Information geometry of divergence functions

S. Amari, A. Cichocki

Research output: Contribution to journalArticlepeer-review

82 Citations (SciVal)


Measures of divergence between two points play a key role in many engineering problems. One such measure is a distance function, but there are many important measures which do not satisfy the properties of the distance. The Bregman divergence, Kullback-Leibler divergence and f-divergence are such measures. In the present article, we study the differential-geometrical structure of a manifold induced by a divergence function. It consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry. The class of Bregman divergences are characterized by a dually flat structure, which is originated from the Legendre duality. A dually flat space admits a generalized Pythagorean theorem. The class of f-divergences, defined on a manifold of probability distributions, is characterized by information monotonicity, and the Kullback-Leibler divergence belongs to the intersection of both classes. The f-divergence always gives the -geometry, which consists of the Fisher information metric and a dual pair of ±α-connections. The -divergence is a special class of f-divergences. This is unique, sitting at the intersection of the f-divergence and Bregman divergence classes in a manifold of positive measures. The geometry derived from the Tsallis q-entropy and related divergences are also addressed.

Original languageEnglish
Pages (from-to)183-195
Number of pages13
JournalBulletin of the Polish Academy of Sciences: Technical Sciences
Issue number1
Publication statusPublished - Mar 2010
Externally publishedYes


  • Divergence functions
  • Information geometry


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