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Matrix Chaos Inequalities and Chaos of Combinatorial Type

Afonso S. Bandeira, Kevin Lucca, Petar Nizić-Nikolac, Ramon van Handel

TL;DR

This paper provides general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner and identifies a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule.

Abstract

Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. Such models have often been studied on a case-by-case basis using ad-hoc methods that can yield suboptimal dimensional factors. In this paper we provide general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner. These inequalities are expressed in terms of flattenings of the coefficients of the matrix chaos. We further identify a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule. This allows us to provide a unified treatment of and improved bounds for matrix chaoses that arise in a variety of applications, including graph matrices, Khatri-Rao matrices, and matrices that arise in average case analysis of the sum-of-squares hierarchy.

Matrix Chaos Inequalities and Chaos of Combinatorial Type

TL;DR

This paper provides general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner and identifies a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule.

Abstract

Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. Such models have often been studied on a case-by-case basis using ad-hoc methods that can yield suboptimal dimensional factors. In this paper we provide general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner. These inequalities are expressed in terms of flattenings of the coefficients of the matrix chaos. We further identify a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule. This allows us to provide a unified treatment of and improved bounds for matrix chaoses that arise in a variety of applications, including graph matrices, Khatri-Rao matrices, and matrices that arise in average case analysis of the sum-of-squares hierarchy.

Paper Structure

This paper contains 35 sections, 19 theorems, 98 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Matrix chaos inequalities
  3. Matrix chaos and decoupling
  4. Flattenings
  5. The main inequalities
  6. Iterated NCK
  7. Iterated strong NCK
  8. Iterated matrix Rosenthal
  9. Iterated strong matrix Rosenthal
  10. Iteration approach
  11. The linear case
  12. Iteration
  13. Chaos of combinatorial type
  14. Definition and guiding example
  15. How to compute norms of flattenings
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $X$ be any matrix chaos as in eq:chaos, and let $Y$ be the decoupled matrix chaos as in eq:decnongaussianchaos defined by the same random variables $h_1,\ldots,h_m$ and matrix coefficients $A_{i_1,\ldots,i_q}$ (where we set $A_{i_1,\ldots,i_q}=0$ when $i_1,\ldots,i_q$ are not distinct). Then we Moreover, this inequality can be reversed provided that the matrix coefficients are assumed to be

Figures (2)

  • Figure 1: Intermediate flattenings that arise from each matrix parameter. Here $\mathcal{B}$ is the (random) tensor of order $3$ associated to the linear chaos $\sum_{i_{\color{NavyBlue} 1}} h_{i_{\color{NavyBlue} 1}} B_{i_{\color{NavyBlue} 1}}$ in \ref{['eq:YaslinearChaos']}.
  • Figure 2: Using Theorem \ref{['thm:gmnormbound']} on graph matrices from Example \ref{['ex:graphmatrices']} yields the following bounds (logarithmic factors omited): $\left\lVert M_\beta\right\rVert \approx \sqrt{n}$, $\left\lVert M_\gamma\right\rVert \approx n$, $\left\lVert M_\delta\right\rVert \approx n^3$.

Theorems & Definitions (53)

  • Theorem 2.1: Decoupling inequalities
  • Remark 2.2: Lower bounding $\mathbb{E}\left\lVert X\right\rVert$
  • Remark 2.3: More general chaoses
  • Theorem 2.4: Iterated NCK
  • Theorem 2.5: Iterated strong NCK
  • Theorem 2.6: Iterated matrix Rosenthal
  • Theorem 2.7: Iterated strong Matrix Rosenthal
  • Example 3.1: Khatri-Rao matrices
  • Definition 3.2
  • Definition 3.3: Matrix Chaos of Combinatorial type
  • ...and 43 more