Locality Sensitive Hashing in Hyperbolic Space

2026-03-20

AI Summary

This work addresses the absence of locality-sensitive hashing (LSH) schemes tailored for approximate nearest neighbor (ANN) search in hyperbolic space by proposing the first native LSH construction for this geometry. The method introduces a two-dimensional hashing scheme based on hyperbolic hyperplane rounding and extends it to higher dimensions via dimensionality reduction combined with local isometric embeddings. Theoretical analysis establishes an upper bound on the performance parameter ρ of ρ ≤ 1/c for d = 2 and ρ ≤ 1.59/c for d ≥ 3, along with a lower bound of ρ ≥ 1/c². This approach achieves, for the first time, sublinear query time and storage complexity for ANN search in hyperbolic space with rigorous theoretical guarantees.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

Abstract

For a metric space $(X, d)$, a family $\mathcal{H}$ of locality sensitive hash functions is called $(r, cr, p_1, p_2)$ sensitive if a randomly chosen function $h\in \mathcal{H}$ has probability at least $p_1$ (at most $p_2$) to map any $a, b\in X$ in the same hash bucket if $d(a, b)\leq r$ (or $d(a, b)\geq cr$). 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 $(r, cr, p_1, p_2)$-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 $ρ=\frac{\log 1/p_1}{\log 1/p_2}$. But LSH for hyperbolic spaces $\mathbb{H}^d$ 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 $ρ\leq 1/c$, based on the hyperplane rounding scheme. For general hyperbolic spaces $(d \geq 3)$, we use dimension reduction from $\mathbb{H}^d$ to $\mathbb{H}^2$ and the 2D hyperbolic LSH to get $ρ\leq 1.59/c$. 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 $ρ\geq 1/c^2$.

Problem

• Locality Sensitive Hashing
• Hyperbolic Space
• Approximate Nearest Neighbor Search
• Metric Space

Innovation

• Locality Sensitive Hashing
• Hyperbolic Space
• Approximate Nearest Neighbor
• Dimension Reduction
• Hash Functions