A Survey on Diffusion Models for Inverse Problems (2410.00083v1)

Abstract: Diffusion models have become increasingly popular for generative modeling due to their ability to generate high-quality samples. This has unlocked exciting new possibilities for solving inverse problems, especially in image restoration and reconstruction, by treating diffusion models as unsupervised priors. This survey provides a comprehensive overview of methods that utilize pre-trained diffusion models to solve inverse problems without requiring further training. We introduce taxonomies to categorize these methods based on both the problems they address and the techniques they employ. We analyze the connections between different approaches, offering insights into their practical implementation and highlighting important considerations. We further discuss specific challenges and potential solutions associated with using latent diffusion models for inverse problems. This work aims to be a valuable resource for those interested in learning about the intersection of diffusion models and inverse problems.

• The paper introduces a taxonomy categorizing diffusion-based methods into four groups: explicit approximations, variational inference, CSGM-type, and asymptotically exact techniques.
• The paper demonstrates that diffusion models can serve as unsupervised priors for inverse problems like image restoration, compressed sensing, and MRI reconstruction without further training.
• The analysis highlights practical benefits such as scalability, flexibility, and efficiency, while advancing theoretical insights in Bayesian inference and numerical approximation.

Overview of Diffusion Models for Solving Inverse Problems

The survey paper, "A Survey on Diffusion Models for Inverse Problems," offers a comprehensive exploration of how diffusion models, particularly pre-trained ones, can be employed to address a variety of inverse problems without necessitating further training. This essay will elucidate the taxonomy introduced by the authors, and the implications of the proposed methodologies in the field of inverse problems.

Introduction to Diffusion Models and Inverse Problems

Diffusion models have seen notable success in generative modeling due to their proficiency in producing high-quality samples. This survey leverages this capability to solve inverse problems by treating diffusion models as unsupervised priors. Inverse problems, such as image restoration, compressed sensing, and MRI reconstruction, fundamentally seek to recover an unknown signal from its corrupted measurements.

Taxonomy of Methods

The authors present a fourfold categorization of methods based on their approach to solving inverse problems using diffusion models:

1. Explicit Approximations for Measurement Matching: These methods approximate the measurement matching score with a closed-form expression. Representative methods under this category include Score-ALD, Score-SDE, ILVR, DPS, and variants like $\Pi$GDM and Moment Matching.
2. Variational Inference: This category includes methods that approximate the true posterior distribution with a simpler, tractable variational distribution. The parameters of this distribution are optimized using methods such as RED-Diff, Blind RED-Diff, Score Prior, and Efficient Score Prior.
3. CSGM-Type Methods: These methods optimize the initialization of the deterministic sampler in the diffusion model’s latent space. Notable methods include DMPlug, SHRED, and expansions using consistency models.
4. Asymptotically Exact Methods: These aim to sample from the true posterior distribution through techniques like Monte Carlo (MCMC) or Sequential Monte Carlo (SMC). Examples include PnP-DM, FPS, PMC, SMCDiff, MCGDiff, and TDS.

Strong Numerical Results and Claims

The paper extensively compares various methods, highlighting differences in their measurement matching and optimization techniques. For example, whereas Score-SDE uses implicit regularization by noising measurements, $\Pi$GDM employs a Gaussian approximation to account for covariance structures in the data. The use of Monte Carlo and variational inference techniques also marks significant advancements in achieving more accurate posterior distributions in practical settings.

Practical and Theoretical Implications

Practical Implications:

• Scalability: Latent diffusion models have shown significant scalability for high-dimensional data, making approaches like Latent DPS and PSLD valuable for large-scale inverse problems.
• Flexibility: Methods like BlindDPS cater to scenarios where the forward operator is not completely known, making them highly applicable in real-world scenarios such as sensor malfunctions or uncalibrated settings.
• Efficiency: Variational methods such as Efficient Score Prior reduce the computational load by leveraging lower-bound approximations and one-step consistency models.

Theoretical Implications:

• Bayesian Inference: The survey underscores the Bayesian approach to inverse problems, enabling robust estimation and uncertainty quantification.
• Approximation Quality: The discussion on explicit approximations showcases that increasingly complex “lifting” matrices can better handle the projection of approximations back into higher-dimensional spaces more accurately.
• Generative Samplers: The CSGM-type methods signify the potential for combining the deterministic nature of ODEs in diffusion models with optimization techniques traditionally used in GANs and VAEs.

Future Developments

In the future, research might focus on creating standardized benchmarks for these methods to critically assess performance across different domains. Moreover, computational considerations will continue to dominate, driving innovations towards reduced memory footprints and faster inference times.

Conclusion

This survey establishes a critical reference point in the intersection of diffusion models and inverse problems. It brings clarity to the dense landscape of different approaches and highlights the nuanced trade-offs and strengths inherent in each category. The implications are far-reaching, suggesting both immediate practical applications and long-term theoretical advancements in solving complex inverse problems.

The intricacies of the surveyed methods and their interconnections lay a solid groundwork for future research, inviting the community to further build upon the promising framework it delineates.