Topological Data Analysis with Bregman Divergences

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. Going beyond Euclidean distance and really beyond metrics, we show that the tools of topological data analysis also apply when we measure distance with Bregman divergences. While these divergences violate two of the three axioms of a metric, they have been found suitable 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. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems