On the Wasserstein Geodesic Principal Component Analysis of probability measures
This work addresses the challenge of analyzing variability in datasets of probability measures, which is incremental as it extends existing GPCA methods to broader classes of distributions.
The paper tackles the problem of performing Geodesic Principal Component Analysis (GPCA) on probability distributions using Wasserstein geometry, achieving a method that works for both Gaussian distributions and more general absolutely continuous measures via neural network parameterization.
This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.