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We introduce novel results for approximate inference on planar graphicalmodels using the loop calculus framework. The loop calculus (Chertkov andChernyak, 2006) allows to express the exact partition function of a graphicalmodel as a finite sum of terms that can be evaluated once the beliefpropagation (BP) solution is known. In general, full summation over allcorrection terms is intractable. We develop an algorithm for the approachpresented in (Certkov et al., 2008) which represents an efficient truncationscheme on planar graphs and a new representation of the series in terms ofPfaffians of matrices. We analyze the performance of the algorithm for thepartition function approximation for models with binary variables and pairwiseinteractions on grids and other planar graphs. We study in detail both the loopseries and the equivalent Pfaffian series and show that the first term of thePfaffian series for the general, intractable planar model, can provide veryaccurate approximations. The algorithm outperforms previous truncation schemesof the loop series and is competitive with other state-of-the-art methods forapproximate inference.

Approximate inference on planar graphs using Loop Calculus and Belief Propagation