Consistency of Archetypal Analysis

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