Rolled Gaussian process models for curves on manifolds

TL;DR

The paper introduces rolled Gaussian process models that lift Euclidean GPs to curves on a manifold $M$ via four maps (unrolling, rolling, unwrapping, wrapping). By rolling mean curves onto $M$ and wrapping Gaussian processes defined on a tangent space, it yields an intrinsic, generative model for manifold-valued curves with tractable inference. It provides a finite-dimensional parametric GP, estimators and hypothesis tests for empirical data, and proves conditions under which rolled means coincide with Fréchet means and that estimators are consistent. The framework is demonstrated on the sphere, SPD matrices, and robot orientation data in $SO(3)$, with numerical experiments and a two-sample mean-curve test in robotics applications. This approach offers a principled, geometry-aware methodology for curve-valued data on manifolds with practical computational tools.

Abstract

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. The resulting model is generative, and is amenable to statistical inference given data as curves on a manifold. We illustrate with examples on the unit sphere, symmetric positive-definite matrices, and with a robotics application involving 3D orientations.

Figures (4)

• Figure 1: A piecewise geodesic curve $\gamma$ on $\mathbb S^2$, red, and its unrolling, blue, into a piecewise linear curve; (a) shows the unrolling $\gamma^{\downarrow}$ into the tangent space $T_{\gamma(0)}\mathbb S^2$ at the initial point of the curve, and (b) shows the unrolling $\gamma_b^{\downarrow}$, incorporating a translation shown in grey, into the tangent space, $T_{b}$, at a prescribed point, $b$. Unrolling, and its inverse, rolling, preserve consecutive inter-point distances on $\gamma$ and $\gamma^\downarrow$.
• Figure 2: Illustration of how curves defined in $\mathbb R^d$ are identified on $M$, via an intermediate tangent space $T_bM$. Red is the mean curve with respect to which the (un)flattening is performed, and blue is a different curve that deviates from the mean curve. The line segment connecting points between the blue and red curves at arbitrary $t$, and the corresponding angle denoted $\theta$, indicate distances and angles preserved by the (un)flattening.
• Figure 3: (a) Curves from the model in $\mathbb R^2$ described in §\ref{['example:rolled:model:on:S2']}. (b) Corresponding rolled and wrapped curves on $\mathbb S^2$.
• Figure 4: For robot SO(3) curves described in §\ref{['sec:robot:SO3']}: (a) unwrapped data, blue, and unrolled fitted mean, red; (b) simulations from the fitted Gaussian process model; (c) bootstrap null distribution for test described in text.

Theorems & Definitions (18)

• Definition 3.1
• Definition 3.2: The four maps
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 3.3: Rolled mean coinciding with the Fréchet mean
• Definition 3.4: Rolled Gaussian process
• Proposition 3.1: Gaussian vector field along $\gamma$
• Proposition 3.2: Equivariance of rolled Gaussian process
• Definition 4.1: Unwrapping coordinates