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Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction

Authors: Luis Scoccola, Hitesh Gakhar, Johnathan Bush, Nikolas Schonsheck, Tatum Rask, Ling Zhou, and Jose A. Perea

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)


Abstract
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a 1-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra-Lenstra-Lovász algorithm from computational number theory.

Cite as

Luis Scoccola, Hitesh Gakhar, Johnathan Bush, Nikolas Schonsheck, Tatum Rask, Ling Zhou, and Jose A. Perea. Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 57:1-57:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{scoccola_et_al:LIPIcs.SoCG.2023.57,
  author =	{Scoccola, Luis and Gakhar, Hitesh and Bush, Johnathan and Schonsheck, Nikolas and Rask, Tatum and Zhou, Ling and Perea, Jose A.},
  title =	{{Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{57:1--57:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.57},
  URN =		{urn:nbn:de:0030-drops-179073},
  doi =		{10.4230/LIPIcs.SoCG.2023.57},
  annote =	{Keywords: dimensionality reduction, lattice reduction, Dirichlet energy, harmonic, cocycle}
}
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