Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces

TL;DR

This work presents a framework for embedding finite subsets of hyperbolic space $\mathbb{H}^D$ into sparse graph metrics with additive distortion $O(\log D)$. It achieves this by discretizing $\mathbb{H}^D$ with a binary tiling into centers of cells, analyzing two discrete models $(\mathbb{B}^D,d_1)$ and $(\mathbb{B}^D,d_2)$, and leveraging compressed quadtrees to build linear-size structures. The resulting results include a graph-metric embedding with $2^{O(D)}n$ vertices, a $O(k\log D)$-additive spanner with $O(n\lambda_k(n))$-size improvements via a transitive-closure, and an AVD-based ANN scheme with near-optimal additive guarantees in $d_1$ and $d_H$. The constructions run in $2^{O(D)}n\log n$ time and provide practical tools for efficient approximate nearest-neighbor queries in hyperbolic data representations, with potential impact on network-analytic embeddings and geometric data structures in hyperbolic spaces.

Abstract

We give an embedding of the Poincaré halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n \cdot λ_k(n)$ edges, where $λ_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for $P$ of size $2^{O(D)}n$. It allows us to answer approximate near-neighbor queries in $2^{O(D)}+O(\log n)$ time, with additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log n$ time.

Figures (9)

• Figure 1: Three models of hyperbolic spaces. (a) The Poincaré halfplane $\mathbb{H}^2$. (b) The first discrete model $(\mathbb{B}^2,d_1)$, with a shortest path of length $d_1(p,q)=5$. (c) The second discrete model $(\mathbb{B}^2,d_2)$, with a shortest path of length $d_2(p,q)=6$.
• Figure 2: Possible moves in $\mathbb{B}^3$. (left) Vertical moves $U(p)$ and $D(p)\in\{q_1,q_2,q_3,q_4\}$. (right) The 8 Horizontal moves $H(p)$, seen from above.
• Figure 3: (a) A counterexample that shows that $d_2$ does not obey the triangle inequality: we have $d_2(p,q)=1$, $d_2(q,r)=1$ and $d_2(p,r)=3$ (b) An example where $d_2(p,q)=5=2+d_1(p,q)$.
• Figure 4: Lemma \ref{['lem:semicircle']}
• Figure 5: Proof of Lemma \ref{['lem:circle']}

Theorems & Definitions (37)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof