MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies

by   Yair Weiss, et al.

Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the MAP solution if the beliefs have no ties. In this paper we extend the setting under which BP can be used to provably extract the MAP. We define Convex BP as BP algorithms based on a convex free energy approximation and show that this class includes ordinary BP with single-cycle, tree reweighted BP and many other BP variants. We show that when there are no ties, fixed-points of convex max-product BP will provably give the MAP solution. We also show that convex sum-product BP at sufficiently small temperatures can be used to solve linear programs that arise from relaxing the MAP problem. Finally, we derive a novel condition that allows us to derive the MAP solution even if some of the convex BP beliefs have ties. In experiments, we show that our theorems allow us to find the MAP in many real-world instances of graphical models where exact inference using junction-tree is impossible.


page 1

page 2

page 3

page 4


α Belief Propagation as Fully Factorized Approximation

Belief propagation (BP) can do exact inference in loop-free graphs, but ...

Primal View on Belief Propagation

It is known that fixed points of loopy belief propagation (BP) correspon...

Dualities in Tree Representations

A characterization of the tree T^* such that BP(T^*)=DFUDS(T), the rever...

A Max-Sum algorithm for training discrete neural networks

We present an efficient learning algorithm for the problem of training n...

Max-Product Belief Propagation for Linear Programming: Applications to Combinatorial Optimization

The max-product belief propagation (BP) is a popular message-passing heu...

Exact Fractional Inference via Re-Parametrization Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms

Inference efforts – required to compute partition function, Z, of an Isi...

Shape Estimation from Defocus Cue for Microscopy Images via Belief Propagation

In recent years, the usefulness of 3D shape estimation is being realized...

Please sign up or login with your details

Forgot password? Click here to reset