Coordinatizing Data With Lens Spaces and Persistent Cohomology

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.