Spherical Coordinates from Persistent Cohomology

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 $X$. We extract a cocycle $\alpha$ from any significant feature and define an associated map $\alpha: VR(X) \to S^2$. We use this map as an infeasible initialization for a variational optimization problem with a unique minimizer, up to rigid motion. We 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 $\alpha$. We show that this process preserves the relevant topological feature of the data and converges to a feasible optimum. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach.