Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models (2307.00619v1)

Abstract: We present the first framework to solve linear inverse problems leveraging pre-trained latent diffusion models. Previously proposed algorithms (such as DPS and DDRM) only apply to pixel-space diffusion models. We theoretically analyze our algorithm showing provable sample recovery in a linear model setting. The algorithmic insight obtained from our analysis extends to more general settings often considered in practice. Experimentally, we outperform previously proposed posterior sampling algorithms in a wide variety of problems including random inpainting, block inpainting, denoising, deblurring, destriping, and super-resolution.

• The paper introduces PSLD, a novel algorithm using latent diffusion models to achieve provable sample recovery in linear inverse problems.
• It leverages foundation models like Stable Diffusion, replacing computationally expensive pixel-space methods with an efficient latent-space approach.
• Extensive experiments on FFHQ, ImageNet, and web images demonstrate superior performance in tasks such as inpainting, denoising, and super-resolution.

Solving Linear Inverse Problems with Latent Diffusion Models: A New Approach

Introduction

In a significant stride toward enhancing the utility of generative models for solving linear inverse problems, recent research presents a novel framework that utilizes latent diffusion models (LDMs). This new approach, pioneered for the first time, allows for leveraging the robust image priors encapsulated in pre-trained LDMs like Stable Diffusion for tasks including inpainting, denoising, and super-resolution, among others. Previously established methodologies were confined to pixel-space diffusion models, thus limiting their application scope. Through comprehensive theoretical analysis and experimental validation, this work not only amplifies the efficacy of using LDMs but also sets new benchmarks across various problem settings.

Theoretical Analysis of the Framework

The research furnishes a detailed theoretical investigation into the algorithm’s performance within a linear model context, elucidating its capability for provable sample recovery in such settings. This analysis paves the way to extend the understanding and applicability of the proposed algorithm to more complex, real-world scenarios often encountered in practice. The novel algorithm, labeled as Posterior Sampling with Latent Diffusion (PSLD), leverages the extensive data and computational investments inherent in foundation models, like Stable Diffusion, thus circumventing the need for task-specific finetuning.

Algorithm and Implementation Details

Central to the PSLD algorithm is the use of latent spaces for diffusion processes. This distinctive approach not only circumvents the high-dimensional challenges associated with pixel-space diffusion methods but also harnesses the powerful priors of pre-trained foundation models. The methodological ingenuity of PSLD lies in its modified objective function that incorporates 'goodness' and 'gluing' adjustments, guiding the diffusion process towards optimal sample recovery. This innovation marks a departure from previous posterior sampling strategies and demonstrates superior performance through extensive simulations.

Experimental Results and Benchmarking

The experimental endeavors encompass both in-distribution data (FFHQ dataset) and out-of-distribution samples (ImageNet and web images), showcasing the algorithm’s robustness and scalability. PSLD consistently outperforms existing posterior sampling algorithms across a range of inverse problems, including various inpainting tasks, super-resolution, and denoising. Notably, the use of the Stable Diffusion model as a foundational generative model significantly contributed to achieving state-of-the-art results, underlining the potential of leveraging large-scale pre-trained models for solving inverse problems.

Practical Implications and Future Perspectives

The introduction of PSLD ushers in a new era for applying latent diffusion models to a broad spectrum of inverse problems. This approach not only expands the utility of LDMs beyond conventional generative tasks but also significantly reduces the computational overhead of model finetuning for specific tasks. Looking forward, the adaptability of PSLD to incorporate evolving foundation models presents a promising avenue for continual improvement. Furthermore, while the current focus is on linear inverse problems, extending this framework to tackle non-linear inverse challenges represents an intriguing direction for future research.

Conclusion

The presented work marks a milestone in the utilization of latent diffusion models for solving linear inverse problems. Through rigorous theoretical analysis and impressive experimental results, this paper not only extends the frontiers of generative modeling applications but also opens up new pathways for future advancements in the field. The PSLD algorithm stands as a testament to the potential of latent diffusion models in surmounting the challenges associated with linear inverse problems, heralding a promising avenue for research and development in generative AI.