通过 DDIM 实现线性逆问题的可证明基于扩散的后验采样
Provable diffusion-based posterior sampling for linear inverse problems via DDIM
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中文总结 AI 辅助
该研究针对线性逆问题，提出\pddim算法，通过对标准DDIM更新进行轻量级逐坐标修改并纳入测量模型，沿测量算子奇异方向分别采样。此算法收敛到贝叶斯后验，在图像恢复任务中性能优于现有方法，实现高效且具后验一致性。
中文摘要 AI 辅助
基于扩散的方法在解决逆问题方面取得了显著的经验成功。然而，许多现有的后验采样器要么缺乏严格的理论保证，要么产生大量的计算开销。我们提出了一种简单有效的算法，称为 \pddim，用于通过 DDIM 型采样器解决具有扩散先验的线性逆问题。我们的方法只需要对标准 DDIM 更新进行轻量级的、逐坐标的修改，同时明确纳入测量模型。关键思想是沿着测量算子的每个奇异方向分别进行后验采样：对于每个方向，当观测信噪比低于相应的扩散信噪比时，采样器遵循学习到的扩散先验，否则切换到基于校准测量的预测器。我们证明了所提出的采样器收敛到基于测量的贝叶斯后验。实验结果表明，所提出的采样器在一系列图像恢复任务中优于现有的基于扩散的后验采样器，在大多数考虑的评估指标上取得了最佳性能。总的来说，我们的结果将有噪声线性逆问题的后验采样转化为简单的逐坐标 DDIM 更新，产生了一种高效、易于实现且具有可证明后验一致性的算法。
英文摘要
Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.
