  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.
