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arXiv · 2510.20978

A Geometric Analysis of PCA

Ayoub El Hanchi·Murat Erdogdu·Chris Maddison
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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 $\pi/4$.

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Ayoub El Hanchi, Murat Erdogdu, Chris Maddison. 2025-10-23. A Geometric Analysis of PCA. https://arxiv.org/abs/2510.20978

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