We consider the following detection problem: given a realization of a symmetric matrix X of dimension n, distinguish between the hypothesis that all upper triangular variables are i.i.d. Gaussians variables with mean 0 and variance 1 and the hypothesis that there is a planted principal submatrix B of dimension L for which all upper triangular variables are i.i.d. Gaussians with mean 1 and variance 1, whereas all other upper triangular elements of X not in B are i.i.d. Gaussians variables with mean 0 and variance 1. We refer to this as the 'Gaussian hidden clique problem'. When L - (1 + ∈)n~(1/2) (∈ ＞ 0), it is possible to solve this detection problem with probability 1 - o_n(1) by computing the spectrum of X and considering the largest eigenvalue of X. We prove that when L ＜ (1 - ∈)n~(1/2) no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as n → ∞. The result above is an immediate consequence of a more general result on rank-one perturbations of k-dimensional Gaussian tensors. In this context we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.
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