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By concisely representing a joint function of many variables as thecombination of small functions, discrete graphical models (GMs) provide apowerful framework to analyze stochastic and deterministic systems ofinteracting variables. One of the main queries on such models is to identifythe extremum of this joint function. This is known as the Weighted ConstraintSatisfaction Problem (WCSP) on deterministic Cost Function Networks and asMaximum a Posteriori (MAP) inference on stochastic Markov Random Fields.Algorithms for approximate WCSP inference typically rely on local consistencyalgorithms or belief propagation. These methods are intimately related tolinear programming (LP) relaxations and often coupled with reparametrizationsdefined by the dual solution of the associated LP. Since the seminal work ofGoemans and Williamson, it is well understood that convex SDP relaxations canprovide superior guarantees to LP. But the inherent computational cost ofinterior point methods has limited their application. The situation hasimproved with the introduction of non-convex Burer-Monteiro style methods whichare well suited to handle the SDP relaxation of combinatorial problems withbinary variables (such as MAXCUT, MaxSAT or MAP/Ising). We compute low rank SDPupper and lower bounds for discrete pairwise graphical models with arbitrarynumber of values and arbitrary binary cost functions by extending aBurer-Monteiro style method based on row-by-row updates. We consider atraditional dualized constraint approach and a dedicated Block CoordinateDescent approach which avoids introducing large penalty coefficients to theformulation. On increasingly hard and dense WCSP/CFN instances, we observe thatthe BCD approach can outperform the dualized approach and provide tighterbounds than local consistencies/convergent message passing approaches.

Efficient semidefinite bounds for multi-label discrete graphical models