Concentration Inequalities for Exchangeable Tensors and Matrix-valued Data

• URL: http://arxiv.org/abs/2601.20152v1
• Date: Wed, 28 Jan 2026 00:55:33 GMT
• Title: Concentration Inequalities for Exchangeable Tensors and Matrix-valued Data
• Abstract summary: We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums.<n>We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability.<n>For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models.
• Score: 10.448428219575403
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

Related papers

• Aggregation of Statistical Evidence under Exchangeability [22.375838048747294]
  We study aggregation of statistical evidence under unknown and potentially complex dependence.<n>We develop a finite-sample power and adaptivity theory for this framework.
  arXiv  Detail & Related papers  (2026-07-17T10:32:05Z)
• Neural Optimal Transport Meets Multivariate Conformal Prediction [58.43397908730771]
  We propose a framework for conditional vectorile regression (CVQR)<n>CVQR combines neural optimal transport with quantized optimization, and apply it to predictions.
  arXiv  Detail & Related papers  (2025-09-29T19:50:19Z)
• Robust and Scalable Variational Bayes [2.014089835498735]
  We propose a robust framework for variational Bayes (VB) that effectively handles outliers and contamination of arbitrary nature in large datasets.<n>Our approach divides the dataset into disjoint subsets, computes the posterior for each subset, and applies VB approximation independently to these posteriors.<n>This novel aggregation method yields the Variational Median Posterior (VM-Posterior) distribution.
  arXiv  Detail & Related papers  (2025-04-16T23:20:43Z)
• Statistical Inference in Classification of High-dimensional Gaussian Mixture [1.2354076490479515]
  We investigate the behavior of a general class of regularized convex classifiers in the high-dimensional limit. Our focus is on the generalization error and variable selection properties of the estimators.
  arXiv  Detail & Related papers  (2024-10-25T19:58:36Z)
• Synergistic eigenanalysis of covariance and Hessian matrices for enhanced binary classification [72.77513633290056]
  We present a novel approach that combines the eigenanalysis of a covariance matrix evaluated on a training set with a Hessian matrix evaluated on a deep learning model. Our method captures intricate patterns and relationships, enhancing classification performance.
  arXiv  Detail & Related papers  (2024-02-14T16:10:42Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Factorized Fusion Shrinkage for Dynamic Relational Data [16.531262817315696]
  We consider a factorized fusion shrinkage model in which all decomposed factors are dynamically shrunk towards group-wise fusion structures. The proposed priors enjoy many favorable properties in comparison and clustering of the estimated dynamic latent factors. We present a structured mean-field variational inference framework that balances optimal posterior inference with computational scalability.
  arXiv  Detail & Related papers  (2022-09-30T21:03:40Z)
• Optimal regularizations for data generation with probabilistic graphical models [0.0]
  Empirically, well-chosen regularization schemes dramatically improve the quality of the inferred models. We consider the particular case of L 2 and L 1 regularizations in the Maximum A Posteriori (MAP) inference of generative pairwise graphical models.
  arXiv  Detail & Related papers  (2021-12-02T14:45:16Z)
• Adversarially-Trained Nonnegative Matrix Factorization [77.34726150561087]
  We consider an adversarially-trained version of the nonnegative matrix factorization. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matrices.
  arXiv  Detail & Related papers  (2021-04-10T13:13:17Z)
• Large Non-Stationary Noisy Covariance Matrices: A Cross-Validation Approach [1.90365714903665]
  We introduce a novel covariance estimator that exploits the heteroscedastic nature of financial time series. By attenuating the noise from both the cross-sectional and time-series dimensions, we empirically demonstrate the superiority of our estimator over competing estimators.
  arXiv  Detail & Related papers  (2020-12-10T15:41:17Z)
• Accounting for Unobserved Confounding in Domain Generalization [107.0464488046289]
  This paper investigates the problem of learning robust, generalizable prediction models from a combination of datasets. Part of the challenge of learning robust models lies in the influence of unobserved confounders. We demonstrate the empirical performance of our approach on healthcare data from different modalities.
  arXiv  Detail & Related papers  (2020-07-21T08:18:06Z)
• Asymptotic Analysis of an Ensemble of Randomly Projected Linear Discriminants [94.46276668068327]
  In [1], an ensemble of randomly projected linear discriminants is used to classify datasets. We develop a consistent estimator of the misclassification probability as an alternative to the computationally-costly cross-validation estimator. We also demonstrate the use of our estimator for tuning the projection dimension on both real and synthetic data.
  arXiv  Detail & Related papers  (2020-04-17T12:47:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.