Implicit Gaussian process representation of vector fields over arbitrary latent manifolds

TL;DR

This paper introduces RVGP, a novel Gaussian process framework for learning vector fields over unknown latent manifolds, enabling super-resolution, inpainting, and disease marker identification from low-density data.

Contribution

RVGP extends Gaussian processes to unknown Riemannian manifolds using eigenfunction-based positional encoding, allowing for accurate vector field reconstruction and analysis.

Findings

RVGP can super-resolve vector fields on manifolds.

It accurately reconstructs neural dynamics from low-density EEG data.

Vector field singularities serve as disease markers.

Abstract

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.