Coordinatizing Data With Lens Spaces and Persistent Cohomology

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.