  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 here H
is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y in
{0,1}^d with dist(x,y) at most r have probability at least p of collision under
a randomly chosen h in H, whereas all pairs x, y in {0,1}^d with dist(x,y) at
least cr have probability at most q of collision. Typically, one considers d
tending to infinity, with c 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 "rho
parameter" rho = ln(1/p)/ln(1/q) is as a function of the parameter c. The
seminal paper of Indyk and Motwani showed that for each c, the extremely simple
family H = {x -> x_i : i in d} achieves rho at most 1/c. The only known lower
bound, due to Motwani, Naor, and Panigrahy, is that rho must be at least .46/c
(minus o_d(1)).
  In this paper we show an optimal lower bound: rho must be at least 1/c (minus
o_d(1)). This lower bound for Hamming space yields a lower bound of 1/c^2 for
Euclidean space (or the unit sphere) and 1/c for the Jaccard distance on sets;
both of these match known upper bounds. Our proof is simple; the essence is
that the noise stability of a boolean function at e^{-t} is a log-convex
function of t.
