Extreme singular values of inhomogeneous sparse random rectangular matrices

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 Erdős-Rényi bipartite graphs for a wide range of sparsity, down to criticality. In particular, for Erdős-Rényi bipartite graphs $\mathcal{G}(n,m,p)$ with $p=\mathit{\omega }(logn)∕n$, and $m∕n\to y\in (0,1)$, our sharp bounds imply that there are no outliers outside the support of the Marčenko-Pastur law almost surely. This result extends the Bai-Yin theorem to sparse rectangular random matrices.