Linear Programming and Belief Propagation with Convex Free Energies Applications in Computer Vision and Computational Biology

The task of finding the most probable explanation (or MAP) in a graphical model comes up in a wide range of applications including image understanding [9], error correcting codes [2] and protein folding [11]. For an arbitrary graph, this problem is known to be NP hard [8] and various approximation algorithms have been proposed (see. e.g [5] for a recent review). Linear Programming (LP) Relaxations are a standard method for approximating combinatorial optimization problems in computer science [1]. They have been used for approximating the MAP problem in a general graphical model by Santos [7]. More recently LP relaxations have been used for error-correcting codes [2], and for protein folding [4]. LP relaxations have an advantage over other approximate inference schemes in that they come with an optimality guarantee when the solution to the linear program is integer, then the LP solution is guaranteed to give the global optimum of the posterior probability. The research described here grew out of our experience in using LP relaxations for problems in computer vision, computational biology and statistical physics. In all these fields, the number of variables in a realistic problem may be on the order of 10 or more. We found that using standard, off-the-shelf LP solvers these problems cannot be solved using standard desktop hardware. A second problem, is that even when we worked on toy problems in which the number of variables was much smaller, we found that a large fraction of the variables in the LP solution were non-integer. This means that the guarantee of optimality is lost, and in practice thresholding the fractional LP solutions gave poor results. The fact that general purpose LP solvers are not suitable for these problems is not that surprising. The linear programs that arise out of LP relaxations for graphical models have a common structure and are a small subset of all possible linear programs. The challenge is to find a LP solver that (1) takes advantage of this special structure and (2) can be used to solve a hierarchy of tighter relaxations in case the standard LP relaxation fails. In this paper, we show that generalized belief propagation with a convex free energy provides such a solver. We show that tree reweighted BP suggested by Wainwright and colleagues [10] is a special case of GBP with a convex free energy but there are many convex free energies that cannot be represented as a tree reweighted free energy. This result has theoretical implications since it shows that the property of solving the LP is distinct from the property of providing a rigorous bound 1E-mail: yweiss@cs.huji.ac.il