Sharp Bounds for the Concentration of the Resolvent in Convex Concentration Settings

TL;DR

This paper establishes sharp probabilistic bounds for the concentration of the resolvent matrix of convex concentrated random matrices, extending understanding of spectral properties in high-dimensional probability.

Contribution

It provides new bounds on the concentration of the resolvent matrix for convex concentrated random matrices, using a novel resolvent series decomposition.

Findings

Bounds on the resolvent concentration with observable diameter

Extension of Talagrand's convex concentration to resolvent matrices

Improved understanding of spectral behavior in high-dimensional random matrices

Abstract

Considering random matrix $X \in \mathcal M_{p,n}$ with independent columns satisfying the convex concentration properties issued from a famous theorem of Talagrand, we express the linear concentration of the resolvent $Q = (I_p - \frac{1}{n}XX^T) ^{-1}$ around a classical deterministic equivalent with a good observable diameter for the nuclear norm. The general proof relies on a decomposition of the resolvent as a series of powers of $X$.