Estimation in the Spiked Wigner Model: A Short Proof of the Replica Formula

Abstract

We consider the problem of estimating the 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 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 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.

Publication
2018 IEEE International Symposium on Information Theory (ISIT)