Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

TL;DR

This paper introduces a systematic method using lattice reduction to generate low-energy, decorrelated torus-valued maps from cohomology classes, improving the analysis of circular coordinates in data.

Contribution

It formalizes geometric correlation between circle-valued maps and develops an algorithm based on lattice reduction to produce decorrelated torus-valued maps from cohomology classes.

Findings

Systematic procedure for constructing low-energy torus maps

Application of LLL algorithm for decorrelation

Computational examples demonstrating effectiveness

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--Lov\'asz algorithm from computational number theory.