A Variational Perspective on Solving Inverse Problems with Diffusion Models

TL;DR

This paper introduces a variational framework for solving inverse image problems with diffusion models, enabling efficient sampling and improved restoration results without retraining for each task.

Contribution

It proposes a novel variational approach that approximates the true posterior in diffusion models, leading to the RED-Diff method with a SNR-based weighting mechanism.

Findings

Outperforms state-of-the-art diffusion-based methods in image restoration tasks.

Provides a new perspective on inverse problems as stochastic optimization.

Enables lightweight and flexible sampling for various inverse tasks.

Abstract

Diffusion models have emerged as a key pillar of foundation models in visual domains. One of their critical applications is to universally solve different downstream inverse tasks via a single diffusion prior without re-training for each task. Most inverse tasks can be formulated as inferring a posterior distribution over data (e.g., a full image) given a measurement (e.g., a masked image). This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. To cope with this challenge, we propose a variational approach that by design seeks to approximate the true posterior distribution. We show that our approach naturally leads to regularization by denoising diffusion process (RED-Diff) where denoisers at different timesteps concurrently impose different structural constraints over the image. To gauge the contribution of denoisers from different timesteps, we propose a weighting mechanism based on signal-to-noise-ratio (SNR). Our approach provides a new variational perspective for solving inverse problems with diffusion models, allowing us to formulate sampling as stochastic optimization, where one can simply apply off-the-shelf solvers with lightweight iterates. Our experiments for image restoration tasks such as inpainting and superresolution demonstrate the strengths of our method compared with state-of-the-art sampling-based diffusion models.