On the Largest and the Smallest Singular Value of Sparse Rectangular Random Matrices

TL;DR

This paper provides estimates for the largest and smallest singular values of sparse rectangular random matrices with specific sparsity and moment conditions, extending understanding of their spectral properties in high-dimensional regimes.

Contribution

It introduces new bounds for singular values of sparse rectangular matrices under particular sparsity and moment assumptions, advancing spectral analysis in high-dimensional probability.

Findings

Derived bounds for largest singular value.

Derived bounds for smallest singular value.

Applicable to matrices with specific sparsity and moment conditions.

Abstract

We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming $\lim_{N,n\to\infty}\frac nN=y\in(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_N\sim \log^{\alpha }N$ for some $\alpha>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf E|X_{jk}|^{4+\delta}\le C<\infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{\frac12-\varkappa}$ with $\varkappa:=\frac{\delta}{2(4+\delta)}$.