PCA of Probability Measures: Sparse and Dense Sampling Regimes

Erell Gachon, Jérémie Bigot, Elsa Cazelles
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:32441-32477, 2026.

Abstract

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-gachon26a, title = {{PCA} of Probability Measures: Sparse and Dense Sampling Regimes}, author = {Gachon, Erell and Bigot, J\'{e}r\'{e}mie and Cazelles, Elsa}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {32441--32477}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/gachon26a/gachon26a.pdf}, url = {https://proceedings.mlr.press/v306/gachon26a.html}, abstract = {A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.} }
Endnote
%0 Conference Paper %T PCA of Probability Measures: Sparse and Dense Sampling Regimes %A Erell Gachon %A Jérémie Bigot %A Elsa Cazelles %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-gachon26a %I PMLR %P 32441--32477 %U https://proceedings.mlr.press/v306/gachon26a.html %V 306 %X A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.
APA
Gachon, E., Bigot, J. & Cazelles, E.. (2026). PCA of Probability Measures: Sparse and Dense Sampling Regimes. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:32441-32477 Available from https://proceedings.mlr.press/v306/gachon26a.html.

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