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The standard approach to max-margin parameter learning for Markov random fields (MRFs) involves incrementally adding the most violated constraints during each iteration of the algorithm. This requires exact MAP inference, which is intractable for many classes of MRF. In this paper, we propose an exact MAP inference algorithm for binary MRFs containing a class of higher-order models, known as lower linear envelope potentials. Our algorithm is polynomial in the number of variables and number of linear envelope functions. With tractable inference in hand, we show how the parameters and corresponding feature vectors can be represented in a max-margin framework for efficiently learning lower linear envelope potentials.

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