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Numerical Aspects of Hyperbolic Geometry

Dorota Celinska-Kopczynska, Eryk Kopczynski

TL;DR

The paper addresses numerical stability across diverse hyperbolic space representations used in applications like networks and machine learning. It conducts a broad, six-test comparison of linear, mixed, reduced, half-plane/half-space, and polar representations, augmented by tessellation-based strategies to counter precision loss. The results show that polar representations, particularly polar2, often yield the best numerical performance, with half-plane invariant also performing well, while certain fixed linear approaches excel in specific walk-related tasks; tessellations such as ${7,3}$ substantially improve robustness. These findings guide practitioners in selecting representations and error-control methods to balance numerical precision with practical considerations in hyperbolic geometry applications.

Abstract

Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation comes with some numerical instability. This paper compares various 2D and 3D hyperbolic geometry representations. To this end, we conduct an extensive simulational scheme based on six tests of numerical precision errors. Our comparisons include the most popular models and less-known mixed and reduced representations. According to our results, polar representation wins, although the halfplane invariant is also very successful. We complete the comparison with a brief discussion of the non-numerical advantages of various representations.

Numerical Aspects of Hyperbolic Geometry

TL;DR

The paper addresses numerical stability across diverse hyperbolic space representations used in applications like networks and machine learning. It conducts a broad, six-test comparison of linear, mixed, reduced, half-plane/half-space, and polar representations, augmented by tessellation-based strategies to counter precision loss. The results show that polar representations, particularly polar2, often yield the best numerical performance, with half-plane invariant also performing well, while certain fixed linear approaches excel in specific walk-related tasks; tessellations such as substantially improve robustness. These findings guide practitioners in selecting representations and error-control methods to balance numerical precision with practical considerations in hyperbolic geometry applications.

Abstract

Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation comes with some numerical instability. This paper compares various 2D and 3D hyperbolic geometry representations. To this end, we conduct an extensive simulational scheme based on six tests of numerical precision errors. Our comparisons include the most popular models and less-known mixed and reduced representations. According to our results, polar representation wins, although the halfplane invariant is also very successful. We complete the comparison with a brief discussion of the non-numerical advantages of various representations.
Paper Structure (16 sections, 4 equations, 6 figures)

This paper contains 16 sections, 4 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic geometry and representations
  3. Spherical geometry
  4. Minkowski hyperboloid (linear representation)
  5. Other models of $\mathbb{H}^2$, and polar representation.
  6. Clifford algebras, and mixed representation.
  7. Reduced representation.
  8. Half-plane representation.
  9. Half-space representation.
  10. Representation variants
  11. Tessellations
  12. Tests
  13. Experimental Results
  14. Comparison based on non-numerical advantages
  15. Conclusions
  16. ...and 1 more sections

Figures (6)

  • Figure 1: The {7,3} tessellation of the hyperbolic plane in the following models: hyperboloid, Poincaré disk, upper half-plane, Beltrami-Klein disk, polar coordinates.
  • Figure 2: Tessellations of $\mathbb{H}^2$ used in our experiments. From left to right: $\{7,3\}$, $\{5,4\}$, $\{8,3\}$, $\{4,5\}$, $\{4,6\}$, $\{6,4\}$, $\{6,6\}$.
  • Figure 3: Aggregate results of our tests, in 2 dimensions (above) and 3 dimensions (below). For each test, the score of the best representation is normalized to 1.
  • Figure 4: Condorcet rankings, in 2 dimensions. Better representations are closer to the bottom.
  • Figure 5: Condorcet rankings, in 3 dimensions.
  • ...and 1 more figures