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Convergent Treereweighted Message Passing for Energy Minimization
 ACCEPTED TO IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE (PAMI), 2006. ABSTRACTACCEPTED TO IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE (PAMI)
, 2006
"... Algorithms for discrete energy minimization are of fundamental importance in computer vision. In this paper we focus on the recent technique proposed by Wainwright et al. [33] treereweighted maxproduct message passing (TRW). It was inspired by the problem of maximizing a lower bound on the energy ..."
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Cited by 489 (16 self)
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Algorithms for discrete energy minimization are of fundamental importance in computer vision. In this paper we focus on the recent technique proposed by Wainwright et al. [33] treereweighted maxproduct message passing (TRW). It was inspired by the problem of maximizing a lower bound on the energy. However, the algorithm is not guaranteed to increase this bound it may actually go down. In addition, TRW does not always converge. We develop a modification of this algorithm which we call sequential treereweighted message passing. Its main property is that the bound is guaranteed not to decrease. We also give a weak tree agreement condition which characterizes local maxima of the bound with respect to TRW algorithms. We prove that our algorithm has a limit point that achieves weak tree agreement. Finally, we show that our algorithm requires half as much memory as traditional message passing approaches. Experimental results demonstrate that on certain synthetic and real problems our algorithm outperforms both the ordinary belief propagation and treereweighted algorithm in [33]. In addition, on stereo problems with Potts interactions we obtain a lower energy than graph cuts.
A comparative study of energy minimization methods for Markov random fields
 IN ECCV
, 2006
"... One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Ran ..."
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Cited by 415 (36 self)
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One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Random Fields (MRF’s), the resulting energy minimization problems were widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the topperforming stereo methods. Unfortunately, most papers define their own energy function, which is minimized with a specific algorithm of their choice. As a result, the tradeoffs among different energy minimization algorithms are not well understood. In this paper we describe a set of energy minimization benchmarks, which we use to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods—graph cuts, LBP, and treereweighted message passing—as well as the wellknown older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching and interactive segmentation. We also provide a generalpurpose software interface that allows vision researchers to easily switch between optimization methods with minimal overhead. We expect that the availability of our benchmarks and interface will make it significantly easier for vision researchers to adopt the best method for their specific problems. Benchmarks, code, results and images are available at
Using linear programming to decode binary linear codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor grap ..."
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Cited by 183 (9 self)
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A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or paritycheck representation of the code. The resulting “LP decoder” generalizes our previous work on turbolike codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including “stopping sets, ” “irreducible closed walks, ” “trellis cycles, ” “deviation sets, ” and “graph covers.” The fractional distance ��— ™ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to ��— ™ P I errors and that there are codes with ��— ™ a @ I A. An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on lowdensity paritycheck (LDPC) codes between LP decoding and the minsum and sumproduct algorithms. Methods for tightening the LP relaxation to improve performance are also provided.
Collective classification in network data
, 2008
"... Numerous realworld applications produce networked data such as web data (hypertext documents connected via hyperlinks) and communication networks (people connected via communication links). A recent focus in machine learning research has been to extend traditional machine learning classification te ..."
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Cited by 178 (32 self)
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Numerous realworld applications produce networked data such as web data (hypertext documents connected via hyperlinks) and communication networks (people connected via communication links). A recent focus in machine learning research has been to extend traditional machine learning classification techniques to classify nodes in such data. In this report, we attempt to provide a brief introduction to this area of research and how it has progressed during the past decade. We introduce four of the most widely used inference algorithms for classifying networked data and empirically compare them on both synthetic and realworld data.
Minimizing nonsubmodular functions with graph cuts  a review
 TPAMI
, 2007
"... Optimization techniques based on graph cuts have become a standard tool for many vision applications. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise Markov Random Fields (MRFs). Currently, there is an accepted view within the computer vision communi ..."
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Cited by 145 (8 self)
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Optimization techniques based on graph cuts have become a standard tool for many vision applications. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise Markov Random Fields (MRFs). Currently, there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of MRF energies (e.g. submodular functions). In this survey we review some results that show that graph cuts can be applied to a much larger class of energy functions (in particular, nonsubmodular functions). While these results are wellknown in the optimization community, to our knowledge they were not used in the context of computer vision and MRF optimization. We demonstrate the relevance of these results to vision on the problem of binary texture restoration.
Decoding ErrorCorrecting Codes via Linear Programming
, 2003
"... Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up ..."
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Cited by 116 (5 self)
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Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full errorcorrecting potential of the system is often very complex, especially for modern codes that approach the theoretical limits of the communication channel. In this thesis we investigate the application of linear programming (LP) relaxation to the problem of decoding an errorcorrecting code. Linear programming relaxation is a standard technique in approximation algorithms and operations research, and is central to the study of efficient algorithms to find good (albeit suboptimal) solutions to very difficult optimization problems. Our new “LP decoders” have tight combinatorial characterizations of decoding success that can be used to analyze errorcorrecting performance. Furthermore, LP decoders have the desirable (and rare) property that whenever they output a result, it is guaranteed to be the optimal result: the most likely (ML) information sent over the
Learning Associative Markov Networks
 Proc. ICML
, 2004
"... Markov networks are extensively used to model complex sequential, spatial, and relational interactions in fields as diverse as image processing, natural language analysis, and bioinformatics. ..."
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Cited by 96 (10 self)
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Markov networks are extensively used to model complex sequential, spatial, and relational interactions in fields as diverse as image processing, natural language analysis, and bioinformatics.
Linear programming relaxations and belief propagation – an empirical study
 Jourmal of Machine Learning Research
, 2006
"... The problem of finding the most probable (MAP) configuration in graphical models comes up in a wide range of applications. In a general graphical model this problem is NP hard, but various approximate algorithms have been developed. Linear programming (LP) relaxations are a standard method in comput ..."
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Cited by 88 (4 self)
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The problem of finding the most probable (MAP) configuration in graphical models comes up in a wide range of applications. In a general graphical model this problem is NP hard, but various approximate algorithms have been developed. Linear programming (LP) relaxations are a standard method in computer science for approximating combinatorial problems and have been used for finding the most probable assignment in small graphical models. However, applying this powerful method to realworld problems is extremely challenging due to the large numbers of variables and constraints in the linear program. TreeReweighted Belief Propagation is a promising recent algorithm for solving LP relaxations, but little is known about its running time on large problems. In this paper we compare treereweighted belief propagation (TRBP) and powerful generalpurpose LP solvers (CPLEX) on relaxations of realworld graphical models from the fields of computer vision and computational biology. We find that TRBP almost always finds the solution significantly faster than all the solvers in CPLEX and more importantly, TRBP can be applied to large scale problems for which the solvers in CPLEX cannot be applied. Using TRBP we can find the MAP configurations in a matter of minutes for a large range of real world problems. 1.