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Coresets for Farthest Point Problems in Hyperbolic Space

Eunku Park, Antoine Vigneron

TL;DR

This work provides the first linear-time construction of coresets of size $O(1/\varepsilon^D)$ for farthest-point problems in fixed-dimensional hyperbolic space $\mathbb{H}^D$, achieving both $(1-\varepsilon)$ multiplicative and additive $\varepsilon$-error. The method separates into small- and large-diameter regimes, using grid bucketing after suitable isometries for the former and a cone-based discretization plus a set of isometries for the latter, underpinned by Gromov-hyperbolic geometry. The resulting coresets enable constant-time approximate farthest-neighbor queries after $O(n/\varepsilon^D)$ preprocessing, and yield efficient approximations for diameter, center, and maximum spanning tree problems in hyperbolic space. The approach advances practical algorithms for hyperbolic embeddings and related geometric problems by combining precise hyperbolic geometry with coreset techniques.

Abstract

We show how to construct in linear time coresets of constant size for farthest point problems in fixed-dimensional hyperbolic space. Our coresets provide both an arbitrarily small relative error and additive error $\varepsilon$. More precisely, we are given a set $P$ of $n$ points in the hyperbolic space $\mathbb{H}^D$, where $D=O(1)$, and an error tolerance $\varepsilon\in (0,1)$. Then we can construct in $O(n/\varepsilon^D)$ time a subset $P_\varepsilon \subset P$ of size $O(1/\varepsilon^D)$ such that for any query point $q \in \mathbb{H}^D$, there is a point $p_\varepsilon \in P_\varepsilon$ that satisfies $d_H(q,p_\varepsilon) \geq (1-\varepsilon)d_H(q,f_P(q))$ and $d_H(q,p_\varepsilon) \geq d_H(q,f_P(q))-\varepsilon$, where $d_H$ denotes the hyperbolic metric and $f_P(q)$ is the point in $P$ that is farthest from $q$ according to this metric. This coreset allows us to answer approximate farthest-point queries in time $O(1/\varepsilon^D)$ after $O(n/\varepsilon^D)$ preprocessing time. It yields efficient approximation algorithms for the diameter, the center, and the maximum spanning tree problems in hyperbolic space.

Coresets for Farthest Point Problems in Hyperbolic Space

TL;DR

This work provides the first linear-time construction of coresets of size for farthest-point problems in fixed-dimensional hyperbolic space , achieving both multiplicative and additive -error. The method separates into small- and large-diameter regimes, using grid bucketing after suitable isometries for the former and a cone-based discretization plus a set of isometries for the latter, underpinned by Gromov-hyperbolic geometry. The resulting coresets enable constant-time approximate farthest-neighbor queries after preprocessing, and yield efficient approximations for diameter, center, and maximum spanning tree problems in hyperbolic space. The approach advances practical algorithms for hyperbolic embeddings and related geometric problems by combining precise hyperbolic geometry with coreset techniques.

Abstract

We show how to construct in linear time coresets of constant size for farthest point problems in fixed-dimensional hyperbolic space. Our coresets provide both an arbitrarily small relative error and additive error . More precisely, we are given a set of points in the hyperbolic space , where , and an error tolerance . Then we can construct in time a subset of size such that for any query point , there is a point that satisfies and , where denotes the hyperbolic metric and is the point in that is farthest from according to this metric. This coreset allows us to answer approximate farthest-point queries in time after preprocessing time. It yields efficient approximation algorithms for the diameter, the center, and the maximum spanning tree problems in hyperbolic space.

Paper Structure

This paper contains 13 sections, 18 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our results.
  3. Our approach.
  4. Comparison with previous work.
  5. Preliminaries
  6. The Poincaré Ball Model
  7. Isometries.
  8. Small Diameter Input
  9. Geometric Lemmas for the Poincaré Ball Model
  10. Gromov Hyperbolic Spaces
  11. Geodesic $\delta$-Hyperbolic Spaces
  12. A Property of Approximate Diametral Pairs
  13. Coreset for Large Diameter Input

Key Result

Proposition 1

Let $h$ be an arbitrary point in $\mathbb{B}^D$ where $D=O(1)$. There is an isometry $\mu_h$ of $(\mathbb{B}^D,d_H)$ such that $\mu_h(h)=O$, and for any point $p \in \mathbb{B}^D$, we can compute $\mu_h(p)$ in $O(1)$ time.

Figures (5)

  • Figure 1: (a) A geodesic $[u,v]$ in the Poincaré ball model $\mathbb{B}^D$. (b) Cone based approach: The input points are in blue. We add the red points to the coreset.
  • Figure 2: Lemma \ref{['lem:theta']}.
  • Figure 3: Proof of Lemma \ref{['lem:anglegeodesic']}.
  • Figure 4: (a) Lemma \ref{['lem:anglegeodesic2']} statement. (b) Lemma \ref{['lem:anglegeodesic2']} proof.
  • Figure 5: A $\delta$-thin geodesic triangle $[u,v,w]$ (left), and the corresponding tripod (right).

Theorems & Definitions (26)

  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 16 more