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

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 $\lambda_{1}(\mathbf{X}_{0})$ of ${\bf X}_{0}$ verifies $\lambda_{1}(\mathbf{X}_{0})>1$, then it is possible to exhibit consistent tests. In this contribution, we prove \textit{a contrario } that under the condition $\lambda_{1}(\mathbf{X}_{0})<1$, there are no consistent tests. Our proof is inspired by previous works devoted to the case of rank 1 matrices ${\bf X}_{0}$.