Topological Data Analysis with Bregman Divergences

TL;DR

This paper extends topological data analysis to non-metric Bregman divergences, enabling the analysis of data with dissimilarity measures like Kullback-Leibler and Itakura-Saito, and provides algorithms for practical computation.

Contribution

It generalizes the TDA framework from metric spaces to Bregman divergences, proving the correctness of Cech and Delaunay complexes and developing efficient algorithms.

Findings

Cech and Delaunay complexes correctly capture homotopy types for Bregman balls.

Vietoris-Rips complexes may fail to approximate the persistence diagram in this setting.

Algorithms are effective and perform well, supported by connections to discrete Morse theory.

Abstract

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.