Topological Data Analysis with Bregman Divergences

摘要

Given a finite set in a metric space, the topological analysis assesses its multi-scale connectivity quantified in terms of a 1-parameter family of homology groups. Going beyond Euclidean distance and really beyond metrics, we show that the basic tools of topological data analysis also apply when we measure distance with Bregman divergences. While these violate two of the three axioms of a metric, they have been found more effective for high-dimensional data. Examples are the Kullback-Leibler divergence, which is commonly used for text and images, and the Itakura-Saito divergence, which is popular for speech and sound.