Smallest singular value of sparse random matrices

TL;DR

This paper extends probability estimates for the smallest singular value of sparse random matrices, relaxing variance conditions and allowing for entries with small variances or null entries, without requiring identical distributions.

Contribution

It introduces new bounds on the smallest singular value for a broader class of sparse matrices by relaxing variance and distribution assumptions.

Findings

Relaxed variance conditions for sparse matrices

Allowed null entries and small variances

Established bounds without identical distribution assumption

Abstract

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries. We also show that it is enough to require boundedness from above of the $r$-th moment, $r > 2$, of the corresponding entries.