On conformal divergences and their population minimizers

Richard Nock*, Frank Nielsen, Shun Ichi Amari

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

36 Citations (Scopus)

Abstract

Total Bregman divergences are a recent tweak of ordinary Bregman divergences originally motivated by applications that required invariance by rotations. They have displayed superior results compared with ordinary Bregman divergences on several clustering, computer vision, medical imaging, and machine learning tasks. These preliminary results raise two important problems. First, report a complete characterization of the left and right population minimizers for this class of total Bregman divergences. Second, characterize a principled superset of total and ordinary Bregman divergences with good clustering properties, from which one could tailor the choice of a divergence to a particular application. In this paper, we provide and study one such superset with interesting geometric features, that we call conformal divergences, and focus on their left and right population minimizers. Our results are obtained in a recently coined (u, v)-geometric structure that is a generalization of the dually flat affine connections in information geometry. We characterize both analytically and geometrically the population minimizers. We prove that conformal divergences (resp. total Bregman divergences) are essentially exhaustive for their left (resp. right) population minimizers. We further report new results and extend previous results on the robustness to outliers of the left and right population minimizers, and discuss the role of the (u, v)-geometric structure in clustering. Additional results are also given.

Original languageEnglish
Article number7163335
Pages (from-to)527-538
Number of pages12
JournalIEEE Transactions on Information Theory
Volume62
Issue number1
DOIs
Publication statusPublished - 1 Jan 2016
Externally publishedYes

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