Inconsistency of diagonal scaling under high-dimensional limit: a replica approach

TL;DR

This paper uses a replica approach to show that diagonal scaling of sample covariance matrices becomes asymptotically inconsistent in high-dimensional settings with Gaussian spike models, contrasting with PCA results.

Contribution

It introduces a non-rigorous replica method analysis revealing inconsistency of diagonal scaling in high dimensions, differing from known PCA behaviors.

Findings

Diagonal scaling is asymptotically inconsistent in high dimensions.

Numerical experiments support the theoretical formulas.

Contrasts with PCA results on similar models.

Abstract

In this note, we claim that diagonal scaling of a sample covariance matrix is asymptotically inconsistent if the ratio of the dimension to the sample size converges to a positive constant, where population is assumed to be Gaussian with a spike covariance model. Our non-rigorous proof relies on the replica method developed in statistical physics. In contrast to similar results known in literature on principal component analysis, the strong inconsistency is not observed. Numerical experiments support the derived formulas.