Diffusion models have shown exceptional performance in solving inverse problems. However, one major limitation is the slow inference time. While faster diffusion samplers have been developed for unconditional sampling, there has been limited research on conditional sampling in the context of inverse problems. In this study, we propose a novel and efficient diffusion sampling strategy that employs the geometric decomposition of diffusion sampling. Specifically, we discover that the samples generated from diffusion models can be decomposed into two orthogonal components: a ``denoised" component obtained by projecting the sample onto the clean data manifold, and a ``noise" component that induces a transition to the next lower-level noisy manifold with the addition of stochastic noise. Furthermore, we prove that, under some conditions on the clean data manifold, the conjugate gradient update for imposing conditioning from the denoised signal belongs to the clean manifold, resulting in a much faster and more accurate diffusion sampling. Our method is applicable regardless of the parameterization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.

提出了一种新颖有效的扩散采样策略，将样本分解为在清洁数据流形上投影得到的“去噪”部分和诱导样本向下一级噪声流形过渡的“噪声”部分，可以更快和更精确地进行采样。在挑战性的实际医学逆成像问题中实现了最先进的重构质量，并且比以前的最先进方法快80倍以上。

通过几何分解的快速扩散采样器用于逆问题