ToroFit: Fitting Wedge Sums of Tori to High-Dimensional Data

Tanishq Bhatia, Jose A. Perea, Tarek M. Zikry

Data arising in scientific applications often exhibits complex nonlinear structure. Identifying and parameterizing said nonlinearities can enable better modeling, enhance predictive analysis, data interpolation, denoising, and anomaly detection. In this paper, we develop theoretical and computational methods to detect and parametrize Euclidean point cloud data lying near a wedge sum of tori. Specifically, we use persistent cohomology and cup products to identify the distinct toroidal components, and on each component, we optimize over low-energy multivariate trigonometric polynomials for best data fit. The ensuing optimization problem reduces to generalized ridge regression, which can be solved efficiently via standard techniques. We demonstrate the utility of the proposed framework in both synthetic and real datasets, including interpolation of low frame-rate recurrent videos, attractor recovery in quasiperiodic dynamical systems, and the parametrization of gene-space cycle data for parasites and humans.