On the limitation of spectral methods: From the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors

TL;DR

This paper investigates the limits of spectral methods in detecting hidden structures in Gaussian matrices and tensors, establishing tight thresholds for detectability based on eigenvalue analysis.

Contribution

It proves that spectral methods cannot detect Gaussian hidden cliques below a certain size, extending the results to rank-one perturbations of Gaussian tensors and establishing fundamental detection limits.

Findings

Spectral methods succeed when the clique size exceeds (1+ε)√n.

Spectral methods fail below (1−ε)√n, with no eigenvalue-based algorithm able to detect the clique.

A lower bound on the signal-to-noise ratio for detection in Gaussian tensor models.

Abstract

We consider the following detection problem: given a realization of a symmetric matrix ${\mathbf{X}}$ 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 where ${\mathbf{X}}$ is the sum of such matrix and an independent rank-one perturbation. This setup applies to the situation where under the alternative, there is a planted principal submatrix ${\mathbf{B}}$ of size $L$ for which all upper triangular variables are i.i.d. Gaussians with mean $1$ and variance $1$, whereas all other upper triangular elements of ${\mathbf{X}}$ not in ${\mathbf{B}}$ are i.i.d. Gaussians variables with mean 0 and variance $1$. We refer to this as the `Gaussian hidden clique problem.' When $L=(1+\epsilon)\sqrt{n}$ ($\epsilon>0$), it is possible to solve this detection problem with probability $1-o_n(1)$ by computing the spectrum of ${\mathbf{X}}$ and considering the largest eigenvalue of ${\mathbf{X}}$. We prove that this condition is tight in the following sense: when $L<(1-\epsilon)\sqrt{n}$ no algorithm that examines only the eigenvalues of ${\mathbf{X}}$ can detect the existence of a hidden Gaussian clique, with error probability vanishing as $n\to\infty$. We prove this result as 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.