Extreme singular values of inhomogeneous sparse random rectangular matrices

TL;DR

This paper introduces a unified method for bounding the extreme singular values of inhomogeneous sparse rectangular matrices using non-backtracking operators, extending classical results to sparse, inhomogeneous cases.

Contribution

It develops probabilistic bounds for singular values of inhomogeneous sparse matrices via non-backtracking operators, extending Bai-Yin law to sparse bipartite graphs.

Findings

Bounds are given in terms of row and column norms of the variance profile.

The method applies to sparse Erdős-Rényi bipartite graphs.

No outliers outside the Marčenko-Pastur support almost surely.

Abstract

We develop a unified approach to bounding the largest and smallest singular values of an inhomogeneous random rectangular matrix, based on the non-backtracking operator and the Ihara-Bass formula for general random Hermitian matrices with a bipartite block structure. We obtain probabilistic upper (respectively, lower) bounds for the largest (respectively, smallest) singular values of a large rectangular random matrix $X$. These bounds are given in terms of the maximal and minimal $\ell_2$-norms of the rows and columns of the variance profile of $X$. The proofs involve finding probabilistic upper bounds on the spectral radius of an associated non-backtracking matrix $B$. The two-sided bounds can be applied to the centered adjacency matrix of sparse inhomogeneous Erd\H{o}s-R\'{e}nyi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erd\H{o}s-R\'{e}nyi bipartite graphs $G(n,m,p)$ with $p=\omega(\log n)/n$, and $m/n\to y \in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Mar\v{c}enko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.