Curved representational Bregman divergences and their applications
This work provides a theoretical extension of Bregman divergences for specialized applications in statistics and information geometry, but it is incremental as it builds on existing divergence concepts.
The paper introduces curved Bregman divergences as Bregman divergences restricted to nonlinear subspaces, showing that barycenters correspond to projections, and applies this to symmetrized Bregman divergences, Kullback-Leibler divergence for circular complex normals, and α-divergences, with an efficient method for intersecting α-divergence spheres.
By analogy to curved exponential families in statistics, we define curved Bregman divergences as Bregman divergences restricted to nonlinear parameter subspaces. 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 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 $α$-divergences are representational curved Bregman divergences with respect to $α$-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 $α$-divergence spheres.