Inference of principal components of noisy correlation matrices with prior information

TL;DR

This paper develops a statistical physics-based approach to infer the top principal component of noisy covariance matrices using prior information, demonstrating improved noise robustness in the spiked covariance model.

Contribution

It introduces a novel replica method framework to incorporate prior information into principal component inference in noisy settings.

Findings

Prior information increases the maximum noise level for successful recovery.

The phase diagram characterizes different regimes of component detectability.

The method outperforms classical approaches without prior knowledge.

Abstract

The problem of infering the top component of a noisy sample covariance matrix with prior information about the distribution of its entries is considered, in the framework of the spiked covariance model. Using the replica method of statistical physics the computation of the overlap between the top components of the sample and population covariance matrices is formulated as an explicit optimization problem for any kind of entry-wise prior information. The approach is illustrated on the case of top components including large entries, and the corresponding phase diagram is shown. The calculation predicts that the maximal sampling noise level at which the recovery of the top population component remains possible is higher than its counterpart in the spiked covariance model with no prior information.