Symmetrized Bregman Divergences and Metrics

While Bregman divergences [3] have been used for several machine learning problems in recent years, the facts that they are asymmetric and does not satisfy triangle inequality have been a major limitation. In this paper, we investigate the relationship between two families of symmetrized Bregman divergences and metrics, which satisfy the triangle inequality. Further, we investigate kmeans-type clustering problems using both families of symmetrized divergences, and give efficient algorithms for the same. The first family, called Generalized Symmetrized Bregman (GSB) divergences, can be derived from any well-behaved convex function. In particular, if φ is a convex function of Legendre type [5], the GSB divergence can be defined as: