Coordinatizing Data With Lens Spaces and Persistent Cohomology

TL;DR

This paper presents a novel framework for constructing coordinates in finite Lens spaces based on persistent cohomology, enabling topological data analysis and dimensionality reduction with minimal landmarks.

Contribution

It introduces a new method to coordinate data in Lens spaces using persistent cohomology and a dimensionality reduction scheme called Lens-PCA, advancing topological data analysis techniques.

Findings

Effective coordinate construction in Lens spaces for data with nontrivial cohomology

Demonstrated pipeline for topological dimensionality reduction

Successful application of Lens-PCA in nonlinear data analysis

Abstract

We introduce here a framework to construct coordinates in \emph{finite} Lens spaces for data with nontrivial 1-dimensional $\mathbb{Z}_q$ persistent cohomology, $q\geq 3$. 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 $S^{2n-1}/\mathbb{Z}_q$ (Lens-PCA: $\mathsf{LPCA}$), and demonstrate the efficacy of the pipeline $PH^1(\;\cdot\; ; \mathbb{Z}_q)$ class $\Rightarrow$ $S^{2n-1}/\mathbb{Z}_q$ coordinates $\Rightarrow$ $\mathsf{LPCA}$, for nonlinear (topological) dimensionality reduction.