Smoothed analysis of the least singular value without inverse Littlewood--Offord theory

High Dimensional Seminar
Wednesday, November 6, 2019 - 3:00pm for 1 hour (actually 50 minutes)
Vishesh Jain – MIT
Galyna Livshyts

We will discuss a novel approach to obtaining non-asymptotic estimates on the lower tail of the least singular value of an $n \times n$ random matrix $M_{n} := M + N_{n}$, where $M$ is a fixed matrix with operator norm at most $O(\exp(n^{c}))$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a random variable with mean 0 and variance 1. This has been previously considered in a series of works by Tao and Vu, and our results improve upon theirs in two ways: 

(i) We are able to deal with $\|M\| = O(\exp(n^{c}))$ whereas previous work was applicable for $\|M\| = O(\poly(n))$. 

(ii) Even for $\|M\| = O(poly(n))$, we are able to extract more refined information – for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(exp(-n^{c}))$, whereas even in the case when $N_n$ is an i.i.d. Bernoulli matrix, the results of Tao and Vu only give inverse polynomial singularity probability.