Invertibility of Sparse Complex Gaussian Matrices

TL;DR

This paper investigates the invertibility of sparse complex Gaussian matrices, extending known results about the least singular value from dense to sparse random matrices, revealing similar probabilistic bounds.

Contribution

It establishes probabilistic bounds on the least singular value for sparse complex Gaussian matrices, generalizing classical results from dense matrices to the sparse case.

Findings

Similar bounds for sparse matrices as in dense cases

Probabilistic estimates for invertibility of sparse matrices

Extension of known results to complex Gaussian ensembles

Abstract

Let $\sigma_n(\cdot)$ denote the least singular value of a $n \times n$ matrix. It is well-known that $\mathbb{P}[\sigma_n(A) \le \varepsilon] \le \varepsilon n$ if $A$ is drawn from the real Ginibre ensemble of $n \times n$ matrices and $\mathbb{P}[\sigma_n(A) \le \varepsilon] \le \varepsilon^2 n^2$ if $A$ is drawn from the complex Ginibre ensemble. In this paper, we will show a similar phenomenon occurs for sparse random matrices.