A Hyperfinite Framework for Score-Based Generative Modeling

2026-08-03

AI Summary

This work addresses the lack of a unified framework bridging discrete and continuous formulations in conventional score-based diffusion models, which typically rely on continuous-time stochastic differential equations. By introducing nonstandard analysis into generative modeling, the authors construct an intrinsic diffusion process on a hyperfinite grid and derive its infinitesimal generator, establishing a correspondence with the Fokker–Planck equation. Leveraging a hyperfinite backward mean identity and Girsanov’s theorem, they rigorously prove that score matching exactly recovers the score function required for the reverse dynamics. Furthermore, they show that when the fourth moment of the increment distribution equals three—characteristic of Gaussianity—the leading dispersive term vanishes, yielding second-order consistency. This framework unifies discrete-grid dynamics, reverse diffusion, score matching, and likelihood optimization within a single theoretical foundation.

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 $κ=3$ 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.

Problem

• score-based generative modeling
• diffusion models
• Nonstandard Analysis
• hyperfinite framework
• Fokker-Planck equation

Innovation

• hyperfinite
• score-based generative modeling
• nonstandard analysis
• reverse-time SDE
• Fokker–Planck equation