Implicit Gaussian process representation of vector fields over arbitrary latent manifolds

TL;DR

RVGP extends Gaussian processes to vector-valued signals on unknown latent manifolds by using the spectrum of the connection Laplacian ${\bm{L}}_c$ as a spectral positional encoding. The method constructs a proximity graph to approximate the manifold and its tangent bundle, uses parallel transport to define a global vector-field regularity, and forms a vector-valued kernel $k_{\mathcal{TM}}$ that respects curvature via the Weitzenböck relationship. It enables super-resolution and inpainting of vector fields while preserving singularities and demonstrates practical impact by reconstructing high-density neural dynamics from low-density EEG and improving Alzheimer's disease classification from resting-state data. This geometry-aware probabilistic framework offers a practical tool for neuroscience and geometric data analysis, robust to sampling density and noise.

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.

Figures (6)

• Figure 1: Construction of vector-valued Gaussian processes on unknown manifolds.A Input consists of samples from a vector field over a latent manifold $\mathcal{M}$. B The manifold is approximated by a proximity graph. Black circles mark two sample points, $i$ and $j$ and their graph neighbourhood. C The tangent bundle is a collection of locally Euclidean vector spaces over $\mathcal{M}$. It is approximated by parallel transport maps between local tangent space approximations. D The eigenvectors of the connection Laplacian are used as positional encoding to define the GP that learns the vector field. E The GP is evaluated as unseen points to predict the smoothest vector field that is consistent with the training data. We use this GP to accurately predict singularities, where sampling is typically sparse.
• Figure 2: Superresolution and inpainting.A Matrix-valued kernel (RVGP) against a scalar-valued kernel (e.g., in Hutchinson2021). B Uniformly distributed samples over the Stanford bunny and torus are interpolated to a higher resolution ($k=50$). C Ablation studies for the Stanford bunny, showing the dependence of alignment of the superresolved vector fields in the test set against data density quantified by the average distance between manifold points $\alpha$ (for $k=50$ fixed) and the number of eigenvectors $k$. The vectorial RVGP representation is compared against a channel-wise representation using an RBF kernel with Laplacian eigenvectors as positional encoding. D Prediction of singularity in masked area. RBF kernel predicts discontinuities along the masked boundary (triangle), vectors that protrude the mesh surface (star) and do not converge to zero magnitude at the singularity. RVGP predicts smoothly varying inpainting ($k=50$).
• Figure 3: Reconstruction of spatiotemporal wave patterns in human EEG. A Snapshot of an alpha wave pattern (8-15 Hz) recorded on low density (64 channel) EEG from a healthy subject projected in two dimensions. B Phase field of an alpha wave. Vector field denotes the the spatial gradient of the voltage signal. C Ground-truth and reconstructed high-density phase field (256 channel) using RVGP, linear and spline interpolation. Streamlines, computed based on the vector field, highlight features of the phase field. RVGP significantly better preserves singularities, i.e., sources, sinks and vortices. D Reconstruction accuracy, measured by the preservation of singularities. E Receiver operating characteristic (ROC) for binary classification of patients with Alzheimer's disease against healthy controls using a linear support vector machine trained on the divergence and vorticity fields. Shaded areas indicate a 95% confidence interval.
• Figure S1: Reconstruction of spatiotemporal wave patterns in 32-channel human EEG. Reconstruction accuracy, measured by the preservation of singularities. The accuracy is 10-fold lower than using 64-channel EEG, meaning that predictive power is lost for all methods at this resolution.
• Figure S2: Superresolution for sparse training data. Same as in Fig. 2A, but for points placed at an average distance of 5% of manifold diameter.