Intrinsic Gaussian Vector Fields: Modeling Tangential Vector Signals on Manifolds

Główne pojęcia

This paper proposes novel Gaussian process models for vector-valued signals on manifolds that are intrinsically defined and account for the geometry of the space, addressing limitations of previous extrinsic approaches.

Streszczenie

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.

Statystyki

The paper does not contain any explicit numerical data or statistics. The focus is on the theoretical development of the Hodge-Matern Gaussian vector field models and their computational aspects.

Cytaty

"Gaussian processes can be scalar- or vector-valued (Alvarez et al., 2012). The important special case of the latter is Gaussian vector fields. These can for example be used to model velocities or accelerations, either as a target in itself or as means for exploring an unknown dynamical system."
"To remedy this, we propose a new approach: fully intrinsic Gaussian vector fields based on the Hodge Laplacian that we name Hodge–Matern Gaussian vector fields."
"By showing how our intrinsic Hodge–Matern Gaussian vector fields improve over their na¨ıve extrinsic counterparts on the two-dimensional sphere, we hope to motivate further research. First, into the development of practical intrinsic Gaussian vector fields on other domains. Second, into applying the proposed models in areas like climate/weather modeling and robotics."

Kluczowe wnioski z

Intrinsic Gaussian Vector Fields on Manifolds

by Daniel Rober... o arxiv.org 04-02-2024

https://arxiv.org/pdf/2310.18824.pdf

Głębsze pytania

How can the proposed Hodge-Matern Gaussian vector field models be extended to handle more general manifold structures beyond the sphere and hypertori considered in the paper

The proposed Hodge-Matern Gaussian vector field models can be extended to handle more general manifold structures beyond the sphere and hypertori by leveraging the concept of product manifolds. When dealing with product manifolds, if the eigenfields of the Hodge Laplacian for the individual factors are known, it is possible to construct the eigenfields for the product manifold. By combining the eigenfields of the individual factors, the eigenfields for the product manifold can be derived. This approach allows for the extension of the Hodge-Matern Gaussian vector field models to more complex manifold structures composed of multiple components.

What are the potential challenges and limitations in applying the intrinsic Hodge-Matern fields to real-world applications like climate modeling or robotics, and how can they be addressed

One potential challenge in applying the intrinsic Hodge-Matern fields to real-world applications like climate modeling or robotics is the computational complexity involved in computing the eigenvalues and eigenfields of the Hodge Laplacian, especially for high-dimensional or complex manifolds. This challenge can be addressed by developing efficient numerical techniques and algorithms for approximating these eigenvalues and eigenfields. Additionally, the interpretability of the results and the generalization of the models to diverse real-world scenarios may pose limitations. To address these challenges, it is essential to validate the models rigorously on diverse datasets and real-world scenarios, ensuring that the models capture the intrinsic geometry of the manifolds accurately and effectively.

Are there other intrinsic operators beyond the Hodge Laplacian that could be leveraged to define alternative classes of Gaussian vector field models on manifolds, and how would they compare to the Hodge-Matern approach

There are other intrinsic operators beyond the Hodge Laplacian that could be leveraged to define alternative classes of Gaussian vector field models on manifolds. One such operator is the Connection Laplacian, which induces a different class of Gaussian vector fields compared to the Hodge Laplacian. The Connection Laplacian takes into account the geometric structure of the manifold in a different way, leading to distinct properties and behaviors of the resulting Gaussian vector fields. By exploring alternative intrinsic operators like the Connection Laplacian, researchers can compare and contrast the performance and characteristics of different classes of Gaussian vector field models on manifolds, providing insights into the most suitable approaches for specific applications.