LSH-Preserving functions and their applications

Locality sensitive hashing (LSH) is a key algorithmic tool that is widely used both in theory and practice. An important goal in the study of LSH is to understand which similarity functions admit an LSH, that is, are LSHable. In this article, we focus on the class of transformations such that given any similarity that is LSHable, the transformed similarity will continue to be LSHable. We show a tight characterization of all such LSH-preserving transformations: they are precisely the probability generating functions, up to scaling. As a concrete application of this result, we study which set similarity measures are LSHable. We obtain a complete characterization of similarity measures between two sets A and B that are ratios of two linear functions of |A B|, |A δB|, |AB|: such a measure is LSHable if and only if its corresponding distance is a metric. This result generalizes the well-known LSH for the Jaccard set similarity, namely, the minwiseindependent permutations, and obtains LSHs for many set similarity measures that are used in practice. Using our main result, we obtain a similar characterization for set similarities involving radicals.