A Hyperfinite Framework for Score-Based Generative Modeling

arXiv:2608.02799 · stat.ML, cs.AI, cs.LG, math.PR · Submitted 2026-08-03 · Read on arXiv

Sunder Ram Krishnan

stat.ML, cs.AI, cs.LG, math.PR

Submitted: 2026-08-03

License: http://creativecommons.org/licenses/by/4.0/

The gist: Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus.

Terminology

Abstract

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker--Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value kappa=3 eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling--while laying foundations for further extensions--that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

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