An RKHS Approach to Estimation with Gaussian Random Field

Abstract

One popular approach to estimating an unknown function from noisy data is the use of a regularized optimization over a reproducing kernel Hilbert space (RKHS). The solution belongs to a nite-dimensional function space. If we assume the additive measurement noise is Gaussian, then there is a well known statistical interpretation that the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random eld with covariance proportional to the kernel associated with the RKHS. In this thesis, we calculate the sharp upper bound of the error of the RKHS estimate (given unit RKHS norm of the underlying function). We also present a statistical interpretation for general loss functions, by assuming the density of prior is in exponential form in terms of RKHS norm and then give some simulation examples.