In this presentation, I will discuss the essence of score-based generative models (SGMs) as entropically regularized Wasserstein proximal operators (WPO) for cross-entropy, elucidating this connection through mean-field games (MFG). The unique structure of SGM-MFG allows the HJB equation alone to characterize SGMs, demonstrated to be equivalent to an uncontrolled Fokker-Planck equation via a Cole-Hopf transform. Furthermore, leveraging the mathematical framework, we introduce an interpretable kernel-based model for the score functions, enhancing the performance of SGMs in terms of training samples and training time. The mathematical formulation of the new kernel-based models, in conjunction with the utilization of the terminal condition of the MFG, unveils novel insights into the manifold learning and generalization properties of SGMs.
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