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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
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Proposes Electrostatic Field Matching (EFM), which places source and target distributions on the plates of a capacitor, learns the field with a neural network, and transports samples along field lines for generation and data-to-data translation.
Problem
Existing electrostatic generative models (PFGM) only handle noise-to-data generation, and no electrostatics-based method handles data-to-data distribution transfer.
Approach
Assign positive charge to the source distribution on one plane and negative charge to the target on a parallel plane, learn the normalized capacitor field with a neural network regressor on interpolated points between plates, and transport samples along field lines; a Gauss-law argument shows the transport is distribution-exact.
Key results
- On a 2D Gaussian to Swiss Roll transfer, EFM recovers the target well at L=6 but poorly at L=30 (qualitative).
- On unpaired colored MNIST 3-to-2 translation, EFM gives visually faithful translations, whereas Flow Matching sometimes fails to preserve digit shape and color (qualitative).
- On CIFAR-10 noise-to-image generation, the paper reports qualitative samples and notes generation quality drops at L=5000 versus L=50 and L=500 (no quantitative metric reported in the text).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the interplate distance from L=6 to L=30 | EFM · Gaussian to Swiss Roll | target distribution recovery quality | Hurt |
| Increase L to 5 000 rather than use L=50 or L=500 | EFM · CIFAR-10 generation | generation quality | Hurt |
| Use L=50 versus L=500 | EFM · CIFAR-10 generation | generation quality | No clear effect |
| Use only forward-oriented field lines | EFM · Colored MNIST generation | target distribution recovery quality | Helped |
| Learn the normalized field instead of the unnormalized field | EFM · Generative modeling and distribution transfer experiments | field approximation performance | Helped |
Surprising findings
- Using large plate distance L degrades the learned transport, so the method is sensitive to this hyperparameter.
Limitations
- Training may be unstable in high dimensions because the field's 1/r^D multiplier becomes tiny.
- Practical implementation uses only forward-oriented lines and stops at the target plate, which can miss parts of the target support.
- Choice of plate distance L lacks a theoretical optimum and strongly affects performance.
Experimental setup
Datasets: 2D Gaussian Swiss Roll colored MNIST CIFAR-10
Benchmarks: Swiss Roll colored MNIST CIFAR-10
Baselines: Flow Matching PFGM
Builds on: Poisson Flow Generative Models Flow Matching Diffusion Bridge Matching
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