Numerical Stability Analysis of Hyperbolic Geometry Representations

Core Concepts

Comparing the numerical stability and precision of various representations of hyperbolic geometry, including linear, mixed, reduced, half-plane/half-space, and generalized polar coordinates, to determine the most accurate and efficient representation.

Abstract

The paper compares different representations of hyperbolic geometry to determine the most numerically stable and precise approach. It considers five main representations: linear, mixed, reduced, half-plane/half-space, and generalized polar coordinates. The authors also evaluate six variants of dealing with numerical errors: invariant, careless, flattened, forced, weakly forced, and binary.
The authors conduct six tests to capture different scenarios that can lead to accumulating numerical imprecisions, such as loop computations, angle and distance calculations, and walking along a path. The results suggest that the polar representation is the best in many cases, although the half-plane invariant is also very successful. The authors also find that fixed linear representations (especially invariant) perform well in game-design-related scenarios.
Additionally, the paper discusses non-numerical advantages of the different representations, such as intuitiveness, ability to represent orientation-reversing isometries, and generalization to other geometries. The authors conclude that the choice of representation should consider both numerical and non-numerical factors.

Stats

The circumference of a hyperbolic circle of radius r is exponential in r, making it difficult to represent points far from the origin accurately with a fixed number of bits.
The Beltrami-Klein disk model maps points in the hyperbolic plane to the inside of the unit disk, where points far from the origin are represented by points very close to the boundary, leading to numerical instability.
In the Poincaré disk model, the mapping of points in the hyperbolic plane to the unit disk is more gradual, allowing for better numerical stability.

Quotes

"Since the circumference of a hyperbolic circle of radius r is exponential in r, any representation based on a fixed number of bits will not be able to distinguish between points in a circle of radius r = Θ(b), even if the pairwise distances between these points are large."
"In the Beltrami-Klein disk model, a point in distance d from C0 is mapped to a point in distance tanh(d) from the center of the disk, which is 1 −Θ(exp(2d)). Floating point numbers cannot express such a slight difference from 1."
"In the Poincaré disk model, this is 1 −Θ(exp(d)), which is more numerically stable."

Key Insights Distilled From

Numerical Aspects of Hyperbolic Geometry

by Dorota Celin... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09039.pdf

Deeper Inquiries

How do the numerical stability and precision of the representations compare when dealing with non-compact hyperbolic geometries, such as those used in certain applications

In the study on hyperbolic geometry representations, the numerical stability and precision of the representations were compared in various scenarios, including non-compact hyperbolic geometries. The representations were evaluated based on their performance in tests that involved accumulating numerical imprecisions, such as the LoopIso, LoopPoint, AngleDist, Distance, Walk, and Close tests.
The results of the study indicated that certain representations, such as the polar representation and the half-plane invariant representation, performed well in maintaining numerical stability in non-compact hyperbolic geometries. These representations were able to handle large distances and angles accurately, which is crucial in applications where precision is essential, such as in social network analysis or machine learning.
On the other hand, representations like the Beltrami-Klein disk model showed limitations in dealing with points far away from the origin, leading to numerical errors due to the exponential growth of distances in hyperbolic space. The study highlighted the importance of tessellations in enhancing numerical precision, especially in scenarios where points are located at significant distances from the origin.
Overall, the study provided valuable insights into the numerical stability of hyperbolic geometry representations in non-compact settings, emphasizing the significance of choosing appropriate representations for specific applications based on their performance in handling numerical precision errors.

What are the potential trade-offs between numerical stability and other desirable properties, such as intuitiveness or ease of implementation, when choosing a representation for a specific application

When choosing a representation for a specific application, there are potential trade-offs between numerical stability and other desirable properties, such as intuitiveness or ease of implementation.

Numerical Stability vs. Intuitiveness: Representations that prioritize numerical stability, such as the polar representation or the half-plane invariant representation, may not always be the most intuitive for users who are more familiar with other models like the Poincaré disk model. In such cases, there might be a trade-off between numerical accuracy and user familiarity.

Numerical Stability vs. Ease of Implementation: Some representations that offer high numerical stability, such as the linear representation or the reduced representation, may require more complex implementation compared to simpler models like the Poincaré disk model. This trade-off between accuracy and implementation complexity needs to be considered based on the specific requirements of the application.

Numerical Stability vs. Generalizability: Representations that excel in numerical stability may not always generalize well to other geometries or applications. For instance, while the half-plane model performs well numerically, it may not be as versatile as other representations in accommodating different types of geometric transformations or structures.

In conclusion, the choice of representation for a specific application involves balancing the need for numerical stability with other factors like intuitiveness, ease of implementation, and generalizability. Understanding the trade-offs between these aspects is crucial in selecting the most suitable representation for the intended use case.

Could the insights from this study on hyperbolic geometry representations be extended to the analysis of numerical stability in representations of other non-Euclidean geometries, such as spherical or elliptic geometry

The insights gained from the study on hyperbolic geometry representations can be extended to the analysis of numerical stability in representations of other non-Euclidean geometries, such as spherical or elliptic geometry.

Applicability of Tessellations: Similar to hyperbolic geometry, tessellations can be used in spherical or elliptic geometry to enhance numerical precision and stability. By associating points with tiles in tessellations, representations can mitigate numerical errors caused by large distances or angles in these geometries.

Representation Trade-offs: Just as in hyperbolic geometry, different representations in spherical or elliptic geometry may exhibit trade-offs between numerical stability and other properties like intuitiveness or ease of implementation. Understanding these trade-offs is essential in selecting the most suitable representation for specific applications.

Generalizability of Insights: The principles and methodologies used in the study of hyperbolic geometry representations, such as comparing various models based on numerical precision tests, can be applied to analyze representations in spherical or elliptic geometry. By conducting similar studies, researchers can identify the most effective representations for different applications in these non-Euclidean geometries.

In summary, the insights from the study on hyperbolic geometry representations provide a valuable framework for evaluating numerical stability in representations of other non-Euclidean geometries, facilitating the selection of optimal models for diverse geometric applications.