On the distortion of locality sensitive hashing

Given a notion of pairwise similarity between objects, locality sensitive hashing (LSH) aims to construct a hash function family over the universe of objects such that the probability two objects hash to the same value is their similarity. LSH is a powerful algorithmic tool for large scale applications and much work has been done to understand LSHable similarities, i.e., similarities that admit an LSH. In this paper we focus on similarities that are provably non-LSHable and propose a notion of distortion to capture the approximation of such a similarity by an LSHable similarity. We consider several well-known non-LSHable similarities and show tight upper and lower bounds on their distortion.