Curved representational Bregman divergences and their applications

By analogy to curved exponential families, we define curved Bregman divergences as restrictions of Bregman divergences to sub-dimensional parameter subspaces, and prove that the barycenter of a finite weighted parameter set with respect to a curved Bregman divergence amounts to the Bregman projection onto the subspace induced by the constraint of the barycenter with respect to the unconstrained full Bregman divergence. We demonstrate the significance of curved Bregman divergences with two examples: (1) symmetrized Bregman divergences and (2) the Kullback-Leibler divergence between circular complex normal distributions. 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.