Non-asymptotic upper bounds for the reconstruction error of PCA

TL;DR

This paper provides non-asymptotic upper bounds for PCA reconstruction error, improving existing bounds and offering insights into excess risk behavior under mild eigenvalue conditions.

Contribution

It introduces unified, tighter non-asymptotic bounds for PCA excess risk, enhancing understanding of reconstruction error beyond traditional subspace distance metrics.

Findings

Bounds unify and improve previous results

Oracle inequalities hold under mild eigenvalue conditions

Excess risk differs from subspace distance measures

Abstract

We analyse the reconstruction error of principal component analysis (PCA) and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. The bounds reveal that the excess risk differs significantly from usually considered subspace distances based on canonical angles. Our approach relies on the analysis of empirical spectral projectors combined with concentration inequalities for weighted empirical covariance operators and empirical eigenvalues.