Clustering in hyperbolic balls

TL;DR

This paper develops a rigorous mathematical framework for clustering in hyperbolic spaces, introducing hyperbolic k-means and EM algorithms for mixture models, advancing unsupervised learning in hyperbolic machine learning.

Contribution

It introduces the first formal definitions and algorithms for clustering and mixture modeling in hyperbolic spaces, enabling new research in hyperbolic machine learning.

Findings

Defined barycenter-based k-means in hyperbolic balls

Developed EM algorithm for hyperbolic mixture distributions

Established foundational methods for unsupervised learning in hyperbolic spaces

Abstract

The idea of representations of the data in negatively curved manifolds recently attracted a lot of attention and gave a rise to the new research direction named {\it hyperbolic machine learning} (ML). In order to unveil the full potential of this new paradigm, efficient techniques for data analysis and statistical modeling in hyperbolic spaces are necessary. In the present paper rigorous mathematical framework for clustering in hyperbolic spaces is established. First, we introduce the $k$-means clustering in hyperbolic balls, based on the novel definition of barycenter. Second, we present the expectation-maximization (EM) algorithm for learning mixtures of novel probability distributions in hyperbolic balls. In such a way we lay the foundation of unsupervised learning in hyperbolic spaces.