Equivalence of Informations Characterizes Bregman Divergences

Abstract: Bregman divergences are a class of distance-like comparison functions which play fundamental roles in optimization, statistics, and information theory. One important property of Bregman divergences is that they cause two useful formulations of information content (in the sense of variability or non-uniformity) in a weighted collection of vectors to agree. In this note, we show that this agreement in fact characterizes the class of Bregman divergences; they are the only divergences which generate this agreement for arbitrary collections of weighted vectors.