# Rolled Gaussian process models for curves on manifolds

arXiv: https://arxiv.org/abs/2503.21980 · HTML: https://mlsanity.com/paper/2503.21980/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Functional Models for Time-Varying Random Objects](https://mlsanity.com/paper/1907.10829/) (2019)
- [Modeling Time-Varying Random Objects and Dynamic Networks](https://mlsanity.com/paper/2104.04628/) (2021)
- [Intrinsic Riemannian Functional Data Analysis](https://mlsanity.com/paper/1812.01831/) (2018)
- [Intrinsic Riemannian Functional Data Analysis for Sparse Longitudinal Observations](https://mlsanity.com/paper/2009.07427/) (2020)
- [Wrapped Gaussian Process Functional Regression Model for Batch Data on Riemannian Manifolds](https://mlsanity.com/paper/2409.03181/) (2024)
- [Phase-Amplitude Separation and Modeling of Spherical Trajectories](https://mlsanity.com/paper/1603.07066/) (2016)
- [Smoothing splines on Riemannian manifolds, with applications to 3D shape space](https://mlsanity.com/paper/1801.04978/) (2018)
- [An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data](https://mlsanity.com/paper/1801.10341/) (2018)
- [Parallel transport on matrix manifolds and Exponential Action](https://mlsanity.com/paper/2408.06054/) (2024)

## Abstract (arXiv)

Given a planar curve, imagine rolling a sphere along that curve without slipping or twisting, and by this means tracing out a curve on the sphere. It is well known that such a rolling operation induces a local isometry between the sphere and the plane so that the two curves uniquely determine each other, and moreover, the operation extends to a general class of manifolds in any dimension. We use rolling to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process with mean m and covariance K, and refer to it as a rolled Gaussian process parameterized by m and K. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We identify conditions on the manifold under which the rolling of m equals the Fr'echet mean of the rolled Gaussian process, propose computationally simple estimators of m and K, and derive their rates of convergence. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.