On the Rank of Random Sparse Matrices

TL;DR

This paper studies the rank properties of random sparse matrices, revealing that dependencies are typically caused by small sets of rows with many zeros, enabling precise co-rank estimation.

Contribution

It provides a novel probabilistic analysis linking row dependencies to zero-rich small subsets, leading to exact co-rank estimates for sparse matrices.

Findings

Dependencies involve few rows with many zeros

Exact co-rank estimates are derived

High probability results for sparse matrices

Abstract

We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.