# Sparse Circular Coordinates via Principal mathbbZ-Bundles

arXiv: https://arxiv.org/abs/1809.09269 · HTML: https://mlsanity.com/paper/1809.09269/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2018.

## Built on by (most cited first, 6 of 6)

- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction](https://mlsanity.com/paper/2212.07201/) (2022)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for circle-valued functions on data with nontrivial underlying topology. We show that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks. It is in this sense that the coordinates are sparse. Several data examples are presented, as well as theoretical results underlying the construction.