Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion

Sampling from the posterior distribution in latent diffusion models for inverse problems is computationally challenging. Existing methods often rely on Tweedie's first-order moments that tend to induce biased results. Second-order approximations are computationally prohibitive making standard reverse diffusion processes intractable for posterior sampling. This paper presents Second-order Tweedie sampler from Surrogate Loss (STSL) a novel sampler offering efficiency comparable to first-order Tweedie while enabling tractable reverse processes using second-order approximation. Theoretical results reveal that our approach utilizing for the trace of the Hessian with only O(1) compute establishes a lower bound through a surrogate loss and enables a tractable reverse process. We show STSL outperforms SoTA solvers PSLD and P2L by reducing neural function evaluations by 4X and 8X respectively while enhancing sampling quality on FFHQ ImageNet and COCO benchmarks. Moreover STSL extends to text guided image editing and mitigates residual distortions in corrupted images. To our best knowledge this is the first work to offer an efficient second order approximation for solving inverse problems using latent diffusion and editing real world images with corruptions.