The paper introduces a new class of Gaussian vector field models called Hodge-Matern Gaussian vector fields, which are intrinsically defined on manifolds and account for the underlying geometry. Key highlights:
Existing approaches to modeling vector fields on manifolds, such as projected Gaussian processes, can introduce undesirable inductive biases. The proposed Hodge-Matern fields aim to address this by being fully intrinsic.
The Hodge-Matern fields are constructed using the Hodge Laplacian, a generalization of the Laplace-Beltrami operator to vector fields. This allows defining divergence-free, curl-free, and harmonic vector field components as specialized kernels.
Computational techniques are developed to efficiently evaluate and sample from Hodge-Matern kernels on important manifolds like the sphere and hypertori, by leveraging the eigenvalues and eigenfields of the Hodge Laplacian.
Experiments on wind data modeling demonstrate that the intrinsic Hodge-Matern fields, especially the divergence-free variant, outperform extrinsic approaches in terms of predictive accuracy and uncertainty quantification.
The paper also discusses potential extensions to meshes and more general manifolds like Lie groups and homogeneous spaces.
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