Summary of Explicit Formulae to Interchangeably Use Hyperplanes and Hyperballs Using Inversive Geometry, by Erik Thordsen et al.
Explicit Formulae to Interchangeably use Hyperplanes and Hyperballs using Inversive Geometry
by Erik Thordsen, Erich Schubert
First submitted to arxiv on: 28 May 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Computational Geometry (cs.CG); Machine Learning (stat.ML)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper presents a novel approach to transforming general Euclidean data into spherical data using inversive geometry. This allows for interchangeable use of discriminative boundaries like separating hyperplanes or hyperballs. The authors provide explicit formulae for embedding and unembedding data, as well as equations to translate inner products and distances between the two spaces. They also introduce a method for enforcing projections onto hemi-hyperspheres and propose an intrinsic dimensionality-based approach to obtain “all-purpose” parameters. Applications in machine learning and vector similarity search demonstrate the usefulness of this cap-ball duality. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary Imagine taking normal data and turning it into special spherical data that’s easy to work with. This paper shows how to do just that using a new way of looking at shapes called inversive geometry. It also reveals a secret connection between two types of boundaries: separating hyperplanes and hyperballs. The authors give simple formulas for changing data from normal to spherical and back again. They even show how to translate important measurements like distances and similarities between the two spaces. This “spherical magic” can be used in machine learning and searching for similar vectors, making it a valuable tool for scientists. |
Keywords
» Artificial intelligence » Embedding » Machine learning
