Our assumption that the loss function decomposes over the edges is crucial to solving this problem.
we use weighted Hamming distance, which counts the number of variables in which a candidate solutionȳ i differs from the target output y i , with different cost for false positives (c + ) and false negatives (c -): The loss-augmented matching problem can then be written as an LP similar to Equation 1 (without the constant term jk cy i,jk ): Hence, without any approximations, we have a continuous optimization problem instead of a combinatorial one: where is the constant term, F i is the appropriate matrix that has a column of features f (x i,jk ) for each edge jk, c i is the vector of the loss terms c + − (c -+ c + )y i,jk and finally Plugging this LP back into our estimation problem, we have where Instead of the derivation in Taskar et al.