Rolled Gaussian process models for curves on manifolds

2025/03/27 by Preston, Simon, Bharath, Karthik, Lopez-Custodio, Pablo +1
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2503.21980

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.

Related

• Manifold embedding for curve registration
• An intrinsic formulation for rolling pseudo-Riemannian manifolds
• Extrinsic Gaussian processes for regression and classification on manifolds
• Gaussians on Riemannian Manifolds: Applications for Robot Learning and\n Adaptive Control
• Gaussian model for closed curves