The Smallest Singular Value of a Shifted Random Matrix

TL;DR

This paper provides improved probabilistic bounds on the smallest singular value of a sum of a random subgaussian matrix and a deterministic matrix with controlled norm, extending previous results by Tao and Vu.

Contribution

It offers a more general and sharper estimate for the smallest singular value of shifted random matrices with deterministic perturbations.

Findings

Enhanced bounds for the smallest singular value of shifted random matrices.

Applicable to matrices with deterministic shifts of norm up to n^γ.

Improves upon previous results by Tao and Vu in generality and tightness.

Abstract

Let $R_n$ be a $n \times n$ random matrix with i.i.d. subgaussian entries. Let $M$ be a $n \times n$ deterministic matrix with norm $\lVert M \rVert \le n^\gamma$ where $1/2<\gamma<1$. The goal of this paper is to give a general estimate of the smallest singular value of the sum $R_n + M$, which improves an earlier result of Tao and Vu.