用近似优化思想解决生成模型在逆问题中的难题
Proximal-Based Generative Modeling for Bayesian Inverse Problems
- 基于近端算子构建新采样机制，避免显式计算似然
- 理论证明可非渐近收敛，且无早期停止偏差
- 适合需要高精度重建的科学计算与逆问题场景
基于得分的扩散模型在生成任务中表现优异，但在逆问题中因时变似然得分难以解析求解而面临根本性瓶颈。为此，我们提出一种新型近端生成建模（PGM）框架，严格规避了显式似然评估。该框架建立在扩散过程中的高斯卷积与非光滑优化中Moreau-Yosida正则化之间的理论等价性之上。由此衍生出由所提Moreau得分驱动的新采样机制，可通过近端算子获得闭式表达。此外，我们引入Moreau得分匹配方法，仅依赖先验分布的样本即可学习近端算子。理论上，PGM消除了基于得分扩散模型固有的早期停止偏差，并实现非渐近收敛。实验表明，PGM在重建质量与采样时间上均显著优于现有最先进方法。
原文摘要 · Abstract (English)
Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.
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