Convergence Analysis of Probability Flow ODE for Score-based Generative Models

Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based...

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Bibliographic Details
Published inarXiv.org
Main Authors Daniel Zhengyu Huang, Huang, Jiaoyang, Lin, Zhengjiang
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 21.07.2024
Subjects
Convergence
Errors
Matching
Mathematical models
Runge-Kutta method
Samplers
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Summary:Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to \(L^2\)-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by \(\mathcal{O}(d^{3/4}\delta^{1/2})\) in the continuous time level, where \(d\) denotes the data dimension and \(\delta\) represents the \(L^2\)-score matching error. For practical implementations using a \(p\)-th order Runge-Kutta integrator with step size \(h\), we establish error bounds of \(\mathcal{O}(d^{3/4}\delta^{1/2} + d\cdot(dh)^p)\) at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.
ISSN:2331-8422