Spherical Coordinates from Persistent Cohomology

TL;DR

This paper introduces a novel method for spherical data parameterization using persistent cohomology and variational optimization, enabling topologically faithful and smooth spherical mappings.

Contribution

It combines persistent cohomology with variational optimization to produce spherical parameterizations that preserve topological features and are computationally feasible.

Findings

Successfully applied to synthetic and real data sets

Produces smooth, topologically accurate spherical maps

Demonstrates effectiveness through numerical experiments

Abstract

We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set $X$ and extract a cocycle $\alpha$ from any significant feature. From this cocycle, we define an associated map $\alpha: VR(X) \to S^2$ and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/M\"{o}bius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of $\alpha$, preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach.