Coordinatizing Data With Lens Spaces and Persistent Cohomology

2019/05/01 by Polanco, Luis, Perea, Jose A.
#Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.00350

Abstract

We introduce here a framework to construct coordinates in finite Lens spaces for data with nontrivial 1-dimensional ℤ_q 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 S^2n-1/ℤ_q (Lens-PCA: LPCA), and demonstrate the efficacy of the pipeline PH¹( ⋅ ; ℤ_q) class ⇒ S^2n-1/ℤ_q coordinates ⇒ LPCA, for nonlinear (topological) dimensionality reduction.

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