Inverse Problems with Diffusion Models: A MAP Estimation Perspective
Sai Bharath Chandra Gutha, Ricardo Vinuesa, Hossein Azizpour
TL;DR
The paper addresses the challenge of solving inverse problems with pre-trained unconditional diffusion priors by formulating a MAP objective over the data prior and measurement model. It introduces a MAP-GA framework that reparameterizes the reverse generation via a consistency-model-based trajectory, enabling tractable gradient-based optimization through a vector-Jacobian product. The approach yields practical algorithms for image restoration (inpainting, deblurring, super-resolution) that outperform several baselines on standard datasets, while providing a principled theoretical link between PF ODE, consistency models, and MAP optimization. This work broadens the applicability of unconditional diffusion priors to conditional generation tasks, offering a flexible, theoretically grounded route for high-quality inverse problem solving in vision.
Abstract
Inverse problems have many applications in science and engineering. In Computer vision, several image restoration tasks such as inpainting, deblurring, and super-resolution can be formally modeled as inverse problems. Recently, methods have been developed for solving inverse problems that only leverage a pre-trained unconditional diffusion model and do not require additional task-specific training. In such methods, however, the inherent intractability of determining the conditional score function during the reverse diffusion process poses a real challenge, leaving the methods to settle with an approximation instead, which affects their performance in practice. Here, we propose a MAP estimation framework to model the reverse conditional generation process of a continuous time diffusion model as an optimization process of the underlying MAP objective, whose gradient term is tractable. In theory, the proposed framework can be applied to solve general inverse problems using gradient-based optimization methods. However, given the highly non-convex nature of the loss objective, finding a perfect gradient-based optimization algorithm can be quite challenging, nevertheless, our framework offers several potential research directions. We use our proposed formulation to develop empirically effective algorithms for image restoration. We validate our proposed algorithms with extensive experiments over multiple datasets across several restoration tasks.