# A Geometric Analysis of PCA

arXiv: https://arxiv.org/abs/2510.20978 · HTML: https://mlsanity.com/paper/2510.20978/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 23 Oct 2025.

## Builds on (most cited first, 8 of 8)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Finite-sample analysis of M-estimators using self-concordance](https://mlsanity.com/paper/1810.06838/) (2018)
- [Minimax Linear Regression under the Quantile Risk](https://mlsanity.com/paper/2406.12145/) (2024)
- [Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Riemannian Steepest Descent](https://mlsanity.com/paper/2209.03480/) (2022)

## Built on by (most cited first, 2 of 2)

- [A Random Matrix Theory of Masked Self-Supervised Regression](https://mlsanity.com/paper/2601.23208/) (2026)
- [Diagonal Attenuation: A Finite-Sample Correction for PCA](https://mlsanity.com/paper/2609.05796/) (2026)

## Abstract (arXiv)

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.