Wasserstein Archetypal Analysis

TL;DR

This paper introduces Wasserstein archetypal analysis (WAA), a novel approach that uses the Wasserstein metric to summarize data with convex polytopes, providing theoretical guarantees and computational methods, especially in low dimensions.

Contribution

The work develops a new Wasserstein-based formulation of archetypal analysis, proves existence and consistency results, and introduces a gradient-based algorithm for practical computation.

Findings

Unique solution in 1D for WAA

Existence of solutions in 2D for absolutely continuous data

Convergence of archetype points with increasing sample size

Abstract

Archetypal analysis is an unsupervised machine learning method that summarizes data using a convex polytope. In its original formulation, for fixed k, the method finds a convex polytope with k vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared Euclidean distance between the data and the polytope is minimal. In the present work, we consider an alternative formulation of archetypal analysis based on the Wasserstein metric, which we call Wasserstein archetypal analysis (WAA). In one dimension, there exists a unique solution of WAA and, in two dimensions, we prove existence of a solution, as long as the data distribution is absolutely continuous with respect to Lebesgue measure. We discuss obstacles to extending our result to higher dimensions and general data distributions. We then introduce an appropriate regularization of the problem, via a Renyi entropy, which allows us to obtain existence of solutions of the regularized problem for general data distributions, in arbitrary dimensions. We prove a consistency result for the regularized problem, ensuring that if the data are iid samples from a probability measure, then as the number of samples is increased, a subsequence of the archetype points converges to the archetype points for the limiting data distribution, almost surely. Finally, we develop and implement a gradient-based computational approach for the two-dimensional problem, based on the semi-discrete formulation of the Wasserstein metric. Our analysis is supported by detailed computational experiments.