Pré-Publication, Document De Travail Année : 2026

PCA of Probability Measures: Sparse and Dense Sampling Regimes

Résumé

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 -α for the empirical covariance operator and the PCA excess risk, where α > 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.

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hal-05489756 , version 1 (02-02-2026)

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  • HAL Id : hal-05489756 , version 1

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Erell Gachon, Jérémie Bigot, Elsa Cazelles. PCA of Probability Measures: Sparse and Dense Sampling Regimes. 2026. ⟨hal-05489756⟩
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