4,162 research outputs found
Distributed Convergence Verification for Gaussian Belief Propagation
Gaussian belief propagation (BP) is a computationally efficient method to
approximate the marginal distribution and has been widely used for inference
with high dimensional data as well as distributed estimation in large-scale
networks. However, the convergence of Gaussian BP is still an open issue.
Though sufficient convergence conditions have been studied in the literature,
verifying these conditions requires gathering all the information over the
whole network, which defeats the main advantage of distributed computing by
using Gaussian BP. In this paper, we propose a novel sufficient convergence
condition for Gaussian BP that applies to both the pairwise linear Gaussian
model and to Gaussian Markov random fields. We show analytically that this
sufficient convergence condition can be easily verified in a distributed way
that satisfies the network topology constraint.Comment: accepted by Asilomar Conference on Signals, Systems, and Computers,
2017, Asilomar, Pacific Grove, CA. arXiv admin note: text overlap with
arXiv:1706.0407
A Low Density Lattice Decoder via Non-Parametric Belief Propagation
The recent work of Sommer, Feder and Shalvi presented a new family of codes
called low density lattice codes (LDLC) that can be decoded efficiently and
approach the capacity of the AWGN channel. A linear time iterative decoding
scheme which is based on a message-passing formulation on a factor graph is
given.
In the current work we report our theoretical findings regarding the relation
between the LDLC decoder and belief propagation. We show that the LDLC decoder
is an instance of non-parametric belief propagation and further connect it to
the Gaussian belief propagation algorithm. Our new results enable borrowing
knowledge from the non-parametric and Gaussian belief propagation domains into
the LDLC domain. Specifically, we give more general convergence conditions for
convergence of the LDLC decoder (under the same assumptions of the original
LDLC convergence analysis). We discuss how to extend the LDLC decoder from
Latin square to full rank, non-square matrices. We propose an efficient
construction of sparse generator matrix and its matching decoder. We report
preliminary experimental results which show our decoder has comparable symbol
to error rate compared to the original LDLC decoder.%Comment: Submitted for publicatio
Polynomial Linear Programming with Gaussian Belief Propagation
Interior-point methods are state-of-the-art algorithms for solving linear
programming (LP) problems with polynomial complexity. Specifically, the
Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where
is the number of unknown variables. Karmarkar's celebrated algorithm is known
to be an instance of the log-barrier method using the Newton iteration. The
main computational overhead of this method is in inverting the Hessian matrix
of the Newton iteration. In this contribution, we propose the application of
the Gaussian belief propagation (GaBP) algorithm as part of an efficient and
distributed LP solver that exploits the sparse and symmetric structure of the
Hessian matrix and avoids the need for direct matrix inversion. This approach
shifts the computation from realm of linear algebra to that of probabilistic
inference on graphical models, thus applying GaBP as an efficient inference
engine. Our construction is general and can be used for any interior-point
algorithm which uses the Newton method, including non-linear program solvers.Comment: 7 pages, 1 figure, appeared in the 46th Annual Allerton Conference on
Communication, Control and Computing, Allerton House, Illinois, Sept. 200
Convergence analysis of the information matrix in Gaussian belief propagation
Gaussian belief propagation (BP) has been widely used for distributed
estimation in large-scale networks such as the smart grid, communication
networks, and social networks, where local measurements/observations are
scattered over a wide geographical area. However, the convergence of Gaus- sian
BP is still an open issue. In this paper, we consider the convergence of
Gaussian BP, focusing in particular on the convergence of the information
matrix. We show analytically that the exchanged message information matrix
converges for arbitrary positive semidefinite initial value, and its dis- tance
to the unique positive definite limit matrix decreases exponentially fast.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0201
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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