Topological Data Analysis with Bregman Divergences

Authors Herbert Edelsbrunner, Hubert Wagner

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Herbert Edelsbrunner
Hubert Wagner

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Herbert Edelsbrunner and Hubert Wagner. Topological Data Analysis with Bregman Divergences. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. 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. 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.
  • Topological data analysis
  • Bregman divergences
  • persistent homology
  • proximity complexes
  • algorithms


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