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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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GrooveSquid.com Paper Summaries

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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