Abstract
When faced with a difficult-to-solve problem, it is standard practice in the optimization community to resort to convex relaxations, that is, approximate the original problem using an easier (convex) problem. The widespread use of convex relaxations can be attributed to two factors: (i) recent advances in convex optimization imply that the relaxation can be solved efficiently; and (ii) the relaxed problem readily lends itself to theoretical analysis that often relieves interesting properties. In this chapter, we will focus on theoretical analysis for MAP estimation of discrete MRFs, with the aim of highlighting the importance of designing tight relaxations. The next chapter will illustrate the efficiency afforded by this approach. We review three standard convex relaxation approaches that have been proposed in the literature:
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