On the Mathematics of Diffusion Models

David McAllester

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arXiv · 2301.11108

On the Mathematics of Diffusion Models

David McAllester

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Abstract

This paper gives direct derivations of the differential equations and likelihood formulas of diffusion models assuming only knowledge of Gaussian distributions. A VAE analysis derives both forward and backward stochastic differential equations (SDEs) as well as non-variational integral expressions for likelihood formulas. A score-matching analysis derives the reverse diffusion ordinary differential equation (ODE) and a family of reverse-diffusion SDEs parameterized by noise level. The paper presents the mathematics directly with attributions saved for a final section.

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David McAllester. 2023-01-25. On the Mathematics of Diffusion Models. https://arxiv.org/abs/2301.11108

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