Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing

TL;DR

This paper establishes a tight lower bound on the exponent for data-dependent Locality-Sensitive Hashing schemes, showing that even data-dependent methods cannot surpass certain theoretical limits in approximate nearest neighbor search.

Contribution

It proves a lower bound on the exponent for data-dependent LSH, matching the known upper bounds and formalizing the complexity constraints of hash functions.

Findings

Lower bound matches the upper bound for $ ho$ in $ ext{ell}_1$ space.

Data-dependent hashing cannot outperform the bound $ ho o 1/(2c-1)$.

Formalizes the notion of succinctness for hash functions to exclude infeasible solutions.

Abstract

We prove a tight lower bound for the exponent $\rho$ for data-dependent Locality-Sensitive Hashing schemes, recently used to design efficient solutions for the $c$-approximate nearest neighbor search. In particular, our lower bound matches the bound of $\rho\le \frac{1}{2c-1}+o(1)$ for the $\ell_1$ space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC'15]. In recent years it emerged that data-dependent hashing is strictly superior to the classical Locality-Sensitive Hashing, when the hash function is data-independent. In the latter setting, the best exponent has been already known: for the $\ell_1$ space, the tight bound is $\rho=1/c$, with the upper bound from [Indyk-Motwani, STOC'98] and the matching lower bound from [O'Donnell-Wu-Zhou, ITCS'11]. We prove that, even if the hashing is data-dependent, it must hold that $\rho\ge \frac{1}{2c-1}-o(1)$. To prove the result, we need to formalize the exact notion of data-dependent hashing that also captures the complexity of the hash functions (in addition to their collision properties). Without restricting such complexity, we would allow for obviously infeasible solutions such as the Voronoi diagram of a dataset. To preclude such solutions, we require our hash functions to be succinct. This condition is satisfied by all the known algorithmic results.