Extremal random matrices with independent entries and matrix superconcentration inequalities

TL;DR

This paper establishes optimal nonasymptotic concentration inequalities for the spectral norm of subgaussian random matrices with independent entries, capturing Tracy-Widom fluctuations and improving previous bounds.

Contribution

It introduces new extremal bounds for matrix moments based on variance patterns, leading to the best possible tail behavior results for such matrices.

Findings

Derived sharp bounds on large moments of Gaussian Wishart matrices.

Established extremal variance patterns maximizing matrix moments.

Provided improved nonasymptotic concentration inequalities at Tracy-Widom scale.

Abstract

We prove nonasymptotic matrix concentration inequalities for the spectral norm of (sub)gaussian random matrices with centered independent entries that capture fluctuations at the Tracy-Widom scale. This considerably improves previous bounds in this setting due to Bandeira and Van Handel, and establishes the best possible tail behavior for random matrices with an arbitrary variance pattern. These bounds arise from an extremum problem for nonhomogeneous random matrices: among all variance patterns with a given sparsity parameter, the moments of the random matrix are maximized by block-diagonal matrices with i.i.d. entries in each block. As part of the proof, we obtain sharp bounds on large moments of Gaussian Wishart matrices.