Kernelised functional Bregman divergences

Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry, being central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to \emph{functional Bregman divergences}, where points are functions rather than finite-dimensional parameter vectors. Other types of functional Bregman divergences are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. We demonstrate a possible future application in two-sample hypothesis testing.