Wasserstein Archetypal Analysis
Katy Craig, Braxton Osting, Dong Wang, and Yiming Xu

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…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Anomaly Detection Techniques and Applications
