Submitted in April, 2014. Constraint Reduction using Marginal Polytope Diagrams for MAP LP Relaxations

Abstract LP relaxation-based message passing algorithms provide an effec-tive 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 relax-ations may be compared and analysed. Furthermore, we propose a new tool called Marginal Polytope Diagrams. Some properties of Marginal Polytope Di-agrams 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 improve-ment in speed over state-of-art algorithms while delivering a competitive, and sometimes higher, quality of solution