  We propose a nonparametric generalization of belief propagation, Kernel
Belief Propagation (KBP), for pairwise Markov random fields. Messages are
represented as functions in a reproducing kernel Hilbert space (RKHS), and
message updates are simple linear operations in the RKHS. KBP makes none of the
assumptions commonly required in classical BP algorithms: the variables need
not arise from a finite domain or a Gaussian distribution, nor must their
relations take any particular parametric form. Rather, the relations between
variables are represented implicitly, and are learned nonparametrically from
training data. KBP has the advantage that it may be used on any domain where
kernels are defined (Rd, strings, groups), even where explicit parametric
models are not known, or closed form expressions for the BP updates do not
exist. The computational cost of message updates in KBP is polynomial in the
training data size. We also propose a constant time approximate message update
procedure by representing messages using a small number of basis functions. In
experiments, we apply KBP to image denoising, depth prediction from still
images, and protein configuration prediction: KBP is faster than competing
classical and nonparametric approaches (by orders of magnitude, in some cases),
while providing significantly more accurate results.
