High-Order Diffusion Solvers for Linear Inverse Problems

Diffusion models (DMs) are effective generative priors but typically require hundreds to thousands of sampling steps. Recent high-order solvers accelerate unconditional generation by exploiting the semilinear structure of diffusion ordinary differential equations (ODEs) and stochastic differential equations (SDEs), integrating linear terms exactly and reducing sampling to 5 to 20 steps while maintaining generation quality. DMs have also become effective generative priors for inverse problems, where measurement conditioning introduces an intractable measurement matching score. Efficient diffusion inverse solvers typically approximate the measurement matching score in various forms. However, to our knowledge, none of these methods can effectively exploit the semilinear structure of the corresponding differential equations to derive conditional high-order solvers. To address this issue, we identify a class of inverse solvers with structured measurement matching scores that incorporate the model prediction while retaining semilinearity. However, the corresponding integral equations contain dense matrices in the linear terms, making direct computation inefficient. We utilize the singular value decomposition of the forward operator to diagonalize the dense matrices in the spectral domain, enabling efficient computation of the linear terms and high-order approximation of the remaining nonlinear terms. We further introduce guidance strength and stochasticity to control measurement influence and sampling noise, respectively. Experiments on CelebA and ImageNet demonstrate that our method achieves improved reconstruction quality over baselines across linear inverse problems with 5 to 10 sampling steps.