A Geometric Analysis of PCA

2025/10/23 by Ayoub El Hanchi, Hanchi, Ayoub El, Murat Erdogdu +3 · 1 voice
#math.ST #stat.ML

paper · pdf · doi:10.48550/arxiv.2510.20978

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 π/4.

Citations

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Discussions

• arxiv.org/abs/2510.20978 'A Geometric Analysis of PCA' - Ayoub El Hanchi, Murat Erdogdu, Chris Maddison Viewing PCA as the problem of estimating a dominant subspace, or as solving a compression proble [bsky, 2 points, 1 comments]

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