On the LSH Distortion of Ulam and Cayley Similarities

TL;DR

This paper investigates the LSH distortion of Ulam and Cayley similarities, revealing sublinear distortion bounds for Ulam and linear bounds for Cayley, advancing understanding of their suitability for hashing-based similarity search.

Contribution

It provides the first bounds on the LSH distortion of Ulam and Cayley similarities, including a sublinear upper bound for Ulam and a tight linear bound for Cayley.

Findings

Ulam similarity has a sublinear LSH distortion of O(n / sqrt(log n))

Lower bound of Ω(n^{0.12}) for Ulam similarity's LSH distortion

Cayley similarity's LSH distortion is Θ(n)

Abstract

Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function $S:\mathcal{X} \times \mathcal{X} \to [0,1]$ is a distribution over hash functions on $\mathcal{X}$ with the property that the probability of collision of any two elements $x,y\in \mathcal{X}$ is exactly equal to $S(x,y)$. However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of $n$ elements. We show that the Ulam similarity admits a sublinear LSH distortion of $O(n / \sqrt{\log n})$; we also prove a lower bound of $\Omega(n^{0.12})$ on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is $\Theta(n)$.