Clustering in Hilbert simplex geometry

TL;DR

This paper introduces Hilbert simplex geometry as a new, computationally efficient way to cluster categorical distributions within the probability simplex, comparing it with existing geometric methods and demonstrating its effectiveness.

Contribution

The work presents a novel Hilbert simplex geometry framework for clustering, providing a new distance measure and benchmarking its performance against traditional information-geometric approaches.

Findings

Hilbert simplex geometry is computationally friendly for clustering.

Benchmark results show competitive performance of Hilbert geometry.

Comparison with Fisher and other divergences highlights advantages and limitations.

Abstract

Clustering categorical distributions in the finite-dimensional probability simplex is a fundamental task met in many applications dealing with normalized histograms. Traditionally, the differential-geometric structures of the probability simplex have been used either by (i) setting the Riemannian metric tensor to the Fisher information matrix of the categorical distributions, or (ii) defining the dualistic information-geometric structure induced by a smooth dissimilarity measure, the Kullback-Leibler divergence. In this work, we introduce for clustering tasks a novel computationally-friendly framework for modeling geometrically the probability simplex: The {\em Hilbert simplex geometry}. In the Hilbert simplex geometry, the distance is the non-separable Hilbert's metric distance which satisfies the property of information monotonicity with distance level set functions described by polytope boundaries. We show that both the Aitchison and Hilbert simplex distances are norm distances on normalized logarithmic representations with respect to the $\ell_2$ and variation norms, respectively. We discuss the pros and cons of those different statistical modelings, and benchmark experimentally these different kind of geometries for center-based $k$-means and $k$-center clustering. Furthermore, since a canonical Hilbert distance can be defined on any bounded convex subset of the Euclidean space, we also consider Hilbert's geometry of the elliptope of correlation matrices and study its clustering performances compared to Fr\"obenius and log-det divergences.