In this paper, we derive variational inference upper-bounds on the log-partition function of pairwise Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of ''hierarchies,'' similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.

We present a new method for calculating approximate marginals for probability distributions defined by graphs with cycles, based on a Gaus- sian entropy bound combined with a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant max- imization problem that can be solved by efficient interior point meth- ods [8]. As with the Bethe approximation and its generalizations [12], the optimizing arguments of this problem can be taken as approximations to the exact marginals. In contrast to Bethe/Kikuchi approaches, our vari- ational problem is strictly convex and so has a unique global optimum. An additional desirable feature is that the value of the optimal solution is guaranteed to provide an upper bound on the log partition function.

We present a new method for calculating approximate marginals for probability distributions defined by graphs with cycles, based on a Gaussian entropy bound combined with a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant maximization problem that can be solved by efficient interior point methods [8]. As with the Bethe approximation and its generalizations [12], the optimizing arguments of this problem can be taken as approximations to the exact marginals. In contrast to Bethe/Kikuchi approaches, our variational problem is strictly convex and so has a unique global optimum. An additional desirable feature is that the value of the optimal solution is guaranteed to provide an upper bound on the log partition function. In experimental trials, the performance of the log-determinant relaxation is comparable to or better than the sum-product algorithm, and by a substantial margin for certain problem classes. Finally, the zero-temperature limit of our log-determinant relaxation recovers a class of well-known semidefinite relaxations for integer programming [e.g., 3].

We present a new method for calculating approximate marginals for probability distributions defined by graphs with cycles, based on a Gaussian entropy bound combined with a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant maximization problem that can be solved by efficient interior point methods [8]. As with the Bethe approximation and its generalizations [12], the optimizing arguments of this problem can be taken as approximations to the exact marginals. In contrast to Bethe/Kikuchi approaches, our variational problem is strictly convex and so has a unique global optimum. An additional desirable feature is that the value of the optimal solution is guaranteed to provide an upper bound on the log partition function. In experimental trials, the performance of the log-determinant relaxation is comparable to or better than the sum-product algorithm, and by a substantial margin for certain problem classes. Finally, the zero-temperature limit of our log-determinant relaxation recovers a class of well-known semidefinite relaxations for integer programming [e.g., 3].

We present a new method for calculating approximate marginals for probability distributions defined by graphs with cycles, based on a Gaussian entropybound combined with a semidefinite outer bound on the marginal polytope. This combination leads to a log-determinant maximization problemthat can be solved by efficient interior point methods [8]. As with the Bethe approximation and its generalizations [12], the optimizing arguments of this problem can be taken as approximations to the exact marginals. In contrast to Bethe/Kikuchi approaches, our variational problemis strictly convex and so has a unique global optimum. An additional desirable feature is that the value of the optimal solution is guaranteed to provide an upper bound on the log partition function. In experimental trials, the performance of the log-determinant relaxation is comparable to or better than the sum-product algorithm, and by a substantial marginfor certain problem classes. Finally, the zero-temperature limit of our log-determinant relaxation recovers a class of well-known semidefinite relaxations for integer programming [e.g., 3].