Tighter continuous relaxations for MAP inference in discrete MRFs: A survey

Abstract

Continuous relaxations are central to map inference in discrete Markov random fields (MRFs). In these methods, the intractable discrete optimization problem is approximated by a continuous relaxation. The relaxation could be based on different approaches, with the linear programming (LP) relaxation being the most well studied. For continuous relaxations, the important considerations are efficiently solving the optimization formulations and improving the approximation quality of the relaxations. In this chapter, we focus on the latter topic, which is referred to as tighter continuous relaxations. We present a comprehensive survey of techniques to achieve a tighter LP relaxation, followed by a discussion of semidefinite programming (SDP) relaxation techniques. We discuss these topics by covering both theoretical motivations and algorithmic considerations. We hope this will be a ready reference for researchers working on tighter relaxations and for practitioners who want to model their problem more accurately and efficiently. The foundation for the theoretical aspects comes from the discrete optimization community through the topics of polyhedral combinatorics and relaxation hierarchies. The structure of discrete MRFs lends a special perspective to these classic topics and we will see how several works have explored this path. Over the years, research efforts in discrete MRFs have characterized graph topologies and/or clique energies that lead to accurate map inference. In recent years, there have been similar efforts, to better understand the settings and their accompanying guarantees, for the algorithms that tighten the continuous relaxations. We will end this chapter by giving a taste for this emerging research topic.