ToroFit: Fitting Wedge Sums of Tori to High-Dimensional Data
Abstract
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.
Related
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- Free maps in critical dimension on low-dimensional tori and closed surfaces
- Strichartz estimates on two-dimensional irrational tori
- Geometry-Adaptive Clustering on High-Dimensional Tori
- The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
BibTeX
@misc{bhatia2026torofit,
title = {{ToroFit: Fitting Wedge Sums of Tori to High-Dimensional Data}},
author = {Tanishq Bhatia and Jose A. Perea and Tarek M. Zikry},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06990},
eprint = {2611.06990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06990},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - ToroFit: Fitting Wedge Sums of Tori to High-Dimensional Data
AU - Tanishq Bhatia
AU - Jose A. Perea
AU - Tarek M. Zikry
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06990
UR - https://arxiv.org/abs/2611.06990
AB - 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.
ER -