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We introduce a method to reduce most higher-order terms of Markov Random Fields with binary labels into lower-order ones without introducing any new variables, while keeping the minimizer of the energy unchanged. While the method does not reduce all terms, it can be used with existing techniques that transformsarbitrary terms (by introducing auxiliary variables) and improve the speed. The method eliminates a higher-order term in the polynomial representation of the energy by finding the value assignment to the variables involved that cannot be part of a global minimizer and increasing the potential value only when that particular combination occurs by the exact amount that makes the potential of lower order. We also introduce a faster approximation that forego the guarantee of exact equivalence of minimizer in favor of speed. With experiments on the same field of experts dataset used in previous work, we show that the roof-dual algorithm after the reduction labels significantly more variables and the energy converges more rapidly.

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