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    "text": "MAP Estimation, Linear Programming and\nBelief Propagation with Convex Free Energies Yair Weiss, Chen Yanover and Talya Meltzer\nSchool of Computer Science and Engineering\nThe Hebrew University of Jerusalem, Jerusalem, Israel\n{yweiss,cheny,talyam}@cs.huji.ac.il Abstract algorithms have been proposed (see. e.g [13, 18] for\nrecent reviews). Finding the most probable assignment Linear Programming (LP) Relaxations are a standard\n(MAP) in a general graphical model is known method for approximating combinatorial optimization\nto be NP hard but good approximations have problems in computer science [1]. They have been\nbeen attained with max-product belief prop- used for approximating the MAP problem in a general\nagation (BP) and its variants. In particular, graphical model by Santos [15]. More recently, LP reit is known that using BP on a single-cycle laxations have been used for error-correcting codes [3]\ngraph or tree reweighted BP on an arbitrary and for protein folding [9]. LP relaxations have an\ngraph will give the MAP solution if the be- advantage over other approximate inference schemes\nliefs have no ties. in that they come with an optimality guarantee – if\nIn this paper we extend the setting under the solution to the linear program is integer, then it\nwhich BP can be used to provably extract is guaranteed to give the global optimum of the MAP\nthe MAP. We define Convex BP as BP algo- problem.\nrithms based on a convex free energy approx- The research described here is based on a remarkimation and show that this class includes or- able recent set of results by Wainwright, Jaakkola\ndinary BP with single-cycle, tree reweighted and Willsky [21, 22] who discussed a variant of beBP and many other BP variants. We show lief propagation called \"tree reweighted belief propathat when there are no ties, fixed-points of gation (TRBP)\". They showed that when the TRBP\nconvex max-product BP will provably give output satisfied certain easy-to-check conditions, one\nthe MAP solution. We also show that convex could provably extract the MAP assignment from the\nsum-product BP at sufficiently small temper- TRBP output. Furthermore, they showed an intriguatures can be used to solve linear programs ing connection between TRBP and LP relaxation.\nthat arise from relaxing the MAP problem. Finally, we derive a novel condition that al- In related work, we have used TRBP on a number of\nlows us to derive the MAP solution even if real world applications [26] and our experience with it\nsome of the convex BP beliefs have ties. In raised a number of questions.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5286v1",
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    "text": "First, TRBP is based\nexperiments, we show that our theorems al- on a distribution over spanning trees of the original\nlow us to find the MAP in many real-world graph. We wanted to know whether the properties of\ninstances of graphical models where exact in- TRBP also hold for other BP variants that are not\nference using junction-tree is impossible. based on spanning trees. Second, in some applications\nthe sufficient conditions given by Wainwright et al. [22]\nfor extracting the MAP do not hold. We wanted to\n1 Introduction know whether one could extend these conditions. In this paper, we show that the answer to both quesThe task of finding the maximum aposteriori assigntions is affirmative. We define a family of algorithms\nment (or MAP) in a graphical model comes up in\ncalled convex BP which refer to belief propagation with\na wide range of applications including image undera convex free energy approximation. We show that\nstanding [19], error correcting codes [3] and protein\ntree reweighted BP suggested by Wainwright and colfolding [27]. For an arbitrary graph, this problem is\nleagues [21] is a special case of convex BP but there are\nknown to be NP hard [16] and various approximation many convex free energies that cannot be represented where the last equation, enforces the consistency of\nas a tree reweighted free energy. This result has the- indicator variables for different potential domains.\noretical implications since it shows that the property\nThis integer program is completely equivalent to the\nof solving the LP is distinct from the property of prooriginal MAP problem, and is hence computationally\nviding a rigorous bound on the free energy, as well as\nintractable. We can obtain the linear programming repractical implications since it provides an expanded\nlaxation by allowing the indicator variables to take on\nfamily of possible LP algorithms.\nnon-integer values. That is, we replace the constraint\nWe also discuss the max-product version of convex qα(xα) ∈{0, 1}. with qα(xα) ∈[0, 1]. This problem\nBP and show that when convex BP has beliefs with- can now be solved efficiently, and if the solutions to\nout ties, the max-product assignment is guaranteed to the LP happen to be integer, we have provably found\nbe the MAP assignment. This gives a unified proof the MAP.\nfor previous results on ordinary BP with a single cycle [4,20,23] and tree reweighted BP [21]. Finally, we 1.2 Belief Propagation and its variants\ngive a new theoretical condition that allows us to provably extract the MAP from convex BP beliefs, even As shown by Yedidia et al. [28], there exist a large\nif they have ties.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
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      "Talya Meltzer"
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    "text": "We illustrate the power of these number of free energy approximations that are based\ntheorems on graphical models with hundreds of vari- on a set of \"double counting numbers\". These double\nables arising from computational biology and error- counting numbers are used to approximate the entropy\ncorrecting codes. of x, denoted ˜H, by means of a linear combination of\nentropies over individual variables i, Hi, and variables\n1.1 MAP and LP relaxation that participate in a factor α, Hα:\nX X\nGiven an observation vector y, we wish to perform ˜H = cαHα + ciHi\ninference on Pr(x|y) which is assumed to factorize into α i\na product of potential functions: Given a set of double counting numbers cα, ci we deY 1 1 αEα(xα) fine the approximate free energy functional. This is\nPr(x|y) = ψα(xα) = e− a functional that takes as input a set of approximate Z Z\nα marginals bα(xα), bi(xi) and uses them to define the\nwhere α is the domain of the potential ψα (the set of average energy and the approximate entropy. The apall variables that participate in the potential) and we proximate free energy at temperature T is simply:\ndefine the \"energy\" Eα(xα) as the negative logarithm F(bα, bi) = U(bα) −T ˜H(bα, bi) (1)\nof the potential. The MAP is the vector x∗which maximizes the pos- where the average energy, U(bα), and the approximate\nterior probability: entropy, ˜H(bα, bi), are given by:\nY X X X\nx∗= arg max ψα(xα) = arg min Eα(xα) U(bα) = bα(xα)Eα(xα)\nx x\nα α α xα X X\nTo define the LP relaxation, we first reformulate the ˜H(bα, bi) = cα bα(xα) ln bα(xα)\nMAP problem as one of integer programming. We in- Xα Xxα\ntroduce indicator variables qi(xi) for each individual + ci bi(xi) ln bi(xi)\nvariable and additional indicator variables qα(xα) for i xi\nall the potential domains. Using these indicator variA special case of approximate free energies is when\nables we define the integer program:\nci = 1 −di, cα = 1, where di is the number of factors\nMinimize: X X that node i participates in (or equivalently, the degree\nqα(xα)Eα(xα) of node i in the factor graph). In this case the approx-\nα xα imate free energy is called the Bethe free energy. Subject to: Given an approximate free energy, there are many possible algorithms that try to minimize it. For concrete-\n∀α, xα qα(xα) ∈{0, 1}\nX ness we give here one possible algorithm, the two-way\n∀α qα(xα) = 1 GBP algorithm [28], but we should emphasize that\nxα all our results hold for any algorithm that converges X\n∀i, xi, α : i ∈α qα(xα) = qi(xi) to stationary points of the approximate free energy\nxα\\i (e.g. [21, 24, 25]). Assuming cα = 1 for all factors, the two-way algorithm is similar to ordinary BP on In summary, we have defined the MAP problem, the\na factor graph, but with an additional \"reweighting\" LP relaxation and a family of belief propagation algostep. As in ordinary BP, we denote the messages sent rithms. The natural questions that arise are:\nfrom factor node α to variable node i by mαi(xi) and\nthe message from variable node i to factor node α by • When can BP algorithms be used to solve the LP\nmiα(xi). The messages are updated as follows: relaxation?",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
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    "text": "X Y\nm0αi(xi) = ψ1/Tα (xα) mjα(xj) • How are the max-product and sum-product algoxα\\i j̸=i rithms related? Y\nm0iα(xi) = mβi(xi)\nβ̸=α • When can BP algorithms be used to provably ex-\nγi γi−1 tract the MAP assignment?\nmαi(xi) ← m0αi(xi) m0iα(xi)\nγi γi−1\nmiα(xi) ← m0iα(xi) m0αi(xi) 2 Convex Free energies\nwith γi = deg(i)1−ci . The max-product belief-propagation As we will show subsequently, a key property in ana- algorithm is the same, but with the sum replaced with\nlyzing approximate free energies is their convexity over a max. Note that when γi = 1 (or, equivalently,\nthe set of constraints.1 Heskes [6, 7] has derived suf- ci = 1 −deg(i)) the above update equations reduce\nficient conditions for an entropy approximation to be to ordinary-BP. From the messages we calculate the\nconvex over the set of constraints. In our setting, we beliefs:\nY can rewrite these conditions as follows:\nbi(xi) ∝ mαi(xi) Definition: An approximate entropy term of the\nY form: X X\nbα(xα) ∝ ψ1/Tα (xα) miα(xi) (2) H = cαHα + ciHi (6)\nα i\nWe emphasize again that this is just one possible algois said to be provably convex if there exist non-negative\nrithm to find stationary points of the approximate free\nnumbers ciα, dα, di such that:\nenergy. In order to deal with any algorithm, we use\nX X X the following characterization of approximate free enH = ciα(Hα −Hi) + dαHα + diHi\nergy stationary points. This characterization follows\ni,α:i∈α α i\ndirectly from differentiating the Lagrangian of the approximate free energy and was used by [8,21,29].\n2.1 Tree Reweighted Free Energies\nObservation: A set of beliefs bα, bi are stationary\npoints of an approximate free energy with double Wainwright and colleagues have introduced an imporcounting numbers cα, ci and temperature T if and only tant subclass of belief propagation algorithms: tree\nif they satisfy: reweighted BP. These are algorithms whose free energy is a linear combination of free energies defined on\n• Admissibility: for all x: spanning trees of the graph. They have shown that\nY Y tree reweighted BP (1) can be used to obtain a rigor-\n(Pr(x))1/T ∝ bcαα (xα) bcii (xi) (3) ous bound on the free energy and (2) gives rise to a\nα i convex free energy approximation. A natural question\nthat arises is whether these two properties of TRBP • Marginalization: The beliefs are positive, sum\nare equivalent – do all BP algorithms that arise from to one and satisfy:\nX convex free energies also give a rigorous bound on the\n∀i, α : i ∈α bα(xα) = bi(xi) (4) free energy. In this section we show that the answer is\nxα\\i negative. In fact tree reweighted BP algorithms represent a small fraction of convex free energy belief propSimilarly, it can be shown that a set of beliefs agation algorithms.\nare fixed-points of the max-product algorithm with\ndouble counting numbers cα, ci if and only if they 1Convexity over the set of constraints means the funcsatisfy the above admissibility condition and max- tion is convex as a function of any beliefs that satisfy the\nmarginalization constraints. This is a weaker assumption\nmarginalization condition: from convexity over any beliefs. Henceforth we refer to this\nweaker assumption as convexity of the entropy approxima-\n∀i, α : i ∈α max bα(xα) = bi(xi) (5) xα\\i tion.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
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    "text": "Tree-reweighted free energies [21] use entropy terms of T = 5 T = 2 T = 0.1\nthe form: X\nHT RBP (µ) = µT HT (7)\nwhere T is a spanning tree in the graph, µT defines a\ndistribution over spanning trees and HT is the entropy\nof that tree. Since HT is convex, so is HT RBP . But not\nevery convex free energy can be written in this way. To see this, note that any tree reweighted entropy can\nbe rewritten:\nX X X\nHT RBP (µ) = ρijHij + (1 − ρij)Hi\n<ij> i j\nFigure 1: Contour plots of the Bethe free energy (top)\nwhere ρij is the edge appearance probability defined\nand a convex free energy (bottom) for a 2D Ising modelby µ. In comparing this to the general entropy apwith uniform external field at different temperatures.proximation (equation 6) we see that tree reweighted\nThe stars indicate local stationary points. Both free\nentropies are missing a degree of freedom (with ci).\nenergies approach the LP as temperature is decreased,In fact, for any TRBP entropy we can add an infibut for the Bethe free energy, a local minimum isnite number of possibile positive combination of sinpresent even for arbitrarily small temperatures.gle node entropies and still maintain convexity. Thus,\nTRBP entropies are a measure zero set of all convex\nentropies. approaches the LP cost (note that the entropy term\nis bounded). If we assume the entropy function toIn some cases, we can even subtract single node enbe convex then the approximate free energy is convextropies from a TRBP entropy and still maintain conand hence any fixed-point corresponds to the globalvexity. For example, the Bethe free energy for a sinminimum.gle cycle can be shown to be convex but it cannot be\nrepresented as tree-reweighted free energy [21]. In par- Note that for any BP algorithm, it is true that the\nticular, it does not give rise to a bound on the free approximate free energy minimization problem apenergy. proaches the LP problem. In particular, this is true for\nordinary BP which minimizes the Bethe free energy.This shows that the family of BP algorithms that proHowever, when the entropy function is non-convex,vide a bound on the free energy is a strict subset of\nthere is no guarantee that fixed-points will correspondthe family of convex BP algorithms.\nto the global optimum. 3 When does sum-product BP solve Figure 1 illustrates the difference. We consider a\nthe LP relaxation? graphical model corresponding to a toroidal grid. The\nnodes are binary and all the pairwise potentials are of\nthe form:Claim: Convex BP=LP Let bα, bi be fixed-point 3 1\nbeliefs from running belief propagation with a convex Ψ =\n1 2\nentropy approximation at temperature T. As T →0\nthese beliefs approach the solution to the linear pro- These potentials correspond to an Ising model with a\ngram. uniform external field – nodes prefer to be similar to\ntheir neighbors and there is a preference for one state\nProof: We know that the BP beliefs are constrained over the other.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
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    "text": "In order to visualize the approximate\nstationary points of the free energy (equation 1). The free energies, we consider beliefs that are symmetric\nminimization of F is done subject to the following con- and identical for all pairs of nodes:\nstraints:\nx y\nbα(xα) ∈ [0, 1] bα =\nX y 1 −(x + 2y)\nbα(xα) = 1\nxα Note that the MAP (and the optimum of the LP) ocX cur at x = 1, y = 0 in which case all nodes are in their\nbα(xα) = bi(xi)\npreferred state. Figure 1 shows the Bethe free energy\nxα\\i\n(top) and a convex free energy (bottom) for this probThe energy term is exactly the LP problem.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
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      "Talya Meltzer"
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    "text": "As we lem for different temperatures. The stars indicate local\ndecrease the temperature, the approximate free energy stationary points. Both free energies approach the LP as temperature is decreased, but for the Bethe free en- we could use the beliefs to find fixed-points of maxergy, a local minimum is present even for arbitrarily product BP at temperature T = 1. But to use maxsmall temperatures. product BP to solve the LP we want to go in the opposite direction, i.e. use max-product BP to define fixedpoints of sum-product BP at small temperatures. It\n4 How are max-product BP and\nturns out that this direction does not always work, as\nsum-product BP related? the following counterexample shows. Example: Consider a graphical model with two nodes Although we have shown that one can use sum-product\nconvex BP to solve the linear program, one needs to be x1, x2 and a pairwise factor:\nable to run the sum-product algorithm at sufficiently 1 1\nsmall temperatures and this may cause serious numer- ψ12(x1, x2) = . 1 0\nical problems. We now show that in certain cases,\none can solve the linear program by running the max- Consider the Bethe approximation for this graph\nproduct algorithm at any temperature. This follows (c12 = 1, c1 = c2 = 0).",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
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    "text": "This entropy approximation is\nfrom the interpertation of the max-product algorithm trivially convex. It is easy to show that for any temas the zero temperature limit of sum-product. perature T, the fixed-points of sum-product BP are: Zero temperature lemma: Suppose 1 1 1 1\n{bα(xα), bi(xi)} are fixed-points of the sum- b12 = 3 1 0 , b1 = b2 = 3(2, 1).\nproduct algorithm at temperature T. Define\nˆbα(xα) ∝bTα(xα) and ˆbi(xi) ∝bTi (xi). Then for any And, again, for any temperature T, the fixed-points of\nT →0, {ˆbα(xα),ˆbi(xi)} approach the conditions for max-product BP are:\nfixed-points of the max-product BP algorithm at\n1 1\ntemperature T = 1. b12 = , b1 = b2 = (1, 1).\n1 0\nProof: Recall that a set of beliefs are fixed-points of\nthe sum-product algorithm if and only if they satisfy In other words, when we run max-product BP we will\nthe admissibility constraint (equation 3) and the mar- get uniform beliefs in both nodes and no matter how\nginalization constraint (equation 4) and they are fixed- small we set T, raising the beliefs to the power 1/T will\npoints of the max-product algorithm if and only if still give uniform beliefs. However, the sum-product\nthey satisfy the admissibility constraint and the max- beliefs are non-uniform for any temperature. Note,\nmarginalization constraint (equation 5). however, that the counterexample still satisfies the\nzero temperature lemma — raising the sum-product\nFor any T, if {bα(xα), bi(xi)} satisfy the admissibil- beliefs to the power T indeed approaches the maxity constraint at temperature T then ˆbα(xα) ∝bTα(xα) product beliefs as T →0.\nand ˆbi(xi) ∝bTi (xi) must satisfy the admissibility conThe counterexample shows the problem with going straint at temperature 1. We just need to show that\nˆbα,ˆbi also satisfy the max-marginalization constraint from max-product beliefs to sum-product beliefs at\nT →0 (which are equivalent to the LP solution) as T →0. Since {bα(xα), bi(xi)} are fixed-points of\n– the max-product beliefs retain the information on the sum-product algorithm, they must satisfy sumthe maximum belief, but have lost the information re- marginalization, and substituting in the definition of\n{ˆbα(xα),ˆbi(xi)} we obtain: garding the number of configurations that attained the\nmaximal value. This motivates the following sufficient\nˆb1/Tα (xα) = ˆb1/Ti (xi) conditions for going from max-product beliefs to sum- product beliefs.\nxα\\i\nGiven a set of beliefs, ˆbα,ˆbi we define the sharpened\nThis can be rewritten: beliefs as follows:\n  T\nqα(xα) ∝ δ(ˆbα(xα) −max ˆbα(xα))\n = ˆbi(xi) X ˆb1/Tα (xα)\nxα\\i qi(xi) ∝ δ(ˆbi(xi) −maxxi ˆbi(xi)) and as T →0 this approaches the max-marginalization To illustrate this definition, a belief vector (0.6, 0.4)\nconstraint. would be sharpened to (1, 0) and a belief vector\n(0.4, 0.4, 0.2) would be sharpened to (0.5, 0.5, 0). The zero temperature lemma suggests that if we could\nrun sum-product BP at arbitrarily small temperatures, Using this definition it can be shown: WEISS ET AL. 421\nȌ14 1 a Ȍ12 1 a\n1 a a 1\na 1 1 a 1 exists an assignment x∗such that bα(x∗α) maximizes\nȌ13 bα(xα) and bi(x∗i ) maximizes bi(xi) then x∗is the\n4 2 a 1 MAP.\nȌ23, Ȍ24 1 a Proof: Since we have fixed-points of max-product BP Ȍ34 a 1 3\n1 a a 1 they are admissible (equation 3). Using the fact that 1 a\na 1 the entropy is provably convex, we can rewrite this as: Y ciα Y Y bα(xα)\nPr(x) ∝ bdαα (xα) bdii (xi) (8)Figure 2: A simple problem for which max-product bi(xi) i,α α i\nconvex BP will converge in a single iteration but the\nbeliefs cannot be used to solve the linear program. a We have rewritten Pr(x) as a product of functions on\nis a real number smaller than 1. xα, xi.",
    "paper_id": "1206.5286",
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    "text": "We want to show that x∗α, x∗i maximize all of\nthese functions. We know that x∗α maximizes bα(xα)\nand x∗i maximizes bi(xi). Therefore, we just need toCorollary: Max-Product Convex BP=LP Let\nworry about the quotients:ˆbα,ˆbi be max-product beliefs at T = 1 for a convex\nBP algorithm. If the sharpened max-product beliefs riα(xα) ≡bα(xα)\nare sum-marginalizable then they are a solution to the bi(xi)\nLP problem. Lemma: Suppose we have a set of beliefs bα, bi thatIn the simple two node example, the sharpened maxare max-marginalizable and there exists x∗such thatproduct beliefs are simply the original beliefs, so they\nare not a solution to the LP problem. In this case, bα(x∗α) maximizes bα(xα) and bi(x∗i ) maximizes bi(xi). Then x∗also maximizes bα(xα)/bi(xi).however, it is easy to \"fix\" the beliefs by defining b1, b2\nas the sum marginals of b12. Figure 2 shows a simple This lemma was proved for the case of pairwise factors\nproblem for which the problem is much harder to fix. in [20] and the generalization for arbitrary factors is\nMax-product convex BP will converge in a single it- straightforward.\neration to beliefs that are proportional to the potenUsing the Lemma we see that x∗maximizes all thetials, but the sharpened beliefs will not be sum marterms in the decomposition (equation 8), since eachginalizeable. Hence they cannot be used to solve the\nterm is either bi(xi), bα(xα) or riα(xi, xα\\i), raised bylinear program. However, sum-product convex BP at\nthe power of a non-negative number.T = 0.0001 gave a solution to the LP. Corollary Convex BP = MAP without ties LetTo summarize, our analysis (as well as that by Kolmogorov and Wainwright [10,11]) shows that the rela- bα, bi be fixed-points of max-product BP with a provably convex entropy function. If there are no ties intion between LP relaxation and max-product convex\nthese beliefs – for every i the maximum of bi(xi) isBP is subtle – although we can always verify postattained at a unique value x∗i – then x∗is the MAP.hoc whether we have obtained the LP solution, and a\nfixed-point corresponding to the LP solution is guar- Proof: Since the beliefs are max-marginalizable the\nanteed to exist, we are not guaranteed to find that fact that there are no ties in the node beliefs implies\nfixed-point. On the other hand, for sum-product con- there are no ties in the factor beliefs. It follows that x∗α\nvex BP the connection to LP is much more direct – maximizes bα(xα) for each α and hence the previous\nat sufficiently small temperatures the BP beliefs will theorem holds.\napproach the LP solution. Both the previous theorem and the corollary were\nproven for the case of TRBP by [21].",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5286v1",
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    "text": "Our proof ex-\n5 When can we extract the MAP tends these results for arbitrary convex BP algorithms.\nfrom max-product convex BP?\n5.1 Dealing with frustrations\nWhereas the previous section focused on using the\nmax-product algorithm to avoid the numerical insta- There are many cases in which it is impossible to find\nbilities associated with sum-product at small temper- an assignment x∗that maximizes all the factor beliefs.\natures, here we show how to use max-product BP di- This happens whenever the beliefs define a frustrated\nrectly to obtain a solution to the MAP problem. cycle (see figure 3). Theorem 1: Convex-BP = MAP without frus- Our final theorem shows that it is possible to extract\ntrations: Let bα, bi be fixed-points of max-product the MAP from convex BP beliefs even if there are frusBP with a provably convex entropy function.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
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    "text": "If there trated cycles. Default Trivial\nBP TRBP\nCBP CBP\n1 2 1 0 1 0 1 32 1 0 0 0\n2 3 2 1 2 1 32 2 32 0 0 0\n1 2 1 0 1 0 1 32 1 0 0 0 Figure 4: Negative double counting numbers −ci for\nfour different free energy approximations on a 3 × 3\na b\ngrid used to illustrate the different algorithms. Figure 3: An illustration of a frustrated cycle.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "The\ntables show pairwise beliefs obtained by a convex BP maximize bi(xi) (otherwise this would contradict maxalgorithm. For the four nodes in (a), it is possible to marginalizability). Hence we can use the lemma again\nfind an assignment x∗that maximizes the pairwise and to show that x∗α must maximize riα(xα).\nsingleton beliefs. Theorem 1 proves that this means\nthat x∗is the global optimum. For the four nodes in Corollary: For pairwise factors, if all nodes on the\n(b) it is impossible to find such an assignment. boundary of the tied nodes, ∂T, have uniform beliefs,\nthen the non-tied beliefs are optimal (that is, x∗NT =\nxMAPNT ). Theorem 2: Convex-BP = MAP with frustraThis is because for uniform beliefs on the bound- tions: Let bα, bi be fixed-points of max-product BP\nary, any assignment xT maximizes the beliefs on the with a provably convex entropy function. The fact that factors are pairwise means the set of non-tied variables and T be the set of tied\nthat it also maximizes all factors that include the variables. We denote the set of tied nodes that have\nboundary nodes. This generalizes a result of Kol- non-tied neighbors as ∂T. Define x∗NT by maximizing\nmogorov and Wainwright [11] for binary nodes. the local beliefs (the maximum here is unique since\nthese are non-tied nodes). Define:\nY Y Y 6 An illustrative example\nbT (xT ) = rciαiα (xα) bdαα (xα) bdii (xi)\ni,α: α⊂T i∈T \\∂T To illustrate the relationship between linear programi∈T \\∂T\nming (LP), ordinary belief propagation (BP), tree\nIf there exists x∗T that maximizes bT (xT ) and for reweighted belief propagation (TRBP) and convex beall regions α that contain both tied and non-tied lief propagation (CBP), we conducted simulations with\nnodes bα(x∗α) maximizes bα(xα) then the assignment a small grid graphical model – 9 nodes, arranged in a\n(x∗T , x∗NT ) is the MAP assignment. 3 × 3 grid. Proof: Using the decomposition equation (equa- One of the difficulties in comparing these different varition 8) we can write: ants of belief propagation comes from the fact that\nthere are many ways to construct TRBP or convex Y Y Y\nPr(x) ∝ rciαiα (xα) bdαα (xα) bdii (xi) BP approximations. We define the default convex BP\ni,α α i approximation based on the following observation. Y Y\n= bT (xT ) · rciαiα (xα) rciαiα (xα) Observation: For any factor graph, the free energy\n1 i,α:i/∈T i,α:i∈∂T approximation given by cα = 1 and ci = −P α:i∈α dα Y Y Y is convex.\n· bdαα (xα) bdii (xi) bdii (xi)\nα̸⊂T i/∈T i∈∂T This follows from the convexity decomposition in section 2 with ciα = dα),1 di = 0 and dα = 0, where dα is We want to show that x∗maximizes all the terms\nthe number of nodes that participate in the factor α.\nin the decomposition. By construction x∗T maximizes\nbT (xT ).",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5286v1",
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    "text": "By the previous lemma, for all regions α ̸⊂T, We consider 4 different approximate free energies\nx∗α maximizes riα(xα). Also, for all i /∈T and α ̸⊂T, which give the double counting numbers ci in figure 4.\nbi(xi) and bα(xα) are maximized by x∗NT . For i ∈∂T, In all of them, all the factors have the same double\nx∗T maximizes bi(xi) due to the assumption that x∗ counting number cα = 1 but they differ in the double\nmaximizes the boundary factors . The only thing to counting numbers ci for the nodes. In ordinary BP,\nworry about are terms of the sort riα(xα), where i ci = 1 −di.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5286v1",
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    "text": "For TRBP we considered two spanning\nis a boundary node. But since the beliefs bα, bi are forests – one for the horizontal edges of the grid, and\nmax-marginalizable for boundary nodes as well and one for the vertical edges. We used the uniform disbα(x∗α) maximizes bα(xα) by assumption, bi(x∗i ) must tribution over these two spanning forests so that the edge appearance probability was 0.5 for all edges.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5286v1",
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    "text": "To of the beliefs are tied and others are not (tied beliefs\nfacilitate comparison with the other approximations, corresponded to fractional solutions to the LP). The\nwe multiplied the entropy approximation by two, so default CBP was fastest (144 iterations) followed by\nthat cα = 1 for all edges and ci = 2 −deg(i) for TRBP (197 iterations) and then trivial CBP (372). We also considered two convex BP ap- the majority of these cases (8 out of 11), ordinary BP\nproximations.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "The default CBP approximation gives did not converge.\nci = −di/2 (since all the factors are pairwise). Finally\nTo summarize, convex BP algorithms have the greatthe trivial approximation ci = 0 is trivially convex\nest practical advantage over ordinary BP in the insince it only sums up positive entropies. For all these\ntermediate regime where the LP is partially fractional\nfree energy approximations we ran the max-product\n- they converge better and allow to provably extract\nalgorithm with a-synchronous updates and a \"dampthe MAP. All convex BP algorithms are equivalent in\nening\" factor of 0.5.\nterms of finding the MAP but convergence rate can\nWe generated 100 samples of these 3×3 \"spin glasses\" vary drastically.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "P\n– the energy was given by E(x) = i Jiixi +P\n<ij> Jijxixj. Jii and Jij were sampled from zero\nmean Gaussians with standard deviations 0.4 and 1.0. 7 Real World Experiments We found that the problems could be subdivided into The experiments reported here were designed to see\nthree classes based on the behavior of the linear pro- how often convex BP will allow us to solve real-world\ngramming relaxation. In the easy regime the linear instances of the MAP problem. Checking the condiprogramming solution is all integer and hence solving tions of theorems 1 and 2 requires finding the MAP in\nthe LP gives the MAP (this happened in 53% of our a reduced graphical model defined over the tied nodes.\nruns). All the convex approximations converged to We use the junction tree algorithm to solve this task\nbeliefs without ties. Consistent with the Max-Product so this becomes infeasible when the subgraph of tied\nConvex BP=LP corollary, the assignments obtained nodes has large induced width.\nby all the approximations were indeed the MAP. Additionally, in all the simulations in the easy regime, or- Our first two datasets are based on real-world graphdinary BP gave the correct answer. However, whereas ical models coming from computational biology. We\nthe convex algorithms come with a MAP certificate, briefly summarize the construction of these datasets\nordinary BP comes with no such theoretical guaran- (see [26] for more details).\ntee. While all algorithms found the right answer in this Proteins are chains of residues, each containing one of\neasy regime, the number of iterations to convergence 20 possible amino acids. All amino acids are connected\nwas different. Ordinary BP converged faster (median together by a common backbone structure, onto which\nnumber of iterations 48), then the default CBP (112 amino-specific side-chains are attached. The probiterations), then TRBP (176 iterations) and finally the lem of predicting the residue side-chain conformations\ntrivial CBP (225 iterations). given a backbone structure is considered of central imIn the hard regime the LP solution is all fractional portance in protein-folding and molecular design and\n(this happened in 36% of the runs). Consistent with has been tackled extensively using a wide variety of\nthe Max-Product Convex BP=LP corollary, all the methods (for a recent review, see [2]).",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
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    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "The typical\nconvex BP algorithms converged in this case to be- way to predict side-chain configurations is to define\nliefs where all the nodes were tied. For this regime, an energy function and a discrete set of possible sideordinary BP never converged. Although the zero- chain conformations, and then search for the minimal\ntemperature lemma guarantees that a fixed-point with energy configuration. Even when the energy function\nties exists for ordinary BP as well, this fixed-point contains only pairwise interactions, the configuration\nwas never found. In this regime, TRBP converged space grows exponentially and it can be shown that\nthe fastest among the convex BP algorithms (median the prediction problem is NP-complete [5].\nnumber of iterations 185), followed by the trivial CBP As a dataset we used 370 X-ray crystal structures with\n(295 iterations) and finally the default CBP (316 it- resolution better than or equal to 2˚A, R factor beerations). However, in terms of finding the MAP, all low 20% and mutual sequence identity less than 50%.\nconvex algorithms were equally useless.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "Each protein consist of a single chain and up to 1,000\nIn the intermediate regime the LP solution is par- residues. Protein structures were acquired from the\ntially integer and partially fractional (this happened Protein Data Bank site (http://www.rcsb.org/pdb).\nin 11% of the runs). Again, all the convex BP al- For each protein, we have built a graphical model usgorithms converged to the same solution where part ing the ROSETTA energy function [12]. The nodes\nof this model correspond to residues, and there are Task LP=IP Thm 1 Thm 2 Failed BP LP CBP + ties\nSide-Chain 1.35% 83.78% 6.76% 8.1% 100\nDesign 0% 2.1% 0% 97.9% 80\n[%]\nTable 1: The percentage of real-world instances solved found 60\nby the different theorems presented in this paper. MAP 40\nLP=IP means that the solution of the LP was integer. 20\nFor the easier problem of side-chain prediction (top)\nwe could find the global optimum for about 92% of the 0 101.6 10 1.4 10 1.2 10 1.0\nBSC crossover probability\ncases. For the harder task of protein design, there are\nso many tied nodes that checking the conditions of the Figure 5: A comparison of success rates as a function\ntheorems becomes infeasible. of crossover probability for a LDPC code on a binary\nsymmetric channel. edges between any two residues that interact [27]; the\npotentials are inversely related to the energy. checks. We simulated sending a codeword over a binary symmetric channel.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
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    "published_date": "2012-06-20",
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    "text": "Each received word defines The protein design problem is the inverse of the prothe local factors in a factor graph, and we used trivial tein folding problem. Given a particular 3D shape,\nconvex BP to find the MAP in this graphical model. we wish to find a sequence of amino-acids that will\nWe repeated this experiment for different signal to be as stable as possible in that 3D shape. Typically\nnoise ratios (SNR). this is done by finding a set of (1) amino-acids and (2)\nrotamer configurations that minimize an approximate Figure 5 shows our results. For high SNR, the probenergy [17].",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
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    ],
    "published_date": "2012-06-20",
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    "text": "While the protein design problem is quite lem is easy and the LP solution is almost always indifferent from side-chain prediction it can be solved teger (success of LP corresponds to a fully integer LP\nusing the same graph structure. The only difference is solution or, equivalently, max-product convex BP havthat now the nodes do not just denote rotamers but ing no ties). However, as the SNR decreases, the LP\nalso the identity of the amino-acid at that location. solution is almost always partially fractional but usThus, the state-space here is significantly larger than ing theorem 1 allows us to find the MAP decoding in\nin the side-chain prediction problem. We, again, used all cases. Thus even though the search space here is\nthe ROSETTA energy function to define the pairwise of size 2204 and the maximal clique in the junctionand local potentials. As a dataset we used 96 X-ray tree includes 134 bits, convex BP allows us to find the\ncrystal structures, 40-180 amino acids long. For each global optimum in a matter of minutes.\nof these proteins, we allowed all residues to assume\nany rotamer of any amino acid. There are, therefore,\n8 Discussion hundreds of possible states for each node.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
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    "published_date": "2012-06-20",
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    "text": "We found that convergence was not an issue – in all Belief Propgation and its variants have shown excellent\nthe experiments convex BP converged in reasonable performance as approximate inference algorithms. In\ntime but the number of ties determines the success this paper we have focused on conditions under which\nof the algorithm. Table 1 shows a breakdown of the the MAP can be provably extracted from BP beliefs.\nsuccess rate for the two problems. In the harder prob- We have shown that previous results – BP on a single\nlem of protein design, the number of ties is so large cycle and TRBP on arbitrary graphs – are special cases\nthat checking the conditions of theorems 1 and 2 is of a wider result on BP with a convex free energy. But in the side-chain problem, even though have also shown that BP with a convex free energy can\nexact inference is NP hard and the search space can be used to solve LP relaxations of the MAP problem.\nbe as large as 10600 (largest clique in junction tree – Finally, we have proven a novel result that allows us\n1060), the number of ties is quite manageable – in over to extract the MAP from convex BP beliefs even when\n90% of the instances we can find the global optimum. there are frustrated cycles. For this data set, ordinary BP also converged 85.14%\nFrom a theoretical perspective, one intriguing result\nof the times, and whenever it converged it found the\narising from our work is the close connection between\nglobal optimum. LP relaxations and a large class of belief propagaOur third dataset was based on a low density parity tion variants (including ordinary BP). Given the large\ncheck code taken from David Mackay's encyclopedia amount of literature on the tightness of LP relaxations\nof sparse graph codes (http://www.inference.phy. for combinatorial problems, this connection may encam.ac.uk/mackay/codes/data.html). We used the able proving correctness of BP variants on a larger\n204.33.484 code which has 204 bits and 102 parity class of problems. From a practical perspective, our theorems proven in are close to optimal for their structures. PNAS,\nsection 5 allow us to go beyond the LP relaxation and 97(19):10383–10388, 2000.\nprovably find the MAP even when the LP relaxation [13] R. Systematic\nis partially fractional. Our experiments on side chain vs. non-systematic algorithms for solving the MPE\ntask. In UAI, 2003.prediction and error correcting code show that using\nthese theorems it is possible to find the MAP on real [14] T. Globally optimal solutions for energy minimization in stereo vision\nworld instances of very large graphical models where\nusing reweighted belief propagation.",
    "paper_id": "1206.5286",
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    "authors": [
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    "text": "In ICCV, 2005.\ntechniques such as junction tree are intractable. On the generation of alternative explailarly, in our work reported in [14] we have used these nations with implications for belief revision. In UAI,\ntheorems to find the global optimum on a number of 1991.\nstereo vision problems. Until our results, only local [16] Y.",
    "paper_id": "1206.5286",
    "title": "MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies",
    "authors": [
      "Yair Weiss",
      "Chen Yanover",
      "Talya Meltzer"
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