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

Antoine Chevreuil; Philippe Loubaton

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arXiv · 1804.04851

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

Antoine Chevreuil·Philippe Loubaton

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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 ${\bf X}_{0}$.

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BibTeX RIS

Antoine Chevreuil, Philippe Loubaton. 2018-04-13. On the detection of low rank matrices in the high-dimensional regime. https://arxiv.org/abs/1804.04851

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