Upper bound for intermediate singular values of random matrices

TL;DR

This paper establishes probabilistic upper bounds on the intermediate singular values of random matrices with independent subgaussian entries, extending to rectangular matrices, and providing insights into their spectral behavior.

Contribution

It introduces new probabilistic bounds for intermediate singular values of subgaussian random matrices, generalizing previous results to rectangular matrices.

Findings

Upper bounds for singular values with high probability

Results applicable to rectangular matrices

Matching lower bounds for singular values

Abstract

In this paper, we prove that an $n\times n$ matrix $A$ with independent centered subgaussian entries satisfies \[ s_{n+1-l}(A) \le C_1t \frac{l}{\sqrt{n}} \] with probability at least $1-\exp(-C_2tl)$. This yields $s_{n-l}(A) \sim \frac{cl}{\sqrt{n}}$ in combination with a known lower bound. These results can be generalized to the rectangular matrix case.