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Constrained Diffusion Implicit Models (2411.00359v1)

Published 1 Nov 2024 in cs.LG and eess.IV

Abstract: This paper describes an efficient algorithm for solving noisy linear inverse problems using pretrained diffusion models. Extending the paradigm of denoising diffusion implicit models (DDIM), we propose constrained diffusion implicit models (CDIM) that modify the diffusion updates to enforce a constraint upon the final output. For noiseless inverse problems, CDIM exactly satisfies the constraints; in the noisy case, we generalize CDIM to satisfy an exact constraint on the residual distribution of the noise. Experiments across a variety of tasks and metrics show strong performance of CDIM, with analogous inference acceleration to unconstrained DDIM: 10 to 50 times faster than previous conditional diffusion methods. We demonstrate the versatility of our approach on many problems including super-resolution, denoising, inpainting, deblurring, and 3D point cloud reconstruction.

Summary

  • The paper presents an innovative approach by incorporating linear constraints directly into the diffusion process to enhance recovery quality in noisy linear inverse problems.
  • It leverages pretrained denoising diffusion implicit models with a projected gradient descent method to align with non-Gaussian noise distributions.
  • Empirical results show significant gains, achieving up to 50x faster inference with improved FID and LPIPS scores in tasks such as inpainting and image reconstruction.

Constrained Diffusion Implicit Models: An Exploration of Linear Inverse Problems

The paper "Constrained Diffusion Implicit Models" presents an innovative approach to solving noisy linear inverse problems by leveraging pretrained diffusion models. This paper focuses on enhancing the denoising diffusion implicit models (DDIM) methodology to enforce constraints directly within the diffusion process, achieving what the authors term as Constrained Diffusion Implicit Models (CDIM).

Overview

Linear inverse problems are pervasive across various domains such as image processing, medical imaging, and signal processing. These problems involve the recovery of signals from linear measurements that may include noise, covering tasks like super-resolution, inpainting, deblurring, and 3D point cloud reconstruction. Traditional methods using diffusion models often require specific training per task or introduce excessive computational overhead, failing to fully recover input observations. CDIM circumvents these issues by intelligently modifying diffusion updates to integrate linear constraints directly.

Methodology

The CDIM framework is built upon the foundation of DDIM, which itself is an extension of the denoising diffusion probabilistic models (DDPM). The central innovation of CDIM is its ability to enforce exact constraints on the residual distribution in the presence of noise, thus broadening the applicability of diffusion models to encompass general noise types, beyond the Gaussian assumption. The proposed method achieves this through a projected gradient descent approach during the diffusion process to ensure consistency with observed measurements. Moreover, CDIM addresses noisy inverse problems by minimizing the Kullback-Leibler (KL) divergence between the empirical distribution of residuals and a predefined noise distribution, allowing for precise empirical alignment even under non-Gaussian noise conditions like Poisson noise.

Experimental Results

The paper showcases a suite of experiments across various tasks and metrics, highlighting CDIM's performance. An impressive aspect of the results is the accelerated inference capability, achieving inference times that are an order of magnitude faster—running from 10 to 50 times faster—compared to existing conditional diffusion methodologies, without compromising on quality. Specific examples include applications in image colorization, denoising, and inpainting completed in as little as three seconds.

From a quantitative perspective, CDIM outperforms several inverse solvers on benchmarks like FFHQ and ImageNet datasets, offering improvements in both FID and LPIPS scores. Demonstrating robustness, CDIM effectively handles cases of Gaussian and Poisson noise, with fidelity in applications illustrated on tasks such as 50% noisy inpainting and sparse point cloud reconstruction.

Implications and Future Directions

The implications of CDIM are substantial, as the method represents a leap forward in efficiently solving linear inverse problems with pretrained diffusion models. By addressing limitations of existing approaches and integrating linear constraints within the diffusion process, CDIM opens pathways for further exploration in both theoretical and practical dimensions. The incorporation of non-Gaussian noise models widens the scope of future investigations to potentially include other types of noise structures and constraints, extending CDIM's versatility and adaptability.

Speculatively, the insights from CDIM can foster advancements in adaptive inference systems, where constraints and noise characteristics can dynamically inform model behavior, thus leading to more generalized and robust AI systems. Further research could explore the extension of this framework to nonlinear inverse problems, potentially overcoming current theoretical calculus limitations inherent in Tweedie’s estimates.

In summary, the paper presents a comprehensive paper that enhances the utility and efficiency of diffusion models in inverse problem-solving, ensuring exact constraints handling and broadening observability in real-world noisy environments. The methodological advancements provided by CDIM can serve as a groundwork for subsequent breakthroughs in AI-driven signal recovery.