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Edit distance, also known as Levenshtein distance, is an essential way tocompare two strings that proved to be particularly useful in the analysis ofgenetic sequences and natural language processing. However, edit distance is adiscrete function that is known to be hard to optimize. This fact hampers theuse of this metric in Machine Learning. Even as simple algorithm as K-meansfails to cluster a set of sequences using edit distance if they are of variablelength and abundance. In this paper we propose a novel metric - soft editdistance (SED), which is a smooth approximation of edit distance. It isdifferentiable and therefore it is possible to optimize it with gradientmethods. Similar to original edit distance, SED as well as its derivatives canbe calculated with recurrent formulas at polynomial time. We prove usefulnessof the proposed metric on synthetic datasets and clustering of biologicalsequences.

Soft edit distance for differentiable comparison of symbolic sequences