Constraint Reduction using Marginal Polytope Diagrams for MAP LP Relaxations

TL;DR

This paper introduces Marginal Polytope Diagrams for analyzing and reducing constraints in MAP LP relaxations, leading to faster algorithms with competitive or improved solution quality.

Contribution

It presents a unified framework for comparing MAP LP relaxations, introduces Marginal Polytope Diagrams, and develops three new message passing algorithms with enhanced speed and solution quality.

Findings

Two algorithms outperform state-of-the-art in speed.

Constraint reduction maintains solution quality.

New framework enables better analysis of LP relaxations.

Abstract

LP relaxation-based message passing algorithms provide an effective tool for MAP inference over Probabilistic Graphical Models. However, different LP relaxations often have different objective functions and variables of differing dimensions, which presents a barrier to effective comparison and analysis. In addition, the computational complexity of LP relaxation-based methods grows quickly with the number of constraints. Reducing the number of constraints without sacrificing the quality of the solutions is thus desirable. We propose a unified formulation under which existing MAP LP relaxations may be compared and analysed. Furthermore, we propose a new tool called Marginal Polytope Diagrams. Some properties of Marginal Polytope Diagrams are exploited such as node redundancy and edge equivalence. We show that using Marginal Polytope Diagrams allows the number of constraints to be reduced without loosening the LP relaxations. Then, using Marginal Polytope Diagrams and constraint reduction, we develop three novel message passing algorithms, and demonstrate that two of these show a significant improvement in speed over state-of-art algorithms while delivering a competitive, and sometimes higher, quality of solution.