Robust Principal Component Analysis in Hilbert spaces

TL;DR

This paper introduces a stable, smooth spectral cutoff approach to PCA in Hilbert spaces, improving robustness by estimating the covariance operator from samples and analyzing approximation quality.

Contribution

It presents a novel spectral cutoff method for PCA in Hilbert spaces, replacing eigenvector projections with smooth eigenvalue functions, and studies its statistical properties.

Findings

Provides bounds on spectral cutoff approximation quality

Analyzes eigenvalue estimation errors from samples

Demonstrates improved stability over classical PCA

Abstract

We propose a stable version of Principal Component Analysis (PCA) in the general framework of a separable Hilbert space. It consists in interpreting the projection on the first eigenvectors as a step function applied to the spectrum of the covariance operator and in replacing it with a smooth cut-off of the eigenvalues. We study the problem from a statistical point of view, so that we assume that we do not have direct access to the covariance operator but we have to estimate it from an i.i.d. sample. We provide some results on the quality of the approximation of our spectral cut-off in terms of the quality of the approximation of the eigenvalues of the covariance operator.