A Geometric Analysis of PCA
Ayoub El Hanchi, Murat A Erdogdu, Chris J. Maddison
Abstract
What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $\pi/4$.
BibTeX
@inproceedings{
hanchi2025a,
title={A Geometric Analysis of {PCA}},
author={Ayoub El Hanchi and Murat A Erdogdu and Chris J. Maddison},
booktitle={The Thirty-ninth Annual Conference on Neural Information Processing Systems},
year={2025},
url={https://openreview.net/forum?id=KEVjRT4haB}
}