Projected Normal Models on Recursively Curved Tori
Abstract
We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus 𝕋n↪ℝn+1. Due to the recursive structure we build an exact uniform sampler by generalizing the process from the curved 2-torus. We generalize projected normal distributions to the curved n-torus and we develop inference theory for these distributions. We utilize both maximum-log-likelihood estimates and energy-based Fréchet estimates for the projected normal distribution for this purpose. This is done by establishing identifiability for isotropic projected normal distributions and by applying theory of maximum likelihood and intrinsic statistics for the curved tori. We numerically verify the geometry, sampler, density transformation, and energy solver. Lastly, we demonstrate the applicability of the proposed model using real data.
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BibTeX
@misc{biswas2026projected,
title = {{Projected Normal Models on Recursively Curved Tori}},
author = {Surojit Biswas and Aron Persson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08974},
eprint = {2610.08974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08974},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Projected Normal Models on Recursively Curved Tori
AU - Surojit Biswas
AU - Aron Persson
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08974
UR - https://arxiv.org/abs/2610.08974
AB - We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus 𝕋^n↪ℝ^n+1. Due to the recursive structure we build an exact uniform sampler by generalizing the process from the curved 2-torus. We generalize projected normal distributions to the curved n-torus and we develop inference theory for these distributions. We utilize both maximum-log-likelihood estimates and energy-based Fréchet estimates for the projected normal distribution for this purpose. This is done by establishing identifiability for isotropic projected normal distributions and by applying theory of maximum likelihood and intrinsic statistics for the curved tori. We numerically verify the geometry, sampler, density transformation, and energy solver. Lastly, we demonstrate the applicability of the proposed model using real data.
ER -