## 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 max-product and tree-reweighted (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-temperature 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 + Σ _{i}h_{i} where the function f is an extended-valued, strictly convex but nonsmooth and the functions h_{i} are extended-valued functions (not necessarily convex). We use tools from convex duality to present the "primal-dual ascent" algorithm which is an extension of the Bregman successive projection scheme and is designed to handle optimization of the general type f + Σ_{i}h_{i}. We then map the fractional-free-energy variational principle for approximate inference onto the optimization formula above and introduce the "norm-product" message-passing algorithm. Special cases of the norm-product include sum-product and max-product (BP algorithms), TRBP and NMPLP algorithms. When the fractional-free-energy is set to be convex (convex-free-energy) the norm-product is globally convergent for the estimation of marginal probabilities and for approximating the LP-relaxation. We also introduce another branch of the norm-product which arises as the "zero-temperature" of the convex-free-energy which we refer to as the "convex-max-product". The convex-max-product is convergent (unlike max-product) and aims at solving the LP- relaxation.

Original language | American English |
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Article number | 5625635 |

Pages (from-to) | 6294-6316 |

Number of pages | 23 |

Journal | IEEE Transactions on Information Theory |

Volume | 56 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2010 |

### Bibliographical note

Funding Information:Manuscript received March 16, 2009; revised June 20, 2010. Date of current version November 19, 2010. This work was supported by the Israeli Science Foundation (ISF) under Grant 519/09. T. Hazan is with Toyota Institute of Technology at Chicago (TTIC), Chicago, IL 60637 USA (e-mail: tamir@ttic.edu). A. Shashua is with the School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: shashua@cs. huji.ac.il). Communicated by H.-A. Loeliger, Associate Editor for Coding Techniques. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2010.2079014

## Keywords

- Approximate inference
- Bethe free energy
- Bregman projection
- Fenchel duality
- Markov random fields (MRF)
- convex free energy
- dual block ascent
- graphical models
- linear programming (LP) relaxation
- max-product algorithm
- maximum a posteriori probability (MAP) estimation
- sum-product algorithm