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183
Decoding by Linear Programming
, 2004
"... This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to rec ..."
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Cited by 1399 (16 self)
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This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1minimization problem (‖x‖ℓ1:= i xi) min g∈R n ‖y − Ag‖ℓ1 provided that the support of the vector of errors is not too large, ‖e‖ℓ0: = {i: ei ̸= 0}  ≤ ρ · m for some ρ> 0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work [5]. Finally, underlying the success of ℓ1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.
MAP estimation via agreement on trees: Messagepassing and linear programming
, 2002
"... We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound ..."
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Cited by 191 (9 self)
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We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a treereweighted maxproduct messagepassing algorithm that is related to but distinct from the maxproduct algorithm. In this way, we establish a connection between a certain LP relaxation of the modefinding problem, and a reweighted form of the maxproduct (minsum) messagepassing algorithm.
MAP estimation via agreement on (hyper)trees: Messagepassing and linear programming approaches
 IEEE Transactions on Information Theory
, 2002
"... We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original ..."
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Cited by 147 (10 self)
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We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is met with equality if and only if the tree problems share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original problem. Next we present and analyze two methods for attempting to obtain tight upper bounds: (a) a treereweighted messagepassing algorithm that is related to but distinct from the maxproduct (minsum) algorithm; and (b) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds. Finally, we discuss the conditions that govern when the relaxation is tight, in which case the MAP configuration can be obtained. The analysis described here generalizes naturally to convex combinations of hypertreestructured distributions.
Graphcover decoding and finitelength analysis of messagepassing iterative decoding of LDPC codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are comp ..."
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Cited by 116 (17 self)
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The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graphcover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graphcover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and messagepassing iterative decoding. Namely, on the one hand it turns out that graphcover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like messagepassing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graphcover decoding can serve as a model to explain the behavior of messagepassing iterative decoding. Understanding the behavior of graphcover decoding is tantamount to understanding
Lagrangian relaxation for MAP estimation in graphical models
 IN: 45TH ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL AND COMPUTING
, 2007
"... We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these const ..."
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Cited by 41 (3 self)
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We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a “duality gap”, and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular treereweighted maxproduct (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by “unwinding” cycles. In addition, we propose a new class of multiscale relaxations that introduce “summary” variables. The potential benefits of such generalizations include: reducing or eliminating the “duality gap” in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.
R.Koetter, Towards LowComplexity LinearProgramming Decoding
 Proc. 4th Int. Symposium on Turbo Codes and Related Topics
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Analysis of absorbing sets and fully absorbing sets of arraybased LDPC codes
 IEEE TRANS. ON INFORMATION THEORY
, 2008
"... The class of lowdensity paritycheck (LDPC) codes is attractive, since such codes can be decoded using practical messagepassing algorithms, and their performance is known to approach the Shannon limits for suitably large blocklengths. For the intermediate blocklengths relevant in applications, how ..."
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Cited by 36 (13 self)
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The class of lowdensity paritycheck (LDPC) codes is attractive, since such codes can be decoded using practical messagepassing algorithms, and their performance is known to approach the Shannon limits for suitably large blocklengths. For the intermediate blocklengths relevant in applications, however, many LDPC codes exhibit a socalled “error floor”, corresponding to a significant flattening in the curve that relates signaltonoise ratio (SNR) to the bit error rate (BER) level. Previous work has linked this behavior to combinatorial substructures within the Tanner graph associated with an LDPC code, known as (fully) absorbing sets. These fully absorbing sets correspond to a particular type of nearcodewords or trapping sets that are stable under bitflipping operations, and exert the dominant effect on the low BER behavior of structured LDPC codes. This paper provides a detailed theoretical analysis of these (fully) absorbing sets for the class of Cp,γ arraybased LDPC codes, including the characterization of all minimal (fully) absorbing sets for the arraybased LDPC codes for γ =2, 3, 4, and moreover, it provides the development of techniques to enumerate them exactly. Theoretical results of this type provide a foundation for predicting and extrapolating the error floor behavior of LDPC codes.
BeliefPropagation for Weighted bMatchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
 in arXiv, http://www.arxiv.org/abs/0709.1190v1
, 2007
"... We consider the general problem of finding the minimum weight bmatching 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. This result is notabl ..."
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Cited by 30 (1 self)
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We consider the general problem of finding the minimum weight bmatching 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. This result is notable in several regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Instead of showing that BP leads to a PTAS, we give a finite bound for the number of iterations after which BP has converged to the exact solution. (3) Variants of the proof work for both synchronous and asynchronous BP; to the best of our knowledge, it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem. (4) It works for both ordinary bmatchings and the more difficult case of perfect bmatchings. (5) Together with the recent work of Sanghavi, Malioutov and Wilskly [41] they are the first complete proofs showing that tightness of LP implies correctness of BP. 1
Adaptive methods for linear programming decoding
 IEEE Trans. Inform. Theory
, 2008
"... Abstract—Detectability of failures of linear programming (LP) decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying LP problem. In this paper, we make a first step in studying this method, and sho ..."
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Cited by 28 (7 self)
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Abstract—Detectability of failures of linear programming (LP) decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying LP problem. In this paper, we make a first step in studying this method, and show that by starting from a simple LP problem and adaptively adding the necessary constraints, the complexity of LP decoding can be significantly reduced. In particular, we observe that with adaptive LP decoding, the sizes of the LP problems that need to be solved become practically independent of the density of the paritycheck matrix. We further show that adaptively adding extra constraints, such as constraints based on redundant parity checks, can provide large gains in the performance. Index Terms—Cutting planes, lowdensity paritycheck (LDPC) codes, linear programming (LP), LP decoding, message passing, maximumlikelihood (ML) decoding, pseudocodewords. I.
Approximate Inference in Graphical Models using LP Relaxations
, 2010
"... Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the vari ..."
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Cited by 27 (1 self)
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Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the variables. We present new approaches to approximate inference based on linear programming (LP) relaxations. Our algorithms optimize over the cycle relaxation of the marginal polytope, which we show to be closely related to the first lifting of the SheraliAdams hierarchy, and is significantly tighter than the pairwise LP relaxation. We show how to efficiently optimize over the cycle relaxation using a cuttingplane algorithm that iteratively introduces constraints into the relaxation. We provide a criterion to determine which constraints would be most helpful in tightening the relaxation, and give efficient algorithms for solving the search problem of finding the best cycle constraint to add according to this criterion.