The Rank and Singular Values of the Inhomogeneous Subgaussian Random Matrices

TL;DR

This paper analyzes the rank and singular values of inhomogeneous subgaussian random matrices, extending previous results by removing the i.i.d. assumption and providing new deviation inequalities.

Contribution

It introduces a method to handle inhomogeneous matrices with different subgaussian moments, extending Rudelson's results to non-i.i.d. matrices using RLCD techniques.

Findings

Probability of low rank is sub-exponential for certain k

Deviation inequality for singular values established

Model accommodates inhomogeneous entries with bounded subgaussian moments

Abstract

Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we get a deviation inequality for the singular values of A. This extends earlier results of Rudelson's paper in 2024 by removing the assumption of the identical distribution of the entries across the matrix. Our model covers inhomogeneous matrices, allowing different subgaussian moments for the entries as long as their subgaussian moments have a standard upper bound. In the past advance, the assumption of i.i.d entries was required due to the lack of least common denominators of the non-i.i.d random matrix. We can overcome this problem using a randomized least common denominator (RLCD) from Livshyts in 2021.