Lower Bounds on Locality Sensitive Hashing

Optimal lower bounds for locality sensitive hashing

We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0, 1} 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} 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} with dist(x, y) ≥ cr have probability at most...

Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing

We prove a tight lower bound for the exponent ρ for data-dependent LocalitySensitive Hashing schemes, recently used to design efficient solutions for the c-approximate nearest neighbor search. In particular, our lower bound matches the bound of ρ ≤ 1 2c−1+o(1) for the l1 space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC’15]. In recent years it emerged that data-dependent h...

Beyond Locality-Sensitive Hashing

We present a new data structure for the c-approximate near neighbor problem (ANN) in the Euclidean space. For n points in R, our algorithm achieves Oc(n + d logn) query time and Oc(n + d logn) space, where ρ ≤ 7/(8c2) + O(1/c3) + oc(1). This is the first improvement over the result by Andoni and Indyk (FOCS 2006) and the first data structure that bypasses a locality-sensitive hashing lower boun...

Locality-Sensitive Hashing

On fast bounded locality sensitive hashing

In this paper, we examine the hash functions expressed as scalar products, i.e., f(x) =< v, x >, for some bounded random vector v. Such hash functions have numerous applications, but often there is a need to optimize the choice of the distribution of v. In the present work, we focus on so-called anti-concentration bounds, i.e. the upper bounds of P [| < v, x > | < α]. In many applications, v is...