Reproducing kernel hilbert spaces in probability and statistics pdf
9.520 - 9/21/2015 - Class 04 - Prof. Lorenzo Rosasco: Reproducing Kernel Hilbert Spaces
Chapter 2 Sampling in Reproducing Kernel Hilbert Space
In functional analysis a branch of mathematics , a reproducing kernel Hilbert space RKHS is a Hilbert space of functions in which point evaluation is a continuous linear functional. The reverse needs not be true. However, there are RKHSs in which the norm is an L 2 -norm, such as the space of band-limited functions see the example below. Such a reproducing kernel exists if and only if every evaluation functional is continuous. James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The subject was eventually systematically developed in the early s by Nachman Aronszajn and Stefan Bergman.
The Annals of Statistics
The notion of Hilbert space embedding of probability measures has recently been used in various statistical applications like dimensionality reduction, homogeneity testing, independence testing, etc. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space RKHS. A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings : we denote this as [gamma]k, indexed by the positive definite pd kernel function k that defines the inner product in the RKHS. In this dissertation, various theoretical properties of [gamma]k and the associated RKHS embedding are presented. First, in order for [gamma]k to be useful in practice, it is essential that it is a metric and not just a pseudometric.