Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization 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. Code is available at https://github.com/HJ-harry/DDS

翻译：Krylov子空间，即通过将给定向量与线性变换矩阵及其连续幂相乘生成的子空间，已在经典优化文献中被广泛研究，用于设计快速收敛于大规模线性逆问题的算法。例如，共轭梯度法（CG），作为最流行的Krylov子空间方法之一，其基本思想是在Krylov子空间中最小化残差误差。然而，随着高性能扩散求解器在逆问题中的最新进展，经典智慧如何与当代扩散模型协同结合尚不明确。在本研究中，我们提出一种新颖且高效的扩散采样策略，该策略协同结合了扩散采样与Krylov子空间方法。具体而言，我们证明：若由Tweedie公式得到的去噪样本的切空间构成一个Krylov子空间，则以来去噪数据初始化的CG可确保数据一致性更新保持在切空间内。这消除了计算流形约束梯度（MCG）的必要性，从而实现了更高效的扩散采样方法。我们的方法适用于任意参数化和设置（即VE、VP）。值得注意的是，在挑战性的真实世界医学逆成像问题中，包括多线圈MRI重建和3D CT重建，我们实现了最先进的重建质量。此外，我们提出的方法推理时间比先前最优方法快80倍以上。代码可在https://github.com/HJ-harry/DDS获取。