Functional data analysis for multivariate distributions through Wasserstein slicing

Han Chen, Hans-Georg Müller

Keywords: Distributional data analysis, multivariate distributional data, Radon transform, Sliced Wasserstein distance

TL;DR: We introduce an invertible Wasserstein slicing map that embeds multivariate distributions into a Hilbert space, enabling functional data analysis with theoretical guarantees and application to real-world data.

Abstract: The modeling of samples of distributions is a major challenge since distributions do not form a vector space. While various approaches exist for univariate distributions, including transformations to a Hilbert space, far less is known about the multivariate case. We utilize a transformation approach to map multivariate distributions to a Hilbert space via a Wasserstein slicing method that is invertible. This approach combines functional data analysis tools, such as functional principal component analysis and modes of variation, with the facility to map back to interpretable distributions. We also provide convergence guarantees for the Hilbert space representations under a broad class of such transforms. The method is illustrated using joint systolic and diastolic blood pressure data.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 19328