The connection between Bayesian estimation of a Gaussian random field and RKHS
This work is incremental, extending known statistical interpretations from quadratic loss to more general losses like absolute value, Vapnik, or Huber in RKHS estimation.
The paper tackles the problem of providing a statistical interpretation for RKHS-based function reconstruction with general loss functions, showing that the RKHS estimate corresponds to the MAP estimate of a Gaussian random field at sampling locations.
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.