Improving Diffusion Models for Inverse Problems using Manifold Constraints (2206.00941v3)

Abstract: Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step followed by a projection-based measurement consistency step, often produce suboptimal results. By studying the generative sampling path, here we show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint, which can be used synergistically with the previous solvers to make the iterations close to the manifold. The proposed manifold constraint is straightforward to implement within a few lines of code, yet boosts the performance by a surprisingly large margin. With extensive experiments, we show that our method is superior to the previous methods both theoretically and empirically, producing promising results in many applications such as image inpainting, colorization, and sparse-view computed tomography. Code available https://github.com/HJ-harry/MCG_diffusion

• The paper introduces the Manifold Constrained Gradient (MCG) as a corrective term to keep diffusion sampling on the data manifold.
• It demonstrates superior performance in image inpainting, colorization, and CT reconstruction with significantly improved metrics.
• The study provides theoretical insights into diffusion model geometry, emphasizing the importance of manifold constraints for generative fidelity.

Improving Diffusion Models for Inverse Problems using Manifold Constraints

The paper "Improving Diffusion Models for Inverse Problems using Manifold Constraints" introduces a novel approach to enhance the performance of diffusion models in solving inverse problems. The central innovation is the incorporation of an additional corrective term, termed as the Manifold Constrained Gradient (MCG), which aims to keep the iterative sampling process closer to the data manifold, thereby reducing error accumulation.

Overview and Contributions

Diffusion models have demonstrated significant potential as both generative models and as solvers for inverse problems across various domains such as image inpainting, colorization, and sparse-view computed tomography (CT). The primary challenge addressed in this paper is the tendency of current diffusion-based inverse problem solvers to deviate from the data manifold during the sampling process, resulting in sub-optimal outputs. This deviation primarily occurs due to the recursive application of reverse diffusion steps followed by projection-based measurement consistency steps.

The authors propose a manifold constraint-inspired correction term that synergistically enhances existing solvers' capabilities. This enhancement is theoretically grounded in the understanding that diffusion processes, when correctly constrained, can maintain the generative path within the desired manifold bounds, thus minimizing divergence-induced errors. The MCG is particularly notable for its ease of implementation and its efficacy in significantly boosting performance.

Experimental Results and Analysis

The paper presents extensive empirical evidence supporting the proposed method's effectiveness. Experiments conducted on tasks such as image inpainting, colorization, and CT reconstruction demonstrate that the MCG significantly outperforms traditional methods, including both state-of-the-art diffusion models and supervised learning-based algorithms.

Key numerical results indicate substantial improvements in metrics like FID and LPIPS for image inpainting, with the proposed MCG method consistently showing superior performance across all tested datasets, including FFHQ and ImageNet. In colorization and CT reconstruction, the method exhibits marked enhancements over previous works, validating its theoretical underpinnings and practical applicability.

Furthermore, the paper provides an analytical perspective on diffusion model geometry, delineating how the proposed MCG correction adeptly manages the tangent components to the data manifold, which are often neglected in standard methods. This theoretical insight is pivotal in explaining the improvement observed across various inverse problems.

Implications and Future Work

The implications of this work are twofold. Practically, the proposed framework offers a robust, unsupervised solution that maintains strong performance across a variety of inverse problems, thereby simplifying the implementation requirements for such tasks in real-world applications. Theoretically, it provides new insights into the geometric nature of diffusion models and the importance of manifold constraints in preserving generative fidelity.

Future work could explore further generalization of the manifold constraints to more complex or high-dimensional data manifolds. Additionally, the integration of this framework with accelerated sampling techniques or other improvements in diffusion models could further enhance efficiency and applicability. Lastly, extending these ideas to other forms of generative models could open new avenues for innovation in machine learning and artificial intelligence.

In conclusion, this paper presents a significant step forward in improving diffusion model performance for inverse problems by elegantly tying together theoretical insights and practical improvements into a cohesive framework.