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Explicit Formulae to Interchangeably use Hyperplanes and Hyperballs using Inversive Geometry

Erik Thordsen, Erich Schubert

TL;DR

The paper tackles the limitation that many algorithms assume spherical geometry or hyperplane boundaries. It develops an inversive-geometry framework that embeds general Euclidean data onto a hypersphere via explicit maps $H_A(X,v,s)$, $S_A(X,v,s)$, and $S(X,v,s)$, and provides an inverse $S^{-1}$ to recover original coordinates. Central contributions include exact Cap-Ball and Cap-Ellipsoid dualities with closed-form center and radius formulas (e.g., $c = S^{-1}\left(\frac{p - \alpha v}{\left\Vert p - \alpha v \right\Vert}, v, s\right)$, $r = s\sqrt{\frac{2\alpha}{b+p_{d+1}}}$, $\alpha = \frac{1-b^2}{2(b+p_{d+1})}$), plus extensions to ellipsoids via Cap-Ellipsoid-Duality and practical guidance for parameter selection using ABID. The framework enables applying sphere-based methods to non-spherical data and vice versa, with kernelizable distance/dot-product translations and useful overhead reductions, demonstrated across machine-learning tasks and vector similarity search. Overall, the approach broadens the applicability of spherical-data techniques and provides concrete, reusable formulas for embedding, dualities, and distance computations in high dimensions.

Abstract

Many algorithms require discriminative boundaries, such as separating hyperplanes or hyperballs, or are specifically designed to work on spherical data. By applying inversive geometry, we show that the two discriminative boundaries can be used interchangeably, and that general Euclidean data can be transformed into spherical data, whenever a change in point distances is acceptable. We provide explicit formulae to embed general Euclidean data into spherical data and to unembed it back. We further show a duality between hyperspherical caps, i.e., the volume created by a separating hyperplane on spherical data, and hyperballs and provide explicit formulae to map between the two. We further provide equations to translate inner products and Euclidean distances between the two spaces, to avoid explicit embedding and unembedding. We also provide a method to enforce projections of the general Euclidean space onto hemi-hyperspheres and propose an intrinsic dimensionality based method to obtain "all-purpose" parameters. To show the usefulness of the cap-ball-duality, we discuss example applications in machine learning and vector similarity search.

Explicit Formulae to Interchangeably use Hyperplanes and Hyperballs using Inversive Geometry

TL;DR

The paper tackles the limitation that many algorithms assume spherical geometry or hyperplane boundaries. It develops an inversive-geometry framework that embeds general Euclidean data onto a hypersphere via explicit maps , , and , and provides an inverse to recover original coordinates. Central contributions include exact Cap-Ball and Cap-Ellipsoid dualities with closed-form center and radius formulas (e.g., , , ), plus extensions to ellipsoids via Cap-Ellipsoid-Duality and practical guidance for parameter selection using ABID. The framework enables applying sphere-based methods to non-spherical data and vice versa, with kernelizable distance/dot-product translations and useful overhead reductions, demonstrated across machine-learning tasks and vector similarity search. Overall, the approach broadens the applicability of spherical-data techniques and provides concrete, reusable formulas for embedding, dualities, and distance computations in high dimensions.

Abstract

Many algorithms require discriminative boundaries, such as separating hyperplanes or hyperballs, or are specifically designed to work on spherical data. By applying inversive geometry, we show that the two discriminative boundaries can be used interchangeably, and that general Euclidean data can be transformed into spherical data, whenever a change in point distances is acceptable. We provide explicit formulae to embed general Euclidean data into spherical data and to unembed it back. We further show a duality between hyperspherical caps, i.e., the volume created by a separating hyperplane on spherical data, and hyperballs and provide explicit formulae to map between the two. We further provide equations to translate inner products and Euclidean distances between the two spaces, to avoid explicit embedding and unembedding. We also provide a method to enforce projections of the general Euclidean space onto hemi-hyperspheres and propose an intrinsic dimensionality based method to obtain "all-purpose" parameters. To show the usefulness of the cap-ball-duality, we discuss example applications in machine learning and vector similarity search.
Paper Structure (8 sections, 5 theorems, 25 equations, 7 figures)

This paper contains 8 sections, 5 theorems, 25 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Hyperspherical Embeddings
  3. Embedded Hyperspherical Caps and Hyperballs
  4. Embedded Hyperspherical Caps and Hyperellipsoids
  5. Parameter Choice
  6. Relevance to Machine Learning
  7. Conclusion
  8. Appendix

Key Result

Theorem 1

For arbitrary $p \in \mathcal{S}_{d}$ and $b \in (-1,1)$ with $b + p_{d+1} > 0$, the image under $S^{-1}(\mathord{{}\_{}}, v, s)$ of the hyperspherical cap $C(p,b)$ is a hyperball in $\mathbb{R}^d$ whenever $v = (0,\ldots,0,1) \in \mathbb{R}^{d+1}$ and $s \in \mathbb{R}_{>0}$. The center and radius

Figures (7)

  • Figure 1: Visualization of the (inverse) stereographic projection from one- and two-dimensional data to two- and three-dimensional spheres. By offsetting the data to a plane in an additional dimension (blue) and inverting the vector norms, i.e., mirroring each point with the unit sphere (green) around the origin (black), we obtain a spherical distribution (orange).
  • Figure 2: A visual proof that the radius in direction $v_{1,\ldots,d}$ is the only one varying from the radius in the spherical case and that its scaling factor is $v_{d+1}$. $i$ and $j$ indicate arbitrary basis vectors orthogonal to $(0,\ldots,0,1)$, not necessarily parallel to coordinate axes. The cosine of $\theta$ equals $v_{d+1}$.
  • Figure 3: Pairwise cosines before and after spherical embedding of mean-centered data. From each distribution we drew 500 samples. For the generated datasets we used 10 dimensional distributions. The visualized cosines belong to 10k random sample pairs. The blue line indicates the identity function.
  • Figure 4: ABID estimates over spherical embeddings with varying $s$ parameter. From each distribution we drew 500 samples. For the generated datasets we used 10 dimensional distributions. The horizontal black line indicates the ABID estimate of the original dataset plus one. The red vertical line indicates the mean vector norm. The blue vertical lines indicate $0.1$ and $10$ times the mean vector norm, respectively.
  • Figure 5: The decision function values for SVDDs and SVMs fitted to the original and embedded data, respectively. Left shows the results for a Gaussian distribution in 10 dimensions, right for three Gaussian blobs in 10 dimensions. The datasets of 5000 points each were iteratively shifted to the center of the SVM induced sphere to obtain an equal shift in distances in all directions. The shifting process is included in the displayed fitting times.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: Hypherspherical Caps
  • Definition 2: Hyperball
  • Theorem 1: Cap-Ball-Duality
  • Proof 1
  • Theorem 2: Cap-Ball-Duality 2
  • Proof 2
  • Corollary 3: Hemispherical Embedding
  • Proof 3
  • Corollary 4
  • Proof 4
  • ...and 5 more