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Simple Norm Bounds for Polynomial Random Matrices via Decoupling

Madhur Tulsiani, June Wu

TL;DR

This work develops a unified, elementary framework to bound spectral norms of random matrices whose entries are low-degree polynomials in independent variables. By iteratively applying decoupling, Hermitian dilation, and matrix Rosenthal-type inequalities, the authors reduce the problem to bounding a handful of deterministic derivative matrices ${\bf F}_{a,b,c}$, yielding bounds that recover many previous results with simpler arguments. A key innovation is improved decoupling for index-structured problems (notably graph-structured indices), which leads to tight bounds controlled by vertex separators in dense and sparse regimes. They also extend the approach to Gaussian inputs and illustrate applications to graph matrices and tensor-network models, including the melon graph, demonstrating practical impact for spectral algorithms and complexity lower bounds. Overall, the paper provides a versatile, scalable method for nonlinear random matrices that broadens the toolkit for spectral analysis in data-dependent settings.

Abstract

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

Simple Norm Bounds for Polynomial Random Matrices via Decoupling

TL;DR

This work develops a unified, elementary framework to bound spectral norms of random matrices whose entries are low-degree polynomials in independent variables. By iteratively applying decoupling, Hermitian dilation, and matrix Rosenthal-type inequalities, the authors reduce the problem to bounding a handful of deterministic derivative matrices , yielding bounds that recover many previous results with simpler arguments. A key innovation is improved decoupling for index-structured problems (notably graph-structured indices), which leads to tight bounds controlled by vertex separators in dense and sparse regimes. They also extend the approach to Gaussian inputs and illustrate applications to graph matrices and tensor-network models, including the melon graph, demonstrating practical impact for spectral algorithms and complexity lower bounds. Overall, the paper provides a versatile, scalable method for nonlinear random matrices that broadens the toolkit for spectral analysis in data-dependent settings.

Abstract

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

Paper Structure

This paper contains 21 sections, 16 theorems, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analyzing concentration via moment estimates.
  3. Efron-Stein inequalities.
  4. Decoupling.
  5. Gaussian polynomial matrices.
  6. Dense graph matrices.
  7. Sparse graph matrices.
  8. Preliminaries and Notation
  9. Sets and indices.
  10. Matrix norms.
  11. Polynomial random matrices.
  12. Partial derivatives.
  13. Non-Hermitian matrices.
  14. Constants.
  15. Decoupling Inequalities
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let ${\bf x} = \{x_i\}_{i = 1}^n$ be a sequence of i.i.d random variables with ${\mathbb E} x_i= 0$, ${\mathbb E} x_i^2= 1$ and $|x_i| \leq L$ for all $1\leq i \leq n$. Let $\{{\bf A}_{{\bf i}}\}_{{\bf i} \in {\mathcal{T}}^d_n}$ be a multi-indexed sequence of deterministic matrices of the same dimen Let $a,b,c\in {\mathbb Z}_{\geq 0}$ and $d = a+b+c$. Then for $2 \leq t \leq \infty,$

Figures (4)

  • Figure 1: A shape $\tau$
  • Figure 2: $\mathbf{F}_{1,0,0}$, $\mathbf{F}_{0,1,0}$ and $\mathbf{F}_{2,3,0}$
  • Figure 3: $\mathbf{F}_{0,0,1}$
  • Figure 4: The Melon Graph

Theorems & Definitions (44)

  • Theorem 1.1: Restatement of \ref{['homo multilinear recursion']}
  • Theorem 1.2: Restatement of \ref{['gaussian recursion']}
  • Definition 2.1: Schatten norm
  • Example 2.2: Rademacher chaos of order 2
  • Remark 2.3
  • Definition 2.4: Permutation Symmetric Property
  • Remark 2.5
  • Definition 2.6: Hermitian Dilation, see Tro15 Sec. 2.1.16
  • Lemma 3.1: Decoupling Inequality, see PG:book Theorem 3.1.1
  • Lemma 3.2
  • ...and 34 more