Fundamental limits of detection in the spiked Wigner model

TL;DR

This paper investigates the fundamental detection limits of a rank-one spike in Wigner matrices, establishing a phase transition threshold for detection and estimation using advanced probabilistic methods.

Contribution

It proves the asymptotic normality of the likelihood ratio below a certain non-spectral threshold, defining the maximal contiguity region for detection and estimation.

Findings

Detection is impossible below the threshold and possible above.

Likelihood ratio is asymptotically normal below the threshold.

Identifies a phase transition for both detection and estimation.

Abstract

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a "spectral" nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. We also provide further information about the performance of the optimal test. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.