Tail Bounds on the Spectral Norm of Sub-Exponential Random Matrices

TL;DR

This paper derives upper tail bounds for the spectral norm of symmetric random matrices with sub-Exponential entries, extending known results from Gaussian matrices using a novel chaining technique.

Contribution

It introduces a new chaining method tailored for sub-Exponential entries to establish spectral norm bounds, filling a gap in deviation inequalities.

Findings

Established upper tail bounds for sub-Exponential matrices

Extended Gaussian deviation results to broader distributions

Developed a novel chaining argument for spectral norm analysis

Abstract

Let $X$ be an $n\times n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-Exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.