Simple Norm Bounds for Polynomial Random Matrices via Decoupling

TL;DR

This paper introduces a straightforward decoupling-based method to bound the spectral norm of polynomial random matrices, simplifying analysis and unifying previous complex bounds in spectral and optimization contexts.

Contribution

The paper develops a novel, simple decoupling and linearization approach for norm bounds of polynomial random matrices, streamlining previous complex techniques.

Findings

Provides a simple, elementary proof for norm bounds

Recovers many previous bounds with less technical complexity

Applicable to matrices arising in spectral and optimization algorithms

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.