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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors present a variational approach L-FIELD to general log-submodular and supermodular distributions. Theoretical contributions include deriving upper and lower bounds on the log-partition function and fully factorized approximate posteriors. The quality of the approximation is tested with respect to the curvature of the function. Empirical results are presented on GMM cuts and MRFs, decomposable functions and facility location modeling.