On the optimality of tree-reweighted max-product message-passing

TL;DR

This paper investigates the properties of tree-reweighted max-product message passing, showing that under certain conditions it can identify parts of the optimal solution, and for submodular functions, it guarantees global optimality.

Contribution

It establishes that weak tree agreement fixed points can identify partial solutions and guarantees global optimality for submodular functions in binary variables.

Findings

WTA fixed points can identify optimal subsets of variables.

For submodular functions, WTA fixed points are globally optimal.

WTA fixed points always maximize the underlying LP relaxation.

Abstract

Tree-reweighted max-product (TRW) message passing is a modified form of the ordinary max-product algorithm for attempting to find minimal energy configurations in Markov random field with cycles. For a TRW fixed point satisfying the strong tree agreement condition, the algorithm outputs a configuration that is provably optimal. In this paper, we focus on the case of binary variables with pairwise couplings, and establish stronger properties of TRW fixed points that satisfy only the milder condition of weak tree agreement (WTA). First, we demonstrate how it is possible to identify part of the optimal solution|i.e., a provably optimal solution for a subset of nodes| without knowing a complete solution. Second, we show that for submodular functions, a WTA fixed point always yields a globally optimal solution. We establish that for binary variables, any WTA fixed point always achieves the global maximum of the linear programming relaxation underlying the TRW method.