Curved representational Bregman divergences and their applications

By analogy to the terminology of curved exponential families in statistics, we define curved Bregman divergences as Bregman divergences restricted to nonlinear parameter subspaces and sub-dimensional Bregman divergences when the restrictions are linear. A common example of curved Bregman divergence is the cosine dissimilarity between normalized vectors. We show that the barycenter of a finite weighted set of parameters under a curved Bregman divergence amounts to the right Bregman projection onto the nonlinear subspace of the barycenter with respect to the full Bregman divergence. We demonstrate the significance of curved Bregman divergences with two examples: (1) symmetrized Bregman divergences, (2) pointwise symmetrized Bregman divergences, and (3) the Kullback-Leibler divergence between circular complex normal distributions. We explain how to reparameterize sub-dimensional Bregman divergences on simplicial sub-dimensional domains. We then consider monotonic embeddings to define representational curved Bregman divergences and show that the $\alpha$-divergences are representational curved Bregman divergences with respect to $\alpha$-embeddings of the probability simplex into the positive measure cone. As an application, we report an efficient method to calculate the intersection of a finite set of $\alpha$-divergence spheres.