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Locality Sensitive Hashing in Hyperbolic Space

Authors Chengyuan Deng , Jie Gao , Kevin Lu, Feng Luo, Cheng Xin

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LIPIcs.SoCG.2026.39.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Random projections and metric embeddings
  • Theory of computation → Nearest neighbor algorithms
Keywords
  • Locality Sensitive Hashing
  • Hyperbolic Geometry
  • Dimension Reduction
  • Approximate Nearest Neighbor Search

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Abstract

For a metric space (X, d), a family ℋ of locality sensitive hash functions is called (r, cr, p₁, p₂) sensitive if a randomly chosen function h ∈ ℋ has probability at least p₁ (at most p₂) to map any a, b ∈ X in the same hash bucket if d(a, b) ≤ r (or d(a, b) ≥ cr). Locality Sensitive Hashing (LSH) is one of the most popular techniques for approximate nearest-neighbor search in high-dimensional spaces, and has been studied extensively for Hamming, Euclidean, and spherical geometries. An (r, cr, p₁, p₂)-sensitive hash function enables approximate nearest neighbor search (i.e., returning a point within distance cr from a query q if there exists a point within distance r from q) with space O(n^{1+ρ}) and query time O(n^ρ) where ρ = (log 1/p₁)/(log 1/p₂). But LSH for hyperbolic spaces ℍ^d remains largely unexplored. In this work, we present the first LSH construction native to hyperbolic space. For the hyperbolic plane (d = 2), we show a construction achieving ρ ≤ 1/c, based on the hyperplane rounding scheme. For general hyperbolic spaces (d ≥ 3), we use dimension reduction from ℍ^d to ℍ² and the 2D hyperbolic LSH to get ρ ≤ 1.59/c. On the lower bound side, we show that the lower bound on ρ of Euclidean LSH extends to the hyperbolic setting via local isometry, therefore giving ρ ≥ 1/c².

Cite As Get BibTex

Chengyuan Deng, Jie Gao, Kevin Lu, Feng Luo, and Cheng Xin. Locality Sensitive Hashing in Hyperbolic Space. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.39

Author Details

Chengyuan Deng
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Jie Gao
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Kevin Lu
  • Department of Mathematics, Rutgers University, Piscataway, NJ, USA
Feng Luo
  • Department of Mathematics, Rutgers University, Piscataway, NJ, USA
Cheng Xin
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA

Funding

The authors would like to acknowledge funding support from NSF through CCF-2118953, DMS-2311064, DMS-2220271, IIS-2229876, CNS-2515159.

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