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, $q\ge 3$. 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 $P{H}^1(\cdot ;{\mathbb{Z}}_q)$ class $\Rightarrow$ $S^{2n-1}/{\mathbb{Z}}_q$ coordinates $\Rightarrow$ $\mathsf{LPCA}$, for nonlinear (topological) dimensionality reduction.