Projected Normal Models on Recursively Curved Tori

2026-10-06

AI Summary

This study addresses the statistical modeling of angular data on curved tori by developing a computationally tractable n-dimensional recursive embedding framework. Methodologically, it generalizes the projected normal distribution to n-dimensional curved tori, enabling exact uniform sampling under the recursive structure and establishing the identifiability of isotropic distributions. For inference, the framework integrates maximum likelihood estimation, energy-based Fréchet estimation, and intrinsic statistical theory. Numerical experiments validate the underlying geometric properties and algorithmic effectiveness, while applications to real-world datasets confirm the model's practical utility. The core contribution of this work lies in providing a unified statistical paradigm that is both theoretically rigorous and computationally feasible for high-dimensional manifold-valued angular data.

Abstract

We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus $\mathbb{T}^n\hookrightarrow\mathbb{R}^{n+1}$. 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.

Problem

• angular data
• curved torus
• projected normal distribution
• statistical inference
• Fréchet estimates

Innovation

• Projected Normal Distribution
• Curved Torus
• Angular Data
• Fréchet Estimation
• Uniform Sampler