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 combine 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.

翻译：Krylov子空间通过将给定向量与线性变换矩阵及其连续幂相乘生成，在经典优化文献中已被广泛研究，用于设计快速收敛的大规模线性逆问题算法。例如，最流行的Krylov子空间方法之一共轭梯度法（CG）基于在Krylov子空间中最小化残差误差的思想。然而，随着高性能扩散求解器在逆问题中的最新发展，经典智慧如何与现代扩散模型协同结合尚不明确。本研究提出一种新颖且高效的扩散采样策略，将扩散采样与Krylov子空间方法协同结合。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Krylov子空间，则以去噪数据初始化的CG可确保数据一致性更新保持在切空间内。这避免了计算流形约束梯度（MCG）的需求，从而得到更高效的扩散采样方法。该方法适用于任意参数化设置（即VE、VP）。值得注意的是，在具有挑战性的真实世界医学逆成像问题（包括多线圈MRI重建和3D CT重建）中，我们实现了最先进的重建质量。此外，所提方法的推理速度比先前最先进方法提升80倍以上。