Learning vector fields when the manifold is unknown
Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying uncertainty in data. Recent work has extended GPs to scalar and vector quantities on non-Euclidean domains, such as the smooth manifolds that appear in computer vision, dynamical systems and neuroscience. However, these approaches assume that the manifold underlying the data is known, which limits their practical use.
RVGP: Gaussian processes on latent Riemannian manifolds
We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. RVGP uses positional encoding with eigenfunctions of the connection Laplacian, an operator associated with the tangent bundle that can be derived from a common graph-based approximation of the data. RVGP is globally regular over the manifold, which allows it to super-resolve and inpaint vector fields while preserving their singularities.
Reconstructing brain dynamics from EEG
We used RVGP to reconstruct high-density neural dynamics from low-density EEG recordings in healthy individuals and in patients with Alzheimer’s disease. Vector field singularities turned out to be important disease markers, and reconstructing them gave a disease-state classification accuracy comparable to that obtained from high-density recordings. RVGP thus overcomes a significant practical limitation in experimental and clinical applications.
See also our related work on how activity is routed across the human cortex.