On the detection of low rank matrices in the high-dimensional regime

TL;DR

This paper investigates the fundamental limits of detecting low-rank matrices in high-dimensional noisy settings, establishing conditions under which detection is impossible, thereby clarifying the phase transition threshold.

Contribution

It proves that when the largest singular value of the low-rank matrix is below a critical threshold, no consistent detection tests exist, extending previous results to more general cases.

Findings

Detection is possible if the largest singular value exceeds 1.

Detection is impossible if the largest singular value is below 1.

The proof simplifies previous approaches and applies to matrices of arbitrary rank.

Abstract

We address the detection of a low rank $n\times n$deterministic matrix $\mathbf{X}_{0}$ from the noisy observation ${\bf X}_{0}+{\bf Z}$ when $n\to\infty$, where ${\bf Z}$ is a complex Gaussian random matrix with independent identically distributed $\mathcal{N}_{c}(0,\frac{1}{n})$ entries. Thanks to large random matrix theory results, it is now well-known that if the largest singular value $\lambda_{1}$ of ${\bf X}_{0}$ verifies $\lambda_{1}>1$, then it is possible to exhibit consistent tests. In this contribution, we prove a contrario that under the condition $\lambda_{1}<1$, there are no consistent tests. Our proof is rather simple, inspired by previous works devoted to the case of rank 1 matrices ${\bf X}_{0}$.