On the Limitation of Spectral Methods: From the Gaussian Hidden Clique Problem to Rank-One Perturbations of Gaussian Tensors

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 + epsilon)root n (epsilon > 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 epsilon)root n no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as n -> infinity. 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.