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 $α$ from any significant feature. From this cocycle, we define an associated map $α: VR(X) \to S^2$ 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/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of $α$, 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.
Generalized Penalty for Circular Coordinate Representation
The Ohio State University（俄亥俄州立大学）；Indiana University（印第安纳大学）；Indiana University Network Science Institute（印第安纳大学网络科学研究所）；Inria Saclay（法国国家信息与自动化研究所萨克莱分部）；CUNY College of Staten Island（纽约市立大学史泰登岛学院）；CUNY Graduate Center（纽约市立大学研究生中心）
