Estimating the principal components of correlation matrices from all their empirical eigenvectors

TL;DR

This paper develops a method to estimate all principal components of a correlation matrix from limited data by leveraging the full spectrum of eigenvectors, using tools from random matrix theory and information theory.

Contribution

It introduces a novel approach that utilizes all eigenvectors of the sample correlation matrix for principal component estimation, incorporating prior information through a spin-glass-like model.

Findings

All eigenmodes of the sample correlation matrix contain useful information.

The method improves estimation when prior knowledge about the principal component is available.

Numerical results demonstrate effectiveness on the spiked covariance model.

Abstract

We consider the problem of estimating the principal components of a population correlation matrix from a limited number of measurement data. Using a combination of random matrix and information-theoretic tools, we show that all the eigenmodes of the sample correlation matrices are informative, and not only the top ones. We show how this information can be exploited when prior information about the principal component, such as whether it is localized or not, is available by mapping the estimation problem onto the search for the ground state of a spin-glass-like effective Hamiltonian encoding the prior. Results are illustrated numerically on the spiked covariance model.