# Coordinatizing Data With Lens Spaces and Persistent Cohomology

arXiv: https://arxiv.org/abs/1905.00350 · HTML: https://mlsanity.com/paper/1905.00350/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2019.

## Built on by (most cited first, 2 of 2)

- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)

## Abstract (arXiv)

We introduce here a framework to construct coordinates in finite Lens spaces for data with nontrivial 1-dimensional mathbbZq persistent cohomology, q≥ 3. Said coordinates are defined on an open neighborhood of the data, yet constructed with only a small subset of landmarks. We also introduce a dimensionality reduction scheme in S2n-1/mathbbZq (Lens-PCA: mathsfLPCA), and demonstrate the efficacy of the pipeline PH1(;·; ; mathbbZq) class Rightarrow S2n-1/mathbbZq coordinates Rightarrow mathsfLPCA, for nonlinear (topological) dimensionality reduction.