Coordinatizing Data With Lens Spaces and Persistent Cohomology

Abstract

We introduce here a framework to construct coordinates in finite Lens spaces for data with nontrivial 1-dimensional Zq persistent cohomology, for q > 2 prime. 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 /Zq (Lens-PCA: LPCA), and demonstrate the efficacy of the pipeline Zq-persistent cohomology ⇒ S 2n−1 /Zq coordinates ⇒ LPCA, for nonlinear (topological) dimensionality reduction.

Generalized Penalty for Circular Coordinate Representation

“…Beyond the S 1 coordinate functions, it is of interest to explore whether the idea of penalized smoothing could be extended to coordinate functions with values in a general topological space other than S 1 . In this direction, we want to explore the idea of generalized penalty functions with Eilenberg-MacLane coordinates, of which S 1 coordinates is a special case (Polanco and Perea 2019). This line of research is motivated by TDA literature extending the circular coordinate framework.…”

Simplicial approximation to CW complexes in practice

“…We can find a related approach in the work of Jose Perea. If X is a topological space and G a topological group, there exists a bijection between Ȟ1 (B; C G ), the first Čech cohomology group of B with coefficients in the sheaf C G , and [X, BG], the homotopy classes of maps from X to the classifying space BG of G. In a series of papers, he showed that some choices for the group G lead to various data analysis methods, such as dimension reduction, or coordinatization of the data [Per18,PP19,Per20].…”