Confliction of the Convexity and Metric Properties in f-Divergences

Mohammadali KHOSRAVIFARD  Dariush FOOLADIVANDA  T. Aaron GULLIVER  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E90-A   No.9   pp.1848-1853
Publication Date: 2007/09/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e90-a.9.1848
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: 
Keyword: 
convexity,  Csiszar f-divergence,  triangle inequality,  identity of indiscernibles,  variational distance,  metric,  

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Summary: 
We prove that the Variational distance (and its positive multiples) is the only f-divergence that satisfies both the identity of indiscernibles and the triangle inequality. Therefore it is the unique f-divergence which serves as a metric. This point is interpreted as a fundamental confliction of the convexity for f(x) with the metric properties for its associated f-divergence. Therefore, we relax the convexity of f(x) and replace it with other constraints to create new metrics.