Consistency of archetypal analysis

TL;DR

This paper proves the consistency of archetypal analysis, showing that archetype points converge to a continuum solution as sample size increases, with additional results for unbounded data distributions.

Contribution

It establishes the first rigorous consistency results for archetypal analysis, including convergence rates and modifications for unbounded data.

Findings

Archetype points converge to a continuum solution with bounded support.

Convergence rate of the optimal objective values is established.

Modified method works for unbounded distributions, with proven consistency.

Abstract

Archetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. 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 distance between the data and the polytope is minimal. In this paper, we prove a consistency result that shows if the data is independently sampled from a probability measure with bounded support, then the archetype points converge to a solution of the continuum version of the problem, of which we identify and establish several properties. We also obtain the convergence rate of the optimal objective values under appropriate assumptions on the distribution. If the data is independently sampled from a distribution with unbounded support, we also prove a consistency result for a modified method that penalizes the dispersion of the archetype points. Our analysis is supported by detailed computational experiments of the archetype points for data sampled from the uniform distribution in a disk, the normal distribution, an annular distribution, and a Gaussian mixture model.