Coordinatizing Data With Lens Spaces and Persistent Cohomology

05/01/2019

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by   Luis Polanco, et al.

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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 PH^1( · ; Z_q) class S^2n-1/Z_q coordinates LPCA, for nonlinear (topological) dimensionality reduction.