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A Variational Perspective on Solving Inverse Problems with Diffusion Models (2305.04391v2)

Published 7 May 2023 in cs.LG, cs.CV, cs.NA, math.NA, and stat.ML

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.

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Citations (91)
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Summary

  • The paper presents a novel variational inference approach that regularizes inverse problem solutions using diffusion models.
  • It demonstrates significant improvements in image fidelity, perceptual quality, and GPU efficiency over conventional methods.
  • A signal-to-noise ratio weighting mechanism underpins efficient stochastic optimization, adapting the method to diverse restoration tasks.

A Variational Approach to Solving Inverse Problems with Diffusion Models

Introduction

The emergence of diffusion models, such as Stable Diffusion, has marked a significant advancement in the field of visual foundation models. Notably, these models serve as a robust prior for sampling in various downstream inverse problems, including image restoration and rendering. However, their application has been limited by the need for universal and adaptive samplers that do not require re-training for each task, alongside the requirement for efficiency and ease of tuning. This paper introduces a variational approach to address these challenges, offering an insightful method for regularization by denoising diffusion process (RED-diff).

Background

Diffusion models have been increasingly applied to inverse problems across diverse domains. Prior approaches have made attempts to develop universal samplers but often struggled with the intractable and multimodal nature of the posterior distribution arising from the nonlinear and recursive backward diffusion process. In response to these challenges, this research leverages variational inference to approximate true posterior distributions effectively. By adopting a principled variational perspective, the introduced method, RED-diff, not only enhances image fidelity and perceptual quality but also exhibits superior GPU efficiency.

Methodology

The core of the proposed method is to approximate the posterior distribution of data given observations through variational inference, utilizing the denoising diffusion model as the data prior and representing the measurement model as a likelihood. The approach leads to regularization by the denoising diffusion process, where denoisers at different timesteps impose structural constraints on the image. A key innovation is the introduction of a weighting mechanism based on signal-to-noise ratio (SNR) to evaluate the contribution of denoisers from different timesteps. This approach effectively formulates sampling as stochastic optimization, facilitated by the application of off-the-shelf solvers with lightweight iterates.

Experiments and Results

The research conducted extensive experiments for various linear and nonlinear image restoration tasks. The variational approach demonstrated superior quality in image fidelity and perceptual quality when compared with state-of-the-art samplers. Furthermore, the method's efficiency was highlighted through its lightweight iterates and GPU-friendly nature. Ablation studies further supported these findings, suggesting that the optimizer parameters, such as learning rate and the number of steps, are effective in tweaking the trade-off between fidelity and perceptual quality.

Implications and Future Directions

This paper's variational perspective on solving inverse problems with diffusion models opens new avenues for research and application in AI. The proposed method, RED-diff, provides a theoretically grounded and computationally efficient approach for leveraging diffusion models in solving a wide range of inverse problems. The success of the weighting mechanism based on denoising SNR presents an exciting area for future exploration, potentially leading to enhancements in the method's adaptability and performance across diverse tasks. As the field continues to evolve, further investigations into optimizing the variational distribution and exploring methods that encourage solution diversity may lead to even more versatile and effective solutions.

Conclusion

The introduction of a variational approach to leverage diffusion models for solving inverse problems represents a significant advancement in the field. By enabling regularization through the denoising diffusion process and formulating sampling as stochastic optimization, the proposed method offers both theoretical insights and practical benefits. The demonstrated superiority in image fidelity and computational efficiency highlights the potential of this approach in advancing the capabilities of AI models in visual domains and beyond.