基于扩散模型的后验采样方法，理论保证收敛性且适用于复杂反问题。
Provable Diffusion Posterior Sampling for Bayesian Inversion
- 通过扩散过程实现从简单分布到复杂后验的概率传输。
- 在Wasserstein-2距离下建立非渐近误差界，确保收敛性。
- 适合需要可靠后验采样的贝叶斯反演任务，如科学计算与图像重建。
我们提出一种新型的基于扩散的后验采样方法，采用即插即用框架。该方法通过扩散过程构建从易采样分布到目标后验的概率传输。为高效初始化采样器，引入粒子的预热策略。后验得分函数通过蒙特卡洛估计近似，样本由Langevin动力学生成，避免了以往工作中常见的启发式近似。驱动Langevin动力学的得分函数从数据中学习，使模型能够捕捉底层先验的丰富结构特征。此外，我们建立了在Wasserstein-2距离下的非渐近误差界，证明该方法即使在复杂、多模态后验分布下仍能收敛。数值实验验证了该方法在多种反问题中的有效性。
原文摘要 · Abstract (English)
We propose a novel diffusion-based posterior sampling method within a plug-and-play framework. Our approach constructs a probability transport from an easy-to-sample distribution to the target posterior via a diffusion process. To initialize the sampler efficiently, we introduce a warm-start strategy for the particles. The posterior score is then approximated using a Monte Carlo estimator in which samples are generated via Langevin dynamics, avoiding the heuristic approximations prevalent in prior work. The score function driving the Langevin dynamics is learned from data, enabling the model to capture rich structural features of the underlying prior. We also establish non-asymptotic error bounds in Wasserstein-2 distance guaranteeing convergence of the proposed method even for complex, multimodal posterior distributions. We corroborate our theoretical findings with numerical experiments demonstrating the effectiveness of the method across a variety of inverse problems.
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