Autoregressive Processes on Riemannian Manifolds
Abstract
This paper introduces a Riemannian autoregressive (R-AR) model of order one for manifold-valued time series. The model is specified through an autoregressive process in the tangent space at a reference point, which is mapped to the manifold via the exponential map. It is characterised by two parameters: a reference point μ, representing the intrinsic central tendency through the Fréchet mean, and an autoregressive parameter ϕ, governing the dependence structure and stationarity properties. When these parameters are unknown, their estimation introduces geometric and probabilistic challenges arising from the intrinsic estimation of μ and from estimating ϕ as the base point varies over the manifold and the associated tangent spaces change. The estimation theory is developed using a strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains, together with ergodic arguments for the strong consistency of the autoregressive parameter estimator. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.
Related
- Smoothness estimation for Whittle–Matérn processes on closed Riemannian manifolds
- Hydrodynamic limit of exclusion processes with killing on weighted Riemannian manifolds via a graph discretization
- Scalable generative modeling of non-Gaussian spatio-temporal fields via autoregressive Gaussian processes
- Spline interpolation on compact riemannian manifolds
- Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
BibTeX
@misc{abuqrais2026autoregressive,
title = {{Autoregressive Processes on Riemannian Manifolds}},
author = {Meshal Abuqrais and Davide Pigoli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.24771},
eprint = {2606.24771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.24771},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Autoregressive Processes on Riemannian Manifolds
AU - Meshal Abuqrais
AU - Davide Pigoli
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2606.24771
UR - https://arxiv.org/abs/2606.24771
AB - This paper introduces a Riemannian autoregressive (R-AR) model of order one for manifold-valued time series. The model is specified through an autoregressive process in the tangent space at a reference point, which is mapped to the manifold via the exponential map. It is characterised by two parameters: a reference point μ, representing the intrinsic central tendency through the Fréchet mean, and an autoregressive parameter ϕ, governing the dependence structure and stationarity properties. When these parameters are unknown, their estimation introduces geometric and probabilistic challenges arising from the intrinsic estimation of μ and from estimating ϕ as the base point varies over the manifold and the associated tangent spaces change. The estimation theory is developed using a strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains, together with ergodic arguments for the strong consistency of the autoregressive parameter estimator. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.
ER -