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.

翻译：扩散模型在解决逆问题方面展现出卓越性能，但主要局限之一是推理速度较慢。尽管针对无条件采样已开发出更快的扩散采样器，但在逆问题条件下的条件采样研究仍十分有限。本研究提出一种新颖高效的扩散采样策略，该策略采用扩散采样的几何分解方法。具体而言，我们发现扩散模型生成的样本可分解为两个正交分量：一个是通过将样本投影到干净数据流形上得到的"去噪"分量，另一个是通过添加随机噪声诱导向下一级低噪声流形转移的"噪声"分量。进一步证明，在干净数据流形的特定条件下，用于施加从去噪信号导出的条件约束的共轭梯度更新属于干净流形，从而实现了更快速、更精确的扩散采样。本方法适用于任意参数化设置（如方差扩展（VE）和方差保持（VP））。值得注意的是，我们在具有挑战性的真实医学逆成像问题（包括多线圈核磁共振成像重建和三维计算机断层扫描重建）中取得了最先进的重建质量。此外，所提方法的推理时间相比先前最先进方法提升超过80倍。