PCA of probability measures

In this talk, we will see how to apply Principal Component Analysis (PCA) to data that come in the form of probability measures rather than standard vectors. A common strategy is to embed the probability measures into a suitable space where PCA can be applied. We will study the estimation of PCA of probability measures in a double asymptotic regime, where $n$ measures are observed, each through $m$ samples. The convergence rates we will derive characterize the relationship between the number of measures and the number of samples. Through numerical experiments, we will demonstrate how carefully reducing the number of samples can significantly lower computational costs while preserving PCA accuracy.