Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

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 su...

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Bibliographic Details
Published inarXiv.org
Main Authors Scoccola, Luis, Gakhar, Hitesh, Bush, Johnathan, Schonsheck, Nikolas, Tatum Rask, Zhou, Ling, Perea, Jose A
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.03.2023
Subjects
Algorithms
Circles (geometry)
Correlation
Dirichlet problem
Homology
Inspection
Number theory
Riemann manifold
Toruses
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Summary: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.
ISSN:2331-8422