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Implicit Gaussian process representation of vector fields over arbitrary latent manifolds
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Introduces RVGP, a Gaussian process over vector fields on unknown Riemannian manifolds using connection Laplacian eigenfunctions from a proximity graph as positional encoding, and applies it to super-resolve sparse EEG vector fields for Alzheimer's classification.
Problem
Modelling vector fields over latent (unknown) Riemannian manifolds with Gaussian processes, where existing vector-valued GPs require the manifold to be known and analytically tractable; also reconstructing high-density EEG from low-density recordings.
Approach
Generalises Matérn GPs to vector fields by using eigenvectors of the connection Laplacian computed from a proximity graph (with local tangent frames and parallel transport) as positional encoding, yielding a matrix-valued kernel that respects tangent-space geometry and preserves singularities.
Key results
- On the bunny mesh with 50% of nodes for training, RVGP gives accurate superresolution with only 10 eigenvectors over a broad range of sampling densities, while the RBF benchmark degrades at lower densities.
- RVGP significantly lowers angular error and curl-divergence MAE versus linear, spline and RBF interpolation for all 61 EEG subjects when reconstructing 256-channel alpha-band fields from 64 channels.
- Linear SVM classification of AD vs controls from RVGP-reconstructed divergence and curl approaches the accuracy obtained from ground-truth high-density EEG, and exceeds the benchmarks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use connection-Laplacian vector-valued RVGP instead of channel-wise RBF interpolation | Vector-field superresolution on the Stanford bunny and torus · Stanford bunny and torus superresolution | vector alignment | Helped |
| Use fewer connection-Laplacian eigenvectors rather than channel-wise RBF positional encoding (ablation) | RVGP superresolution · High-resolution surface, α=1.5% | vector alignment | Helped |
| Reduce sampling density (ablation) | RVGP and channel-wise RBF superresolution · Stanford bunny superresolution | vector alignment | Mixed |
| Use RVGP rather than RBF interpolation for masked-region inpainting | Vector-field inpainting · Masked vortex singularity | topological consistency | Helped |
| Use RVGP rather than channel-wise interpolation | EEG vector-field reconstruction · 64-channel EEG to 256-channel reconstruction | angular error | Helped |
| Use 32-channel rather than 64-channel EEG input (ablation) | EEG reconstruction methods · 32-channel EEG reconstruction | reconstruction error | Hurt |
| Use RVGP-reconstructed rather than low-density input features for classification | Linear SVM for Alzheimer's disease versus healthy controls · Alzheimer's disease vs healthy controls | classification accuracy | Helped |
| Add Gaussian geometric noise to manifold points (ablation) | RVGP superresolution · Off-manifold test and training points | prediction accuracy | No clear effect |
Surprising findings
- RBF kernel interpolation performed most similarly to RVGP on the EEG scalp, attributed to the homogeneous curvature of the human head.
- 32-channel EEG reconstruction errors were too large for all methods to be practically relevant, suggesting 64 channels is an empirical limit for resolving small-scale singularities.
Limitations
- Requires sufficient sampling density and a suitable similarity metric to build an intrinsic manifold approximation, especially in high dimensions.
- Benefit over channel-wise modelling diminishes when manifold curvature is homogeneous.
- Assumes tangent spaces of uniform dimension; non-tangent vector fields require an additional rotation step.
Experimental setup
Datasets: Stanford bunny mesh toroidal surface mesh 256-channel resting-state EEG (61 subjects: 33 AD, 28 controls)
Benchmarks: Stanford bunny superresolution torus inpainting 256-channel EEG reconstruction
Metrics: inner product alignment of vectors angular error divergence and curl mean absolute error singularity preservation ROC AUC for AD classification
Baselines: RBF kernel (channel-wise) linear interpolation spline interpolation scalar-valued GP with Laplacian eigenmaps
Builds on: Matérn Gaussian processes on Riemannian manifolds graph Laplacian positional encoding vector heat method connection Laplacian
Closest papers by meaning
- Residual Deep Gaussian Processes on Manifolds (ICLR 2025 · similarity 0.90)
- Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels (NeurIPS 2021 · similarity 0.89)
- Manifold GPLVMs for discovering non-Euclidean latent structure in neural data (NeurIPS 2020 · similarity 0.89)
- Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry (NeurIPS 2025 · similarity 0.86)
- Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems (NeurIPS 2024 · similarity 0.85)
- Scalable Gaussian Processes with Latent Kronecker Structure (ICML 2025 · similarity 0.85)
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