Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric

TL;DR

This paper introduces a method for summarizing families of probability measures using principal geodesic analysis within the Wasserstein geometry, enabling efficient data representation and interpretation.

Contribution

It extends PCA concepts to probability measures under optimal transport, proposing scalable algorithms with relaxed geodesics and regularization for practical applications.

Findings

Effective summarization of probability measures using geodesic curves.

Scalable algorithms demonstrated on image shape and color histogram data.

Bridging Euclidean PCA concepts with Wasserstein geometry.

Abstract

Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this problem under the optimal transport (Wasserstein) geometry, using curves that are restricted to be geodesic segments under that metric. We show that concepts that play a key role in Euclidean PCA, such as data centering or orthogonality of principal directions, find a natural equivalent in the optimal transport geometry, using Wasserstein means and differential geometry. The implementation of these ideas is, however, computationally challenging. To achieve scalable algorithms that can handle thousands of measures, we propose to use a relaxed definition for geodesics and regularized optimal transport distances. The interest of our approach is demonstrated on images seen either as shapes or color histograms.