Explicit Formulae to Interchangeably use Hyperplanes and Hyperballs using Inversive Geometry

TL;DR

This paper introduces explicit formulae using inversive geometry to convert between hyperplanes and hyperballs, enabling flexible discriminative boundaries and data transformations in Euclidean and spherical spaces for machine learning applications.

Contribution

It provides novel explicit formulae for embedding Euclidean data into spherical spaces and mapping between hyperspherical caps and hyperballs, facilitating boundary interchangeability.

Findings

Explicit formulae for embedding Euclidean data into spherical data.

Duality between hyperspherical caps and hyperballs with mapping equations.

Applications demonstrated in machine learning and vector similarity search.

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.