The singular values and vectors of low rank perturbations of large rectangular random matrices

TL;DR

This paper analyzes how low-rank perturbations affect the extreme singular values and vectors of large rectangular random matrices, revealing phase transitions and convergence properties in the large matrix limit.

Contribution

It provides a rigorous proof of the almost sure convergence of singular values and vectors under low-rank perturbations, extending previous eigenvalue results to singular values.

Findings

Extreme singular values converge almost surely.

Phase transition depends on perturbation singular values.

Finite-size fluctuations are characterized.

Abstract

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalue aspect of the problem, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transforms that linearizes rectangular additive convolution in free probability theory. The large matrix limit of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform. We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the finite $n$ fluctuations above these non-random limits.