Estimation in the spiked Wigner model: A short proof of the replica formula

TL;DR

This paper provides a concise proof of the replica-symmetric formula for the mutual information in the spiked Wigner model, clarifying the fundamental limits of high-dimensional PCA under low signal-to-noise ratios.

Contribution

It offers a simple, transparent proof of the replica formula using Gaussian interpolations and concentration arguments, also extendable to spiked tensor models.

Findings

Confirmed the replica-symmetric formula for mutual information

Demonstrated the proof's applicability to tensor models

Clarified the fundamental limits of estimation in the model

Abstract

We consider the problem of estimating a rank-one perturbation of a Wigner matrix in a setting of low signal-to-noise ratio. This serves as a simple model for principal component analysis in high dimensions. The mutual information per variable between the spike and the observed matrix, or equivalently, the normalized Kullback-Leibler divergence between the planted and null models are known to converge to the so-called {\em replica-symmetric} formula, the properties of which determine the fundamental limits of estimation in this model. We provide in this note a short and transparent proof of this formula, based on simple executions of Gaussian interpolations and standard concentration-of-measure arguments. The \emph{Franz-Parisi potential}, that is, the free entropy at a fixed overlap, plays an important role in our proof. Furthermore, our proof can be generalized straightforwardly to spiked tensor models of even order.