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Integration-free Kernels for Equivariant Gaussian Process Modelling
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Characterizes stochastic equivariance of vector-valued Gaussian random fields via kernel equivariance and builds integration-free equivariant matrix-valued kernels by projecting inputs onto fundamental regions of group actions.
Problem
Building equivariant matrix-valued GP kernels requires costly group integrations (Haar integration), making GP inference with equivariant vector-valued kernels computationally prohibitive, and existing alternatives (e.g., Helmholtz kernel) are equivariant only in the posterior mean.
Approach
Prove that stochastic equivariance of a centred second-order vector random field is equivalent to kernel equivariance, then construct integration-free equivariant kernels by mapping inputs to a fundamental region via a section (group element) and applying a base kernel with left/right multiplication by representation matrices.
Key results
- On SO(2) vector fields, the integration-free kernel K_Pi gives RMSE 0.11 (F1) and 0.13 (F2) versus 0.27/0.33 for squared exponential, and posterior computation takes about 55 seconds versus 45 hours for the integral-based kernel at 500 test points and 100 training points.
- On water dipole moments, the integration-free equivariant GP significantly outperforms the baseline squared-exponential GP and invariant variants across all training set sizes.
- In the ocean experiment, combining equivariant and non-equivariant kernels yields better LogS than single non-equivariant kernels for most mixing values.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the integration-free equivariant kernel KΠ instead of KSE | Synthetic SO(2) vector-field GP · F1 | RMSE: 0.27 [0.11] -> 0.11 [0.06]* | Helped |
| use the integration-free equivariant kernel KΠ instead of KSE | Synthetic SO(2) vector-field GP · F2 | RMSE: 0.33 [0.06] -> 0.13 [0.05] | Helped |
| use KΠ instead of the double-integral equivariant kernel KR | Synthetic SO(2) vector-field GP · F2 | RMSE: 0.22 [0.2] -> 0.13 [0.05] | Helped |
| use KΠ instead of KR | Synthetic SO(2) vector-field GP · F1 | LogS: -5.95 [0.46]* -> -6.03 [0.51] | No clear effect |
| replace integration-based KR evaluation with integration-free KΠ evaluation | Equivariant GP posterior computation · 500 test locations given 100 training points | computation time: 45 hours -> 55 seconds | Helped |
| use a disconnected, partitioned fundamental region instead of the connected region (ablation) | KΠ · Synthetic SO(2) field F2 | RMSE: 0.13 [0.05] -> 0.49 | Hurt |
| use a disconnected, partitioned fundamental region instead of the connected region (ablation) | KΠ · Synthetic SO(2) field F2 | LogS: -7.41 [0.81] -> −2.56 | Hurt |
| include translation, permutation, and rotation equivariance in the kernel | Water-molecule dipole GP · Water molecule dipole moments | RMSE | Helped |
| combine non-equivariant and equivariant kernels instead of using a single non-equivariant kernel | Ocean velocity GP, GP 5 · Gulf of Mexico drifter velocities | LogS | Helped |
| use the combined equivariant and non-equivariant kernel instead of the single non-equivariant kernel | Ocean velocity GP, GP 5 · Gulf of Mexico drifter velocities | RMSE | Mixed |
Surprising findings
- A disconnected fundamental region (alternating partition) substantially degrades performance and produces discontinuous posterior means.
- The Helmholtz GP does not give rotation-equivariant posterior samples despite its equivariant posterior mean.
Limitations
- Discontinuities of the kernel and poor choice of fundamental region can degrade performance
- Cubic complexity of exact GP inference limits scale; sparse GP extension is only sketched
- Choice of fundamental region and section is not unique and requires problem-specific construction
Experimental setup
Datasets: synthetic SO(2)-equivariant vector fields Gulf Drifters Open dataset water molecule dipole moments (850 configurations, quantum chemistry) N-Methylformamide dipole moments (21,000 molecules)
Benchmarks: synthetic SO(2) vector fields F1 and F2 Gulf of Mexico ocean drifter data water molecule dipole moment dataset
Metrics: RMSE LogS computation time bias
Baselines: squared exponential matrix-valued kernel Helmholtz kernel Haar double-integral equivariant kernel K_R invariant/permutation/translation kernels for dipole task
Builds on: Reisert & Burkhardt kernel equivariance Ginsbourger et al. fundamental region invariant kernels Helmholtz kernel (Berlinghieri et al.) Haar integration equivariant kernels
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