Smoothed analysis of symmetric random matrices with continuous distributions

TL;DR

This paper proves that symmetric matrices formed by adding a deterministic symmetric matrix to a symmetric random matrix with continuous, potentially heavy-tailed entries are invertible with high probability, with bounds independent of the deterministic part.

Contribution

It establishes a universal invertibility bound for symmetric matrices with continuous distribution entries, regardless of the deterministic component or tail heaviness.

Findings

Invertibility bound of O(n^2) with high probability

Bound independent of the deterministic matrix D

No moment assumptions required on the entries of R

Abstract

We study invertibility of matrices of the form $D+R$ where $D$ is an arbitrary symmetric deterministic matrix, and $R$ is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that $|(D+R)^{-1}| = O(n^2)$ with high probability. The bound is completely independent of $D$. No moment assumptions are placed on $R$; in particular the entries of $R$ can be arbitrarily heavy-tailed.