4,932 research outputs found
Maximum Weight Matching via Max-Product Belief Propagation
Max-product "belief propagation" is an iterative, local, message-passing
algorithm for finding the maximum a posteriori (MAP) assignment of a discrete
probability distribution specified by a graphical model. Despite the
spectacular success of the algorithm in many application areas such as
iterative decoding, computer vision and combinatorial optimization which
involve graphs with many cycles, theoretical results about both correctness and
convergence of the algorithm are known in few cases (Weiss-Freeman Wainwright,
Yeddidia-Weiss-Freeman, Richardson-Urbanke}.
In this paper we consider the problem of finding the Maximum Weight Matching
(MWM) in a weighted complete bipartite graph. We define a probability
distribution on the bipartite graph whose MAP assignment corresponds to the
MWM. We use the max-product algorithm for finding the MAP of this distribution
or equivalently, the MWM on the bipartite graph. Even though the underlying
bipartite graph has many short cycles, we find that surprisingly, the
max-product algorithm always converges to the correct MAP assignment as long as
the MAP assignment is unique. We provide a bound on the number of iterations
required by the algorithm and evaluate the computational cost of the algorithm.
We find that for a graph of size , the computational cost of the algorithm
scales as , which is the same as the computational cost of the best
known algorithm. Finally, we establish the precise relation between the
max-product algorithm and the celebrated {\em auction} algorithm proposed by
Bertsekas. This suggests possible connections between dual algorithm and
max-product algorithm for discrete optimization problems.Comment: In the proceedings of the 2005 IEEE International Symposium on
Information Theor
Belief propagation for optimal edge cover in the random complete graph
We apply the objective method of Aldous to the problem of finding the
minimum-cost edge cover of the complete graph with random independent and
identically distributed edge costs. The limit, as the number of vertices goes
to infinity, of the expected minimum cost for this problem is known via a
combinatorial approach of Hessler and W\"{a}stlund. We provide a proof of this
result using the machinery of the objective method and local weak convergence,
which was used to prove the limit of the random assignment problem.
A proof via the objective method is useful because it provides us with more
information on the nature of the edge's incident on a typical root in the
minimum-cost edge cover. We further show that a belief propagation algorithm
converges asymptotically to the optimal solution. This can be applied in a
computational linguistics problem of semantic projection. The belief
propagation algorithm yields a near optimal solution with lesser complexity
than the known best algorithms designed for optimality in worst-case settings.Comment: Published in at http://dx.doi.org/10.1214/13-AAP981 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Belief Propagation for Linear Programming
Belief Propagation (BP) is a popular, distributed heuristic for performing
MAP computations in Graphical Models. BP can be interpreted, from a variational
perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to
solve a special class of Linear Programming (LP) problems. For this class of
problems, MAP inference can be stated as an integer LP with an LP relaxation
that coincides with minimization of the BFE at ``zero temperature". We
generalize these prior results and establish a tight characterization of the LP
problems that can be formulated as an equivalent LP relaxation of MAP
inference. Moreover, we suggest an efficient, iterative annealing BP algorithm
for solving this broader class of LP problems. We demonstrate the algorithm's
performance on a set of weighted matching problems by using it as a cutting
plane method to solve a sequence of LPs tightened by adding ``blossom''
inequalities.Comment: To appear in ISIT 201
Correlation Decay in Random Decision Networks
We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
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