Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Finding Interesting Associations without Support Pruning
IEEE Transactions on Knowledge and Data Engineering
Similarity Search in High Dimensions via Hashing
VLDB '99 Proceedings of the 25th International Conference on Very Large Data Bases
High-dimensional computational geometry
High-dimensional computational geometry
Fast Pose Estimation with Parameter-Sensitive Hashing
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Computational applications of noise sensitivity
Computational applications of noise sensitivity
Locality-sensitive hashing scheme based on p-stable distributions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Entropy based nearest neighbor search in high dimensions
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Efficient algorithms for substring near neighbor problem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Randomized algorithms and NLP: using locality sensitive hash function for high speed noun clustering
ACL '05 Proceedings of the 43rd Annual Meeting on Association for Computational Linguistics
Google news personalization: scalable online collaborative filtering
Proceedings of the 16th international conference on World Wide Web
Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions
Communications of the ACM - 50th anniversary issue: 1958 - 2008
Lower Bounds on Locality Sensitive Hashing
SIAM Journal on Discrete Mathematics
A Geometric Approach to Lower Bounds for Approximate Near-Neighbor Search and Partial Match
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A locality-sensitive hash for real vectors
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Spherical lsh for approximate nearest neighbor search on unit hypersphere
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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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 ↦ xi : i ∈ [d]} achieves ρ ≤ 1/c. The only known lower bound, due to Motwani et al. [2007], is that ρ must be at least ( e1/c - 1)/(e1/c + 1) ≥ .46/c (minus od(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 od(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.