Optimal lower bounds for locality-sensitive hashing (except when q is tiny)

We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}^d under the Hamming distance. Recall that H is said to be an (r,cr,p, q)-sensitive hash family if all pairs x,y ∈ {0,1}^d with dist(x,y) ≤ r have probability at least p of collision under a randomly chosen h ∈ H, whereas all pairs x,y ∈ {0,1}^d with dist(x,y) ≥ cr have probability at most q of collision. Typically, one considers d → ∞, with c > 1 fixed and q bounded away from 0. For its applications to approximate nearest-neighbor search in high dimensions, the quality of an LSH family H is governed by how small its ρ parameter ρ = ln(1/p)/ln(1/q) is as a function of the parameter c. The seminal paper of Indyk and Motwani [1998] showed that for each c ≥ 1, the extremely simple family H = {x → x_i: i ∈ [d]} achieves ρ ≤ 1/c. The only known lower bound, due to Motwani et al. [2007], is that ρ must be at least (e^1/c - 1)/(e^1/c + 1) ≥.46/c (minus o_d(1)). The contribution of this article is twofold. (1) We show the "optimal" lower bound for ρ: it must be at least 1/c (minus o_d(1)). Our proof is very simple, following almost immediately from the observation that the noise stability of a boolean function at time t is a log-convex function of t. (2) We raise and discuss the following issue: neither the application of LSH to nearest-neighbor search nor the known LSH lower bounds hold as stated if the q parameter is tiny. Here, "tiny" means q = 2^-Θ(d), a parameter range we believe is natural.