Communications in Mathematical Sciences

Volume 6 (2008)

Number 4

Metrics defined by Bregman Divergences

Pages: 915 – 926



P. Chen

Y. Chen

M. Rao


Bregman divergences are generalizations of the well known Kullback-Leibler divergence. They are based on convex functions and have recently received great attention. We present a class of "squared root metrics" based on Bregman divergences. They can be regarded as natural generalization of Euclidean distance. We provide necessary and sufficient conditions for a convex function so that the square root of its associated average Bregman divergence is a metric.


Metrics, Bregman divergence, convexity

2010 Mathematics Subject Classification

26D10, 94A15

Published 1 January 2008