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Spherical coordinates from persistent cohomology

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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 \(\alpha : VR(X) \rightarrow S^2\) 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 \(\alpha \), 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.

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Acknowledgements

The authors would like to thank Chad Giusti for many helpful discussions, and Jose Perea for an enlightening conversation about the paper. We are also grateful to several anonymous referees, whose detailed and thoughtful feedback led to significant improvements in the clarity and cohesiveness of the paper. NC Schonsheck’s work is supported by the Air Force Office of Scientific Research under award number FA9550-21-1-0266 and SC Schonsheck’s work is supported by the US National Science Foundation grant DMS-1912747.

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  1. Nikolas C. Schonsheck and Stefan C. Schonsheck have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, 15 Orchard Road, Newark, 19716, Delaware, USA

    Nikolas C. Schonsheck

  2. TETRAPODS Institute of Data Science, University of California Davis, 1 Shields Avenue, Davis, 95616, California, USA

    Nikolas C. Schonsheck & Stefan C. Schonsheck

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  1. Nikolas C. Schonsheck
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Appendix A: Algorithm

Appendix A: Algorithm

Algorithm 1
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Energy Minimization

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Schonsheck, N.C., Schonsheck, S.C. Spherical coordinates from persistent cohomology. J Appl. and Comput. Topology 8, 149–173 (2024). https://doi.org/10.1007/s41468-023-00141-w

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