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Positive Semidefinite Matrix Supermartingales

Hongjian Wang, Aaditya Ramdas

TL;DR

The paper develops a comprehensive theory of positive semidefinite (PSD) matrix supermartingales and backward submartingales under the Loewner order, establishing matrix Ville-type inequalities and Doob-style convergence results that generalize scalar nonnegative martingale theory to the matrix setting. It introduces forward PSD matrix supermartingales, matrix test supermartingales, and matrix e-values/e-processes, including randomized variants that yield time-uniform and stopping-time concentration bounds expressed in the Loewner order. The framework is applied to spectral concentration, exchangeable matrices, and a range of tail conditions (including Chernoff-type, empirical Bernstein, and Catoni-style bounds), often outperforming scalarization approaches. The results enable flexible sequential testing for matrix-valued hypotheses (e.g., covariance testing) and provide practical tools for matrix concentration with dependent and exchangeable data, with broad relevance to statistics, econometrics, and theoretical computer science.

Abstract

We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative supermartingales and backward nonnegative submartingales, whose convergence and maximal inequalities are the theoretical foundations for a wide and ever-growing body of results in statistics, econometrics, and theoretical computer science. Our results lead to new concentration inequalities for either martingale-dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-standard Chernoff bounds to self-normalized heavy-tailed settings. Further, these inequalities are usually expressed in the Loewner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.

Positive Semidefinite Matrix Supermartingales

TL;DR

The paper develops a comprehensive theory of positive semidefinite (PSD) matrix supermartingales and backward submartingales under the Loewner order, establishing matrix Ville-type inequalities and Doob-style convergence results that generalize scalar nonnegative martingale theory to the matrix setting. It introduces forward PSD matrix supermartingales, matrix test supermartingales, and matrix e-values/e-processes, including randomized variants that yield time-uniform and stopping-time concentration bounds expressed in the Loewner order. The framework is applied to spectral concentration, exchangeable matrices, and a range of tail conditions (including Chernoff-type, empirical Bernstein, and Catoni-style bounds), often outperforming scalarization approaches. The results enable flexible sequential testing for matrix-valued hypotheses (e.g., covariance testing) and provide practical tools for matrix concentration with dependent and exchangeable data, with broad relevance to statistics, econometrics, and theoretical computer science.

Abstract

We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative supermartingales and backward nonnegative submartingales, whose convergence and maximal inequalities are the theoretical foundations for a wide and ever-growing body of results in statistics, econometrics, and theoretical computer science. Our results lead to new concentration inequalities for either martingale-dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-standard Chernoff bounds to self-normalized heavy-tailed settings. Further, these inequalities are usually expressed in the Loewner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.
Paper Structure (34 sections, 27 theorems, 121 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 34 sections, 27 theorems, 121 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Matrix Analysis
  4. Scalar Nonnegative Supermartingales and Statistical Applications
  5. Matrix Concentration Inequalities
  6. Spectral Bounds
  7. Ahlswede-Winter Bounds
  8. Forward PSD Supermartingales
  9. Convergence, Optional Stopping, and Maximal Inequalities
  10. Matrix Test Supermartingales
  11. Examples: Conditional Mean Testing Problems
  12. Matrix e-Values and e-Processes
  13. Backward PSD Submartingales
  14. Submartingales and Maximal Inequalities: From Forward to Backward
  15. Application: Concentration of Exchangeable Matrices
  16. ...and 19 more sections

Key Result

Lemma 3.1

(1) An $\mathcal{S}_d$-valued adapted process $\{ \mathbf{Y} _n \}$ is a supermartingale if and only if for every non-random vector $\mathbf{v} \in \mathbb R^d$ or $\mathbf{v} \in \mathbb Q^d$, the scalar-valued process $\{ \mathbf{v}^\mathsf {T} \mathbf{Y} _n \mathbf{v} \}$ is a supermartingale.

Figures (2)

  • Figure 1: Empirical findings with commuting i.i.d. random matrices. Left. Comparison of the cumulative distribution functions of rejection times of the sequential tests corresponding to the matrix- and scalar-valued test processes under the same alternative distribution. Right. Evolution of the inverse of the effective rank of the matrix-valued test process under the alternative.
  • Figure 2: Empirical findings with non-commuting i.i.d. random matrices for covariance testing. Left. Comparison of the cumulative distribution functions of rejection times of the sequential tests corresponding to the matrix- and scalar-valued test processes under the same alternative distribution. Middle. Evolution of the inverse of the effective rank of the matrix-valued test process under the alternative. Right. Comparison between the trace of the matrix-valued test process, and the scalar-valued test process under the same alternative distribution.

Theorems & Definitions (74)

  • Lemma 3.1
  • Theorem 3.2: Matrix Supermartingale Convergence Theorem
  • proof
  • Theorem 3.3: Matrix Optional Stopping
  • proof
  • Theorem 3.4: Matrix Ville's Inequality
  • proof
  • Lemma 3.5: Incremental Construction of Matrix Supermartingales
  • proof
  • Example 3.6: Matrix MGF Supermartingale
  • ...and 64 more