Wasserstein proximal operators describe score-based generative models and resolve memorization
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913228882182144 |
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| author | Zhang, Benjamin J. Liu, Siting Li, Wuchen Katsoulakis, Markos A. Osher, Stanley J. |
| author_facet | Zhang, Benjamin J. Liu, Siting Li, Wuchen Katsoulakis, Markos A. Osher, Stanley J. |
| contents | We focus on the fundamental mathematical structure of score-based generative models (SGMs). We first formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled partial differential equations: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Second, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. In addition, the WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Finally, our mathematically informed, interpretable kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06162 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein proximal operators describe score-based generative models and resolve memorization Zhang, Benjamin J. Liu, Siting Li, Wuchen Katsoulakis, Markos A. Osher, Stanley J. Machine Learning We focus on the fundamental mathematical structure of score-based generative models (SGMs). We first formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled partial differential equations: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Second, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. In addition, the WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Finally, our mathematically informed, interpretable kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications. |
| title | Wasserstein proximal operators describe score-based generative models and resolve memorization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2402.06162 |