# Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

arXiv: https://arxiv.org/abs/2212.07201 · HTML: https://mlsanity.com/paper/2212.07201/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 32 references that are in the map.
- First posted on arXiv 14 Dec 2022.

## Builds on (most cited first, 3 of 3)

- [(Quasi)Periodicity Quantification in Video Data, Using Topology](https://mlsanity.com/paper/1704.08382/) (2017)
- [Decoding of neural data using cohomological feature extraction](https://mlsanity.com/paper/1711.07205/) (2017)
- [Sparse Circular Coordinates via Principal mathbbZ-Bundles](https://mlsanity.com/paper/1809.09269/) (2018)

## Built on by (most cited first, 5 of 5)

- [Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning](https://mlsanity.com/paper/2307.15778/) (2023)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

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.