Tightening LP Relaxations for MAP using Message Passing

TL;DR

This paper introduces a dual message-passing algorithm that iteratively improves LP relaxations for MAP inference, effectively handling complex problems where standard relaxations are insufficient.

Contribution

It proposes a novel dual message-passing approach for adaptive cluster selection in LP relaxations, enhancing MAP inference in challenging graphical models.

Findings

Successfully applied to protein sidechain placement

Outperforms standard LP relaxations in complex problems

Efficiently reuses solutions during cluster addition

Abstract

Linear Programming (LP) relaxations have become powerful tools for finding the most probable (MAP) configuration in graphical models. These relaxations can be solved efficiently using message-passing algorithms such as belief propagation and, when the relaxation is tight, provably find the MAP configuration. The standard LP relaxation is not tight enough in many real-world problems, however, and this has lead to the use of higher order cluster-based LP relaxations. The computational cost increases exponentially with the size of the clusters and limits the number and type of clusters we can use. We propose to solve the cluster selection problem monotonically in the dual LP, iteratively selecting clusters with guaranteed improvement, and quickly re-solving with the added clusters by reusing the existing solution. Our dual message-passing algorithm finds the MAP configuration in protein sidechain placement, protein design, and stereo problems, in cases where the standard LP relaxation fails.