Geometry-Adaptive Clustering on High-Dimensional Tori
Abstract
Multivariate angular data lie on a torus, but clustering methods often use the flat torus. We introduce curved-torus energy (CTE), a position-dependent dissimilarity based on restricted Riemannian path energy and minimization over winding classes. CTE has a certified finite winding reduction, an exact flat endpoint, a geodesic upper bound, and stability with respect to the geometry parameter. For label-free geometry selection, we propose CEHS-HM, a held-out criterion for selecting the warping parameter at a specified clustering resolution. We establish a finite-grid population-score regret bound and also consider joint geometry-resolution selection over finite candidate grids. Controlled experiments on 𝕋3 show that CTE agrees with the flat method at zero warping and becomes more useful as geometric warping increases. Additional experiments show that the CEHS-selected geometry should be viewed as a data-adaptive clustering choice rather than as an estimator of a generative deformation parameter. Under a product-von-Mises mechanism, the selector tends toward the flat endpoint when curvature is unnecessary. On ILE protein torsion data, the proposed method identifies a stable five-mode representation of joint backbone-side-chain conformation.
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BibTeX
@misc{biswas2026geometryadaptive,
title = {{Geometry-Adaptive Clustering on High-Dimensional Tori}},
author = {Surojit Biswas and Soumyadip Paik and Buddhananda Banerjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24079},
eprint = {2610.24079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24079},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Geometry-Adaptive Clustering on High-Dimensional Tori
AU - Surojit Biswas
AU - Soumyadip Paik
AU - Buddhananda Banerjee
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24079
UR - https://arxiv.org/abs/2610.24079
AB - Multivariate angular data lie on a torus, but clustering methods often use the flat torus. We introduce curved-torus energy (CTE), a position-dependent dissimilarity based on restricted Riemannian path energy and minimization over winding classes. CTE has a certified finite winding reduction, an exact flat endpoint, a geodesic upper bound, and stability with respect to the geometry parameter. For label-free geometry selection, we propose CEHS-HM, a held-out criterion for selecting the warping parameter at a specified clustering resolution. We establish a finite-grid population-score regret bound and also consider joint geometry-resolution selection over finite candidate grids. Controlled experiments on 𝕋^3 show that CTE agrees with the flat method at zero warping and becomes more useful as geometric warping increases. Additional experiments show that the CEHS-selected geometry should be viewed as a data-adaptive clustering choice rather than as an estimator of a generative deformation parameter. Under a product-von-Mises mechanism, the selector tends toward the flat endpoint when curvature is unnecessary. On ILE protein torsion data, the proposed method identifies a stable five-mode representation of joint backbone-side-chain conformation.
ER -