisfy Assumption[3\.2](https://arxiv.org/html/2608.10587#S3.Thmtheorem2)\. Let𝛃^\\hat\{\\bm\{\\beta\}\}denote the estimator in \(4\) and defineϵ=‖𝛃^ℛ‖2‖𝛃^𝒦‖2\\epsilon=\\dfrac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\. Then,

ϵ→0,as​δ→∞\.\{\\epsilon\}\\rightarrow 0,\\qquad\\text\{as\}\\,\\delta\\rightarrow\\infty\.

###### Proof\.

Recall

ϵ=‖𝜷^ℛ‖2‖𝜷^𝒦‖2≤1λ​\(‖𝒃ℛ‖2‖𝜷^𝒦‖2\+‖M𝒦​ℛ‖2\)\.\\epsilon=\\frac\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\\leq\\frac\{1\}\{\\lambda\}\\left\(\\frac\{\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}\+\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}\\right\)\.By Lemma[C\.9](https://arxiv.org/html/2608.10587#A3.Thmtheorem9),‖𝒃ℛ‖2=O​\(\(Δ​\(δ\)\+1\)​δ−2\)\+O​\(\(Δ​\(δ\)\+1\)​e−cB​δ\)\.\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}=O\\\!\\bigl\(\(\\Delta\(\\delta\)\+1\)\\delta^\{\-2\}\\bigr\)\+O\\\!\\bigl\(\(\\Delta\(\\delta\)\+1\)e^\{\-c\_\{B\}\\delta\}\\bigr\)\.By Lemma[C\.13](https://arxiv.org/html/2608.10587#A3.Thmtheorem13),‖𝜷^𝒦‖2=Ω​\(Δ​\(δ\)δ\)\.\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}=\\Omega\\\!\\left\(\\frac\{\\Delta\(\\delta\)\}\{\\delta\}\\right\)\.Therefore,

‖𝒃ℛ‖2‖𝜷^𝒦‖2=O​\(Δ​\(δ\)\+1Δ​\(δ\)​δ−1\)\+O​\(Δ​\(δ\)\+1Δ​\(δ\)​δ​e−cB​δ\)\.\\displaystyle\\frac\{\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}=O\\\!\\left\(\\frac\{\\Delta\(\\delta\)\+1\}\{\\Delta\(\\delta\)\}\\delta^\{\-1\}\\right\)\+O\\\!\\left\(\\frac\{\\Delta\(\\delta\)\+1\}\{\\Delta\(\\delta\)\}\\delta e^\{\-c\_\{B\}\\delta\}\\right\)\.SinceΔ​\(δ\)≥cΔ\>0\\Delta\(\\delta\)\\geq c\_\{\\Delta\}\>0, we have

Δ​\(δ\)\+1Δ​\(δ\)=O​\(1\)\.\\frac\{\\Delta\(\\delta\)\+1\}\{\\Delta\(\\delta\)\}=O\(1\)\.Hence,

‖𝒃ℛ‖2‖𝜷^𝒦‖2=O​\(δ−1\)\+O​\(δ​e−cB​δ\)\.\\frac\{\\\|\\bm\{b\}\_\{\\mathcal\{R\}\}\\\|\_\{2\}\}\{\\\|\\hat\{\\bm\{\\beta\}\}\_\{\\mathcal\{K\}\}\\\|\_\{2\}\}=O\(\\delta^\{\-1\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\)\.By Lemma[C\.10](https://arxiv.org/html/2608.10587#A3.Thmtheorem10),‖M𝒦​ℛ‖2=O​\(δ−1\)\+O​\(e−cB​δ\)\.\\\|M\_\{\\mathcal\{KR\}\}\\\|\_\{2\}=O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\)\.Combining all,

ϵ​\(δ\)≤1λ​\(O​\(δ−1\)\+O​\(δ​e−cB​δ\)\+O​\(δ−1\)\+O​\(e−cB​δ\)\)\.\\epsilon\(\\delta\)\\leq\\frac\{1\}\{\\lambda\}\\Bigl\(O\(\\delta^\{\-1\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\)\+O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}\\delta\}\)\\Bigr\)\.Therefore,

ϵ​\(δ\)=O​\(δ−1\)\+O​\(δ​e−cB​δ\),\\epsilon\(\\delta\)=O\(\\delta^\{\-1\}\)\+O\(\\delta e^\{\-c\_\{B\}\\delta\}\),for some fixed0<cB′<cB0<c^\{\\prime\}\_\{B\}<c\_\{B\}\. Thus,

ϵ​\(δ\)=O​\(δ−1\)\+O​\(e−cB′​δ\)\.\\epsilon\(\\delta\)=O\(\\delta^\{\-1\}\)\+O\(e^\{\-c\_\{B\}^\{\\prime\}\\delta\}\)\.In particular,ϵ​\(δ\)→0\\epsilon\(\\delta\)\\to 0asδ→∞\\delta\\to\\infty\. ∎

## Appendix DExperiment Details

### D\.1Computation infrastructure

Experiments were run on a single compute node with an 192 cores AMD Genoa CPU, 336 GB RAM, and Red Hat Enterprise Linux 9\.8 \(Plow\)\. We used Python 3\.11\.5, NumPy 1\.26\.4, scikit\-learn 1\.5\.2, and SciPy 1\.12\.0\.

### D\.2Simulation Data Setup

Specifically, we simulated=500d=500features overn=2000n=2000time points according to

Yt=μ0\+ϕ​Yt−1\+\(a​cos⁡\(2​π​\(t\+δ\)T\)\+b​sin⁡\(2​π​\(t\+δ\)T\)\)\+εt,Y\_\{t\}=\\mu\_\{0\}\+\\phi Y\_\{t\-1\}\+\\left\(a\\cos\\left\(\\frac\{2\\pi\(t\+\\delta\)\}\{T\}\\right\)\+b\\sin\\left\(\\frac\{2\\pi\(t\+\\delta\)\}\{T\}\\right\)\\right\)\+\\varepsilon\_\{t\},whereYt∈ℝdY\_\{t\}\\in\\mathbb\{R\}^\{d\},μ0∈ℝd\\mu\_\{0\}\\in\\mathbb\{R\}^\{d\}denotes the mean baseline vector, andϕ\\phiis the AR\(1\) coefficient\. The error term satisfiesεt∼𝒩d​\(0,Σ0\)\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\_\{d\}\(0,\\Sigma\_\{0\}\), whereΣ0\\Sigma\_\{0\}is thed×dd\\times din\-control variance–covariance matrix\. The final term is a first\-order Fourier component that induces seasonality with periodTT, amplitude parametersaaandbb, and phase shiftδ\\delta\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/cor_matrix.png)Figure 7:Heatmap of the correlation matrix among thed=500d=500simulated features in Phase I\.Variables are ordered by correlation blocks, yielding a clear block\-diagonal pattern with strong within\-block dependence \(\>0\.8\>0\.8\) and comparatively weaker between\-block correlations \(lighter off\-diagonal regions,<0\.5<0\.5\)\. The colour bar encodes Pearson correlation coefficients on\[−1,1\]\[\-1,1\], with diagonal elements equal to 1\. There are 15 blocks in total with sizes of 53, 16, 73, 62, 22, 25, 4, 23, 54, 3, 31, 39, 3, 65 and 27, respectively\.The covariance structureΣ0\\Sigma\_\{0\}is constructed to exhibit block\-wise dependence \(i\.e\., correlation blocks\) to reflect clustered feature relationships\. Variables are ordered by correlation blocks, yielding a clear block\-diagonal pattern with strong within\-block dependence \(\>0\.8\>0\.8\) and comparatively weaker between\-block correlations \(lighter off\-diagonal regions,<0\.5<0\.5\)\. There are 15 blocks in total with sizes of 53, 16, 73, 62, 22, 25, 4, 23, 54, 3, 31, 39, 3, 65 and 27, respectively\. An example of a correlation matrix is shown in Figure[7](https://arxiv.org/html/2608.10587#A4.F7)\.

### D\.3Monitoring Method Performance

The monitoring performance of the AI\-based methods is measured by false alarm probability \(FAP\), conditional expected delay \(CED\) and recall\.

FAP assesses Phase I performance, and all models are set to achieve a comparable FAP level\. Under a pre\-defined FAP, CED measures how quickly the method can detect the change, while recall rate denotes the fraction of signals detected in the total Phase II data\. Both summarize Phase II detection performance, and lower FAP and CED indicate better performance\.

For all scenarios we use, the FAP of the monitoring model is controlled as around0\.010\.01, and the averaged actual FAP over 1000 runs is about0\.01150\.0115\. CED and recall are reported in Table[1](https://arxiv.org/html/2608.10587#A4.T1)\.

Table 1:Monitoring performance
### D\.4Hyperparameters

Table[2](https://arxiv.org/html/2608.10587#A4.T2)summarises all hyperparameters used in the simulation study\. All parameters are held fixed across methods unless otherwise stated\. BREAD and LIME share the same ridge penaltyλ\\lambda, kernel widthσ\\sigma, and sample sizennto ensure a controlled comparison\.

Table 2:Hyperparameter settings\.
### D\.5Computational cost

Table[3](https://arxiv.org/html/2608.10587#A4.T3)reports the average time per explanation, measured over the faithfulness experiments \(1000 runs, Scenario 1, shift=−5=\-5\)\. BREAD and LIME are comparable in runtime, as both require the same number of model evaluations with equivalent surrogate fitting cost\. The naive method is approximately four times slower than the both, as it constructs a502502coalitions, including full and empty, and solves a\(502×501\)\(502\\times 501\)weighted least\-squares system, following a KernelSHAP\-style leave\-one\-out strategy, which dominates the runtime at high dimension\.

Table 3:Run time
### D\.6Qualitative Results

Figure[8](https://arxiv.org/html/2608.10587#A4.F8)illustrates an example of diagnosing a signal by the proposed model and LIME under Scenario 1\. In Figure[8\(a\)](https://arxiv.org/html/2608.10587#A4.F8.sf1), the feature importance is distributed nearly symmetrically around zero, which implies that the signal is driven by many features with similar contribution\. By contrast, in Figure[8\(b\)](https://arxiv.org/html/2608.10587#A4.F8.sf2), there is a subset of features with notably high importance, indicating that these features contribute most to the monitoring statisticT2T^\{2\}\. If we choose the top\-53 features with the highest importance in this case, the proposed method is able to give the features with a shift\.

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/qualitative_result_lime.png)\(a\)LIME
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/qualitative_result_bread.png)\(b\)BREAD

Figure 8:Signal diagnosis results\.Normalised feature importance is shown for two cases\. Horizontal bars represent individual features ordered by importance\. Values to the left of zero indicate negative contributions to the monitoring statistic, whereas values to the right indicate positive contributions\. \(a\) LIME\. It shows an approximately symmetric distribution about zero\. \(b\) BREAD\. It is positively skewed with an extended high\-importance tail, overlapping with the actual shifted features \(ground truth\)\.
### D\.7Simulation Results

The detailed simulation results are shown in Table[4](https://arxiv.org/html/2608.10587#A4.T4), which were also displayed in Figure 3 and 4\.

Table 4:Explanation faithfulness under different scenarios and mean shift \(Bold indicates the best performance\)\.
### D\.8Case Study Data Visualisation

![Refer to caption](https://arxiv.org/html/2608.10587v1/images/escplots_00.png)\(a\)Escalator 00
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/escplots_08.png)\(b\)Escalator 08
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/escplots_13.png)\(c\)Escalator 13
![Refer to caption](https://arxiv.org/html/2608.10587v1/images/escplots_14.png)\(d\)Escalator 14

Figure 9:Energy consumption data of selected monitored escalators across training, validation, and test data\.This figure presents the energy consumption patterns \(Wh\) of four escalators \(Escalators 00, 08, 13, and 14\) over time, segmented into training, validation, and test periods\. The x\-axis represents the time index, while the y\-axis indicates energy consumption\. Each subplot corresponds to a different escalator, highlighting distinct operational behaviours and energy profiles\.Here, we illustrate the energy data of Escalators 00, 08, 13, and 14\. Escalator 00 represents the normal escalator behaviour, while escalators 08, 13, and 14 depict a period of non\-energy reflecting abnormal operating conditions possibly caused by a fault or accidental shutdowns\.