## Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference (2008)

Citations: | 23 - 7 self |

### BibTeX

@MISC{Hazan08norm-productbelief,

author = {Tamir Hazan and Amnon Shashua},

title = {Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference},

year = {2008}

}

### Years of Citing Articles

### OpenURL

### Abstract

Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified message-passing algorithm architecture. In particular we generalize the Belief Propagation (BP) algorithms of sum-product and maxproduct and tree-rewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on ”convex-free-energy” and Linear-Programming (LP) relaxation as a zero-temprature of a convex-free-energy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of f + ∑ i hi

### Citations

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Citation Context ...he belief-propagation (BP) algorithms [33] come in two varieties: the sum-product algorithm for computing marginal probabilities and the max-product algorithm for computing the MAP assignment. Citing =-=[50]-=-, the centrality of inference using graphical models and the utility of the BP algorithms for solving them is reflected in the fact thatPRESENTED IN PART AT THE CONFERENCE ON UNCERTAINTY IN ARTIFICIA... |

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Citation Context ...ine sets C1∩...∩Cn and the sub-problem in Eqn. 22 corresponds to the projection of µ i onto the affine set Ci. Hildreth [22] extended the problem with open half spaces Ci = {x | a⊤ i x ≤ bi}. Bregman =-=[5]-=- extended Hildreth’s problem setup by including any strictly convex function f. The special case of Entropy projections was introduced later by Csiszar [9], as I-projections. Dykstra [11], [10] was th... |

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Citation Context ... half spaces Ci = {x | a⊤ i x ≤ bi}. Bregman [5] extended Hildreth’s problem setup by including any strictly convex function f. The special case of Entropy projections was introduced later by Csiszar =-=[9]-=-, as I-projections. Dykstra [11], [10] was the first to introduce general convex sets Ci (i.e., going beyond affine sets or half-spaces) but limited the treatment to f representing the Euclidean norm ... |

185 | Correctness of local probability propagation in graphical models with loops
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Citation Context ... such a case, the MAP assignment of xi corresponds to the index of highest entry of bi(xi). In general convergence is not guaranteed, and the MAP assignment can be recovered only for specific graphs, =-=[46]-=-, [2]. A. Inference using a Variational Principle The BP algorithms apply to tree-structured factor graphs yet are well defined for general factor graphs but without convergence or accuracy guarantees... |

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Citation Context ...ever, a convergent message passing algorithm for the general class of convex free energies is still lacking. The existing algorithms either employ damping heuristics to ensure convergence in practice =-=[44]-=- or focus on a sub-class of free energies where the entropy term is a positive combination of joint entropies [21]. The MAP assignment problem has been shown to be approximated by a Linear-Programming... |

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Citation Context ...results were reported in [16]. Figure 6 compares the running time of the convex-sumproduct algorithm with a general convex solver performing conditional gradient descent on the primal energy function =-=[4]-=- which uses linear programming to find feasible search directions. We ran the algorithms on n × n grids where n = 2, 3, ..., 10. The stopping criteria for all algorithms was the same and based on a pr... |

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Citation Context ... of free energies where the entropy term is a positive combination of joint entropies [22]. The MAP assignment problem has been shown to be approximated by a Linear-Programming (LP) relaxation scheme =-=[43]-=- with message-passing algorithmic attempts as a solution [45], [18]. All attempts so far guarantee convergence only under special cases (such as binary variables). A double-loop of message passing usi... |

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Citation Context ...-passing algorithms have been independently derived under different disciplines. Those include the Viterbi algorithm [42], Gallager’s sum-product algorithm for decoding low-density parity check codes =-=[15]-=-, the turbo-decoding algorithm [3], the Kalman filter for signal processing [25], and the transfer-matrix approach in statistical mechanics [1]. The BP algorithms are exact, i.e., the resulting margin... |

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Citation Context ... written on top of the inference package by Talya Meltzer available at http://www.cs.huji.ac.il/∼talyam/. generated by direct optimization of the non-convex Bethe free energy using the CCCP algorithm =-=[49]-=-. The input graph for those experiments was the Ising model on a two dimensional ∑ 8 × 8 grid. The distribution has the form p(x) ∝ e ij∈E θijxixj+θixi , where θij, θi are parameters, xi ∈ {±1}, and E... |

97 |
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Citation Context ...to obtain each of its possible ni values. In this paper, we will focus on both inference problems with the objective of introducing a unifying algorithmic thrust. Exact inference is, however, NP-hard =-=[37]-=- thus introducing the need to derive algorithms for approximate inference. One of the most popular class of methods for inference over (factor) graphs are message-passing algorithms which pass message... |

84 | Fixing max-product: Convergent message passing algorithms for MAP LP-relaxations
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Citation Context ...of joint entropies [22]. The MAP assignment problem has been shown to be approximated by a Linear-Programming (LP) relaxation scheme [43] with message-passing algorithmic attempts as a solution [45], =-=[18]-=-. All attempts so far guarantee convergence only under special cases (such as binary variables). A double-loop of message passing using a proximal minimization technique proposed recently by [32] is c... |

82 | Loopy belief propagation and gibbs measure
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Citation Context ...t is well-defined and often gives surprisingly good approximate results for graphical models with cycles. However, in this context there are no convergence guarantees (except under some special cases =-=[39]-=-, [19], [31]) and the algorithms fail to converge in many cases of interest. During the past decade there has been much progress in putting forward a framework for approximate inference using variatio... |

66 |
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Citation Context ...e vector µ i onto the convex set Ci. In that case, following some algebraic manipulations (such as eliminating µ i among other manipulations) the scheme (with A = 0) reduces to the well known Dykstra =-=[11]-=- (also goes under different names such as Hildreth, Bregman, Csiszar, Han) successive projection algorithm which has its origins in the work of Von-Neumann [30]. Further historical details can be foun... |

65 |
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Citation Context ... the marginal consistency constraints to better approximate the full probability simplex constraints. This effort includes Kikuchi free energy, region graphs and other hyper-graph based methods [50], =-=[26]-=-. The second thrust looks for convergence guaranteed message-passing algorithms by extending the Bethe free energy to form a wider class of functions, known as convex free energies, which are convex i... |

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Citation Context ...up in a wide range of applications covering a variety of disciplines. Those include digital communications (error correcting codes [12]), computer vision [36], medical diagnosis [24], protein folding =-=[47]-=-, computer graphics [13], [8], clustering [34], as well as other broad disciplines which include signal processing, artificial intelligence and statistical physics [14], [25]. Probabilistic inference ... |

59 | On the uniqueness of loopy belief propagation fixed points
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Citation Context ...ell-defined and often gives surprisingly good approximate results for graphical models with cycles. However, in this context there are no convergence guarantees (except under some special cases [37], =-=[20]-=-, [29]) and the algorithms fail to converge in many cases of interest. During the past decade there has been much progress in putting forward a framework for approximate inference using variational pr... |

54 | On optimality of tree-reweighted maxproduct message-passing
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Citation Context ... ∑ α∈N(i) cα. These max-product type algorithms are not guaranteed to converge, but whenever they converge one can extract an optimal solution for a pairwise linear program with binary variables, cf. =-=[27]-=- theorem 4 and [29] corollary 2. The third family corresponds to concave entropy approximation with cα, ci, ciα ≥ 0. These convex-maxproduct algorithms are guaranteed to converge to the global optimum... |

49 | Tree-based reparameterization for approximate estimation on graphs with cycles
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Citation Context ...s for any factor graph. An important member of this class is the tree-reweighted (TRW) free energy which consist of a linear combination of free energies defined on spanning trees of the factor graph =-=[43]-=-. It is notable that for this specific member of convex free energies a convergent message-passing algorithm, applicable to pairwise factors only, has been recently introduced [16]. However, a converg... |

47 | Maximum weight matching via maxproduct belief propagation
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Citation Context ...a case, the MAP assignment of xi corresponds to the index of highest entry of bi(xi). In general convergence is not guaranteed, and the MAP assignment can be recovered only for specific graphs, [46], =-=[2]-=-. A. Inference using a Variational Principle The BP algorithms apply to tree-structured factor graphs yet are well defined for general factor graphs but without convergence or accuracy guarantees. The... |

45 | MAP estimation, linear programming and belief propagation with convex free energies
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Citation Context ...ation of joint entropies [21]. The MAP assignment problem has been shown to be approximated by a Linear-Programming (LP) relaxation scheme [45] with message-passing algorithmic attempts as a solution =-=[47]-=-, [17]. All attempts so far guarantee convergence only under special cases (such as binary variables) with the exception of [17] which we will show arises as a special case of our algorithm. A double-... |

36 |
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Citation Context ...ions covering a variety of disciplines. Those include digital communications (error correcting codes [12]), computer vision [38], medical diagnosis [23], protein folding [49], computer graphics [13], =-=[8]-=-, clustering [36], as well as other broad disciplines which include signal processing, artificial intelligence and statistical physics [14], [24]. Probabilistic inference comes in two distinct forms a... |

34 | Message-passing for graph-structured linear programs: Proximal methods and rounding schemes
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Citation Context ...inary variables) with the exception of [17] which we will show arises as a special case of our algorithm. A double-loop of message passing using a proximal minimization technique proposed recently by =-=[34]-=- is convergent but at a considerable computational expense. In this paper, we derive a class of approximate inference message-passing algorithms, which we call norm-product algorithms, using the notio... |

33 | Y.: Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation
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Citation Context ... max-product type algorithms are not guaranteed to converge, but whenever they converge one can extract an optimal solution for a pairwise linear program with binary variables, cf. [27] theorem 4 and =-=[29]-=- corollary 2. The third family corresponds to concave entropy approximation with cα, ci, ciα ≥ 0. These convex-maxproduct algorithms are guaranteed to converge to the global optimum for a pairwise lin... |

30 | Fractional belief propagation
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Citation Context ...e in this paper but for the sake of clarity we leave it outside the current scope. The Bethe approximation Hbethe(b) of the entropy can be written in a more general form, known as fractional entropy, =-=[48]-=- ∑ ¯cαH(bα) + ∑ ¯ciH(bi), (2) α i xα where ¯ci = 1 − ∑ α∈N(i) ¯cα. Thus when the coefficients ¯cα = 1 for all factor nodes we obtain the Bethe approximation. A convex free energy is based on a result ... |

28 | Convexity arguments for efficient minimization of the Bethe and Kikuchi free energies
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Citation Context ...nting validity of marginals). When the factor graph has cycles the Bethe energy is non-convex and although it is possible to derive convergent algorithms to a local minima of the Bethe function [49], =-=[22]-=- the computational cost is large and thus has not gained popularity. To overcome the difficulty with the non-convexity of the Bethe approximation, several authors have introduced a class of approximat... |

25 |
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Citation Context ...primal solution is to find the projection of b0 onto the intersection of the affine sets C1∩...∩Cn and the sub-problem in Eqn. 22 corresponds to the projection of µ i onto the affine set Ci. Hildreth =-=[23]-=- extended the problem with open half spaces Ci = {x | a⊤ i x ≤ bi}. Bregman [5] extended Hildreth’s problem setup by including any strictly convex function f. The special case of Entropy projections w... |

24 | Sucient conditions for convergence of loopy belief propagation
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Citation Context ...fined and often gives surprisingly good approximate results for graphical models with cycles. However, in this context there are no convergence guarantees (except under some special cases [39], [19], =-=[31]-=-) and the algorithms fail to converge in many cases of interest. During the past decade there has been much progress in putting forward a framework for approximate inference using variational principl... |

23 |
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Citation Context ...., λ ∗ n. Also, consider the primal sequence generated by ∇f ∗ (− ∑ i λi) computed from the dual sequence, then this primal sequence is bounded and its limit point is the optimal solution x∗ . Proof: =-=[40]-=-. APPENDIX B THE PRIMAL-DUAL BLOCK ASCENT ALGORITHM We describe an algorithm for solving programs of the form f(b) + ∑ hi(b) while solving sub-problems which consists of f(b) and a single function hi(... |

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Citation Context ...less task. Problems involving inference using graphical models comes up in a wide range of applications covering a variety of disciplines. Those include digital communications (error correcting codes =-=[12]-=-), computer vision [36], medical diagnosis [24], protein folding [47], computer graphics [13], [8], clustering [34], as well as other broad disciplines which include signal processing, artificial inte... |

16 | Convergent messagepassing algorithms for inference over general graphs with convex free energies
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Citation Context ...sible ∑ (cα + α ∑ i∈N(α) ciα − 1) 2 , (38) which is a least-squares criteria for uniformity of ¯cα. We refer to the two least-squares scheme as L2 convex free energy approximation. In an earlier work =-=[18]-=-, we also used the maximum entropy approach where the criterion function minimizes ∑ α ¯cα ln ¯cα. Further investigation for constructing good convex free energy approximations can be found in [30]. T... |

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Citation Context ...raphical models comes up in a wide range of applications covering a variety of disciplines. Those include digital communications (error correcting codes [12]), computer vision [38], medical diagnosis =-=[23]-=-, protein folding [49], computer graphics [13], [8], clustering [36], as well as other broad disciplines which include signal processing, artificial intelligence and statistical physics [14], [24]. Pr... |

15 |
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Citation Context ...s, in the particular case when f(b) = ˆ f(b), i.e., is essentially smooth, and hi(b) = δCi (b) (the indicator function of convex set Ci), the update step (b) for Algorithm 1 is a ”Bregman” projection =-=[6]-=- of the vector µ i onto the convex set Ci. In that case, following some algebraic manipulations (such as eliminating µ i among other manipulations) the scheme (with A = 0) reduces to the well known Dy... |

14 | Learning and inferring image segmentations using the GBP typical cut algorithm
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Citation Context ...variety of disciplines. Those include digital communications (error correcting codes [12]), computer vision [36], medical diagnosis [24], protein folding [47], computer graphics [13], [8], clustering =-=[34]-=-, as well as other broad disciplines which include signal processing, artificial intelligence and statistical physics [14], [25]. Probabilistic inference comes in two distinct forms and typically invo... |

13 | Convergent propagation algorithms via oriented trees
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Citation Context ...of the factor graph [43]. It is notable that for this specific member of convex free energies a convergent message-passing algorithm, applicable to pairwise factors only, has been recently introduced =-=[16]-=-. However, a convergent message passing algorithm for the general class of convex free energies is still lacking. The existing algorithms either employ damping heuristics to ensure convergence in prac... |

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Citation Context ...rk [18], we also used the maximum entropy approach where the criterion function minimizes ∑ α ¯cα ln ¯cα. Further investigation for constructing good convex free energy approximations can be found in =-=[30]-=-. The desire towards uniformity, besides being used extensively in probabilistic settings, is motivated by the success of the Bethe free energy where ¯cα = 1. The Bethe free energy is non-convex for f... |

10 | The Dykstra algorithm with Bregman projections
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Citation Context ...cial case of Algorithm 8 when hi = δCi , where Ci is a convex set, and f is essentially smooth, i.e., A = 0, can be mapped (by eliminating step 2(a)) to a successive Bregman projection algorithm [6], =-=[7]-=- which is also known under the names of Dykstra, Hildreth, Han and Csiszar. This class of iterative projection schemes has a long history starting from Von-Neumann in the 50s [32] who introduced the c... |

10 |
An iterative procedure for obtaining iprojections onto the intersection of convex sets. The Annals of Probability, 13:975–984
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Citation Context ...Bregman [5] extended Hildreth’s problem setup by including any strictly convex function f. The special case of Entropy projections was introduced later by Csiszar [9], as I-projections. Dykstra [11], =-=[10]-=- was the first to introduce general convex sets Ci (i.e., going beyond affine sets or half-spaces) but limited the treatment to f representing the Euclidean norm and the KL divergence. The view of the... |

8 |
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Citation Context ... = 0) reduces to the well known Dykstra [11] (also goes under different names such as Hildreth, Bregman, Csiszar, Han) successive projection algorithm which has its origins in the work of Von-Neumann =-=[30]-=-. Further historical details can be found in Appendix B. Another useful property of the algorithm that it is well defined for non-convex primal energies. Specifically, we can establish the following r... |