9,604 research outputs found
Consensus Propagation
We propose consensus propagation, an asynchronous distributed protocol for
averaging numbers across a network. We establish convergence, characterize the
convergence rate for regular graphs, and demonstrate that the protocol exhibits
better scaling properties than pairwise averaging, an alternative that has
received much recent attention. Consensus propagation can be viewed as a
special case of belief propagation, and our results contribute to the belief
propagation literature. In particular, beyond singly-connected graphs, there
are very few classes of relevant problems for which belief propagation is known
to converge.Comment: journal versio
Pairwise MRF Calibration by Perturbation of the Bethe Reference Point
We investigate different ways of generating approximate solutions to the
pairwise Markov random field (MRF) selection problem. We focus mainly on the
inverse Ising problem, but discuss also the somewhat related inverse Gaussian
problem because both types of MRF are suitable for inference tasks with the
belief propagation algorithm (BP) under certain conditions. Our approach
consists in to take a Bethe mean-field solution obtained with a maximum
spanning tree (MST) of pairwise mutual information, referred to as the
\emph{Bethe reference point}, for further perturbation procedures. We consider
three different ways following this idea: in the first one, we select and
calibrate iteratively the optimal links to be added starting from the Bethe
reference point; the second one is based on the observation that the natural
gradient can be computed analytically at the Bethe point; in the third one,
assuming no local field and using low temperature expansion we develop a dual
loop joint model based on a well chosen fundamental cycle basis. We indeed
identify a subclass of planar models, which we refer to as \emph{Bethe-dual
graph models}, having possibly many loops, but characterized by a singly
connected dual factor graph, for which the partition function and the linear
response can be computed exactly in respectively O(N) and operations,
thanks to a dual weight propagation (DWP) message passing procedure that we set
up. When restricted to this subclass of models, the inverse Ising problem being
convex, becomes tractable at any temperature. Experimental tests on various
datasets with refined or regularization procedures indicate that
these approaches may be competitive and useful alternatives to existing ones.Comment: 54 pages, 8 figure. section 5 and refs added in V
Kernel Belief Propagation
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
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