A Hyperfinite Framework for Score-Based Generative Modeling

• URL: http://arxiv.org/abs/2608.02799v1
• Date: Mon, 03 Aug 2026 18:51:44 GMT
• Title: A Hyperfinite Framework for Score-Based Generative Modeling
• Abstract summary: We develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis.<n>We show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics.<n>Under suitable assumptions, we derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives.
• Score: 0.8122270502556375
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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