Intrinsic Gaussian Vector Fields on Manifolds

TL;DR

This work addresses the challenge of modeling vector-valued signals on non-Euclidean domains by introducing fully intrinsic Gaussian vector fields on manifolds through the Hodge–Matérn framework. By leveraging the Hodge Laplacian and its heat kernel, the authors construct kernels that respect the manifold geometry and enable divergence-free, curl-free, and composite inductive biases, avoiding artifacts from extrinsic projections. They provide practical computation tools, explicit eigenstructures for the sphere and hypertori, and extend the methodology to product manifolds and potential extensions to meshes and Lie groups. Experimental results on ERA5 wind data demonstrate that divergence-free and compositional kernels yield improved predictive performance and uncertainty quantification, highlighting the approach’s relevance for climate modeling and robotics.

Abstract

Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when uncertainty quantification is needed. In the manifold setting, vector-valued signals can behave very differently from scalar-valued ones, with much of the progress so far focused on modeling the latter. The former, however, are crucial for many applications, such as modeling wind speeds or force fields of unknown dynamical systems. In this paper, we propose novel Gaussian process models for vector-valued signals on manifolds that are intrinsically defined and account for the geometry of the space in consideration. We provide computational primitives needed to deploy the resulting Hodge-Matérn Gaussian vector fields on the two-dimensional sphere and the hypertori. Further, we highlight two generalization directions: discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie groups, and homogeneous spaces. Finally, we show that our Gaussian vector fields constitute considerably more refined inductive biases than the extrinsic fields proposed before.

Figures (10)

• Figure 1: Vector fields on manifolds are not just vector functions, for them the vectors are always tangential.
• Figure 2: Comparing an intrinsic Matérn kernel ($\nu=\infty$) of \ref{['eqn:matern_on_manifolds']} to an extrinsic one, the restriction of a Euclidean Matérn kernel to the manifold. Note the latter induces high correlation between the points across the minor axis of the ellipse, despite them being far from each other in terms of the intrinsic distance.
• Figure 3: GP regression from a single observation (the red vector) for a very large length scale $\kappa$. Black vectors represent the prediction $\mu_{f \mid \v{y}}(\cdot)$, the background color shows the uncertainty $\lVert*\rVert{k_{f \mid \v{y}}(\cdot,\cdot)}$: yellow for high, blue for low. On (a) and (b) we use a projected Matérn GP of hutchinson2021; on (c) and (d) we use our Hodge--Matérn Gaussian vector field, $\nu = \infty$ in both cases. Unnaturally, uncertainty in (a) and (b) is non-monotonous with respect to the distance on the sphere: it is considerably lower at the antipode than halfway to it.
• Figure 4: Eigenfunctions on the sphere $\mathbb{S}_2$ (represented by color) and the respective eigenfields.
• Figure 5: Intrinsic Gaussian vector field samples on $\mathbb{S}_2$.

Theorems & Definitions (58)

• Definition 1
• Proposition 1
• Theorem 1
• Theorem 1
• Theorem 1
• Proposition 1
• Remark 2
• Theorem 2
• Proposition 3
• Proposition 3