Rolled Gaussian process models for curves on manifolds

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 an (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. 

I will describe how rolling may be used to construct an analogue of a Gaussian process on a manifold starting from a Euclidean Gaussian process parameterised by a mean and a covariance function. Rolling thus prescribes a simple generative model for a dataset of curves on a manifold. The key benefit is that standard estimates for the mean and covariance functions can be used once the curves are unrolled onto a Euclidean space. I will discuss asymptotic properties of such estimators, detail how the curvature of the manifold influences convergence rates, and demonstrate use of the model in a robotics application involving 3D orientations.