Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems (2303.05754v3)

Abstract: Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS

• The paper introduces a novel method integrating diffusion models with Krylov subspace techniques to significantly accelerate inverse problem solving.
• It achieves over 80x reduction in inference time and enhanced image quality in complex tasks such as multi-coil MRI and 3D CT reconstructions.
• Leveraging Tweedie’s formula, the methodology maintains data consistency within the tangent space while eliminating computationally intensive steps.

An Overview of Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems

The paper under consideration, "Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems," introduces a novel method to enhance the efficiency and effectiveness of solving inverse problems in large-scale scenarios, particularly in the medical imaging domain. The authors combine classical Krylov subspace methods with modern diffusion models to create a diffusion sampling strategy that addresses the slow inference times typically associated with diffusion models.

The core innovation lies in integrating diffusion models with the Krylov subspace approach, particularly the conjugate gradient (CG) method. The primary insight is that if the tangent space, determined by a denoised sample from a diffusion model, forms a Krylov subspace, then initiating the CG method from this point maintains data consistency within the tangent space. This approach obviates the need for computing the computationally intensive manifold-constrained gradient (MCG), which is typically used in diffusion model-based inverse problem solvers (DIS).

Numerical Performance and Generalization

The authors demonstrate state-of-the-art performance in complex medical imaging tasks, such as multi-coil MRI and 3D CT reconstructions. Notably, the proposed method not only improves the reconstruction quality but also drastically reduces inference times by over 80 times compared to previous state-of-the-art methods. This performance gain is achieved with as few as 20 to 50 neural function evaluations (NFEs), a significant reduction from the traditional requirements of diffusion models.

Methodological Insights

The methodology involves leveraging Tweedie's formula to generate a denoised sample, ensuring that the CG updates remain within the defined tangent space of this denoised signal. This leads to an efficient diffusion sampling process, applicable to both variance-preserving and variance-exploding diffusion settings. The paper emphasizes that the new sampling strategy, termed Decomposed Diffusion Sampling (DDS), achieves superior quality with reduced computation time, demonstrating adaptability across various medical imaging tasks.

Practical and Theoretical Implications

Practically, the authors highlight the approach's potential to handle varying data acquisition schemes inherent in medical imaging modalities without the model becoming overly specialized to particular data types. Theoretically, the paper bridges a gap between classical optimization methods and modern AI-driven diffusion models, suggesting that hybrid approaches can bring significant computational advantages.

Future Directions

This research opens several avenues for further exploration. First, it suggests potential improvements in fields requiring efficient large-scale inverse problem-solving. Additionally, the integration of classical numerical methods with AI-driven approaches could be expanded beyond medical imaging to other domains dealing with large datasets and complex inverse problems. Finally, the paper invites future work on optimization schemes that can further exploit the geometric properties of denoised samples in diffusion processes.

In conclusion, this paper presents a substantial advancement in accelerating the resolution of large-scale inverse problems, showcasing a compelling example of how classical and modern techniques can be synergistically employed to achieve practical efficiency and effectiveness in AI applications.