Clustering in Hilbert simplex geometry

Abstract

Clustering categorical distributions in the 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 this clustering task a novel computationally-friendly framework for modeling the probability simplex termed Hilbert simplex geometry. In the Hilbert simplex geometry, the distance function is described by a polytope. We discuss the pros and cons of those different statistical modelings, and benchmark experimentally these geometries for center-based k-means and k-center clusterings. We show that Hilbert metric in the probability simplex satisfies the property of information monotonicity. Furthermore, since a canonical Hilbert metric distance can be defined on any bounded convex subset of the Euclidean space, we also consider Hilbert's projective geometry of the elliptope of correlation matrices and study its clustering performances.

Taxicab Correspondence Analysis and Taxicab Logratio Analysis: A Comparison on Contingency Tables and Compositional Data

“…This has the practical-useful consequence that a lot of classical multivariate statistical theory can easily be applied in CoDA. But other geometries in the simplex have been studied, see for instance Nielsen and Sun (2017), which to our knowledge have not been applied in CoDA; TSVD falls in this category.…”

Inductive Geometric Matrix Midranges

“…For example, see the adaptation of K-means clustering to several similarity measures considered by Stanitsas et al (2017), which include the affine-invariant Riemannian (AIRM) and log-Euclidean metrics. See also Nielsen and Sun (2017) for an account of clustering in Hilbert simplex geometry. In this work, we consider K-means clustering with the Thompson distance d ∞ as the similarity measure, and employ the IMR to calculate centroids for clusters.…”

Manifold Structured Prediction

“…In the experiment we considered = 1e − 5. The geodesic induced by the Fisher metric is, d(y, y ) = arccos m i=1 y i y i [26]. This geodesic comes from applying the map π : ∆ m → S m−1 , π(y) = ( √ y 1 , .…”