Electrostatic Models for Score Matching

Laura Appleby, Diana Cai, Lawrence K. Saul, Daniel D. Lee

Keywords: score matching, score-based generative modeling, electrostatics, energy-based models, curl-free vector fields, Fisher divergence, structured score estimation

TL;DR: We formulate score matching as electrostatics, modeling scores as electric fields to yield curl-free fields whose divergences are charge densities, giving interpretable, competitive estimators with fewer parameters and fast closed-form training.

Abstract:

We introduce an electrostatic formulation of score-based generative modeling in which the score function is modeled as the electric field of a static charge distribution. As the gradient of the log-density with respect to the data, the score function satisfies a number of vector calculus constraints. The advantage of electric charge models is that Maxwell’s equations guarantee curl-free, conservative electric fields by construction, and the divergence of the field is efficiently computed as the local charge density of the model. We parameterize the score as a superposition of electric fields from uniformly charged slabs, producing piecewise linear fields and piecewise quadratic potentials with a natural connection to energy-based models and product-of-experts models. Under the Fisher divergence,
training reduces to a quadratic optimization problem with closed-form regularized solutions, avoiding iterative gradient descent over the score model. Experiments on synthetic data and MNIST latent score estimation show competitive performance with neural and kernel-based baselines while using fewer parameters, enabling fast closed-form training, and offering clear interpretability.

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Submission Number: 167