The connection between Bayesian estimation of a Gaussian random field and RKHS

Abstract—Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regu-larization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers. Index Terms—kernel based regularization; Gaussian processes; representer theorem; reproducing kernel Hilbert spaces; regular