Curved representational Bregman divergences and their applications
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 $\alpha$-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.
Code (0)
등록된 구현이 없습니다.
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
A scaled Bregman theorem with applications
Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms. This paper explores the use of Bregman divergences to establish reductions between such algorithms and their a…
BIG-bench Machine LearningClusteringSymplectic Bregman divergences
We present a generalization of Bregman divergences in symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel-Young…
On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds
We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi s…
Generalising maximum mean discrepancy: kernelised functional Bregman divergences
Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, param…
A note on the quasiconvex Jensen divergences and the quasiconvex Bregman divergences derived thereof
We first introduce the class of strictly quasiconvex and strictly quasiconcave Jensen divergences which are oriented (asymmetric) distances, and study some of their properties. We then define the strictly quasiconvex Bre…