On the detection of low rank matrices in the high-dimensional regime
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We address the detection of a low rank n× ndeterministic matrix X_0 from the noisy observation X_0+ Z when n→∞, where Z is a complex Gaussian random matrix with independent identically distributed N_c(0,1/n) entries. Thanks to large random matrix theory results, it is now well-known that if the largest singular value λ_1(X_0) of X_0 verifies λ_1(X_0)>1, then it is possible to exhibit consistent tests. In this contribution, we prove a contrario that under the condition λ_1(X_0)<1, there are no consistent tests. Our proof is inspired by previous works devoted to the case of rank 1 matrices X_0.
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