Communications in Mathematical Sciences

Volume 6 (2008)

Number 4

Metrics defined by Bregman divergences: Part 2

Pages: 927 – 948



P. Chen

Y. Chen

M. Rao


Bregman divergences have played an important role in many research areas. Divergence is a measure of dissimilarity and by itself is not a metric. If a function of the divergence is a metric, then it becomes much more powerful. In Part 1 we have given necessary and sufficient conditions on the convex function in order that the square root of the averaged associated divergence is a metric. In this paper we provide a min-max approach to getting a metric from Bregman divergence. We show that the "capacity" to the power 1/e is a metric.


Metrics; Bregman divergence; triangle inequality; Kullback-Leibler divergence; Shannon entropy; capacity

2010 Mathematics Subject Classification

26D10, 94A15

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