Solving Inverse Problems via Diffusion Optimal Control

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Henry Li et el.

TL;DR
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Current diffusion-based inverse problem solvers frame signal recovery as probabilistic sampling, encountering issues such as intractable likelihood functions, strict score network reliance, and poor initial guess prediction. These methods often suffer from sensitivity to discretization and approximation errors, hindering their accuracy and robustness.

This research proposes an innovative solution by transforming the generative process into a discrete optimal control problem. A diffusion-based optimal controller, inspired by iterative Linear Quadratic Regulator (iLQR), is developed, capable of handling diverse forward operators (super-resolution, inpainting, etc.). The resulting algorithm is shown to overcome prior limitations by accurately recovering the idealized posterior sampling equation. The method demonstrates significant improvement in solving various inverse problems, achieving state-of-the-art image reconstruction results.

Key Takeaways
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Why does it matter?
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This paper is crucial for researchers working on inverse problems, diffusion models, and optimal control. It offers a novel framework that significantly improves the performance and robustness of existing methods. This opens new avenues for research into more efficient and accurate solutions for various inverse problems across many domains. The combination of these techniques provides a state-of-the-art benchmark for image reconstruction.

Visual Insights
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This figure compares two different approaches to solving inverse problems using diffusion models. The left side shows a probabilistic posterior sampler which estimates the initial state (x0) and uses this approximation to guide the sampling process. The right side illustrates the proposed optimal control-based sampler, where the initial state is calculated exactly at each step, providing higher quality gradients and a more accurate trajectory update. This results in better accuracy and stability across different number of steps.

This table presents a quantitative comparison of different methods for solving various inverse problems on the FFHQ 256x256-1K dataset. The performance is measured using two metrics: Fréchet Inception Distance (FID) and Learned Perceptual Image Patch Similarity (LPIPS). Lower values indicate better performance for both metrics. The table includes results for various methods, including different variants of the proposed Diffusion Optimal Control method, as well as several baselines like Diffusion Posterior Sampling (DPS) and Plug-and-Play ADMM (PNP-ADMM). The results are broken down by inverse problem type (super-resolution, inpainting, deblurring).

In-depth insights
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Diffusion Optimal Ctrl
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The heading ‘Diffusion Optimal Control’ suggests a novel approach to solving inverse problems. It cleverly combines the strengths of diffusion models, known for their generative capabilities, with optimal control theory, a powerful framework for guiding dynamic systems. This approach likely addresses limitations of existing diffusion-based inverse problem solvers by framing the generative process as a discrete optimal control problem. This reframing allows for more precise control over the sampling process, potentially leading to improved reconstruction quality and robustness to noise. The use of optimal control theory likely enables the method to handle a broader range of inverse problems by directly optimizing the system’s trajectory toward the desired solution, rather than relying solely on score function approximations. This direct optimization could mitigate issues like poor initial prediction quality, a common weakness of probabilistic sampling methods. The algorithm’s generality, as suggested by the heading, might mean it’s applicable to diverse tasks such as super-resolution, inpainting, and deblurring. Overall, ‘Diffusion Optimal Control’ points towards a powerful, potentially state-of-the-art, method for inverse problems leveraging the best aspects of both generative modeling and control theory.

Posterior Sampling
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Posterior sampling, in the context of diffusion models for inverse problems, aims to generate samples from a target posterior distribution representing the desired solution given observed data. The core challenge lies in the intractability of the conditional likelihood function, making direct sampling infeasible. Existing methods often resort to approximating the conditional score function, which introduces significant errors and limits accuracy. The paper proposes an alternative perspective, shifting away from direct posterior sampling to an optimal control framework. This approach leverages the iterative nature of diffusion processes, framing the inverse problem as a discrete optimal control episode. By formulating a cost function that reflects the distance from the desired solution, the method elegantly avoids explicit calculation of the often-intractable conditional likelihood. Instead, it directly learns an optimal control strategy to guide the diffusion process towards the desired posterior, resulting in significant performance improvements. This strategy makes the method robust to the accuracy of score network approximations, overcoming a key limitation of conventional probabilistic approaches. Ultimately, this novel approach offers a more accurate, robust, and efficient method for solving inverse problems using diffusion models.

High-Dim Control
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The section on ‘High-Dim Control’ in this research paper tackles the significant computational challenges associated with applying optimal control methods, specifically the iterative Linear Quadratic Regulator (iLQR), to high-dimensional systems. This is a critical issue because many real-world problems, such as image processing and reconstruction (the focus of this paper), naturally involve high-dimensional data. The core problem is the sheer size of the matrices involved in calculating gradients and Hessians, leading to memory constraints and prohibitive computational costs. The paper addresses this by introducing three key innovations: First, it leverages randomized low-rank matrix approximations, significantly reducing memory requirements and computational complexity. Second, a matrix-free approach is used, avoiding explicit matrix formation to further reduce costs. Finally, an adaptive Adam optimizer replaces the typical backtracking line search, accelerating convergence. These strategies are crucial for making optimal control applicable to realistically sized inverse problems. The analysis highlights the trade-offs inherent in these choices; for example, low-rank approximations introduce approximation error, and the choice of optimizer influences performance. The overall impact of these techniques is a significant improvement in efficiency, making it feasible to apply optimal control to high-dimensional problems that were previously intractable, therefore extending its applicability to complex scenarios and real-world datasets.

Inverse Problem
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Inverse problems, where the goal is to infer an unobserved cause from its observed effect, are a central theme in many scientific fields. The challenge lies in the ill-posed nature of these problems, often involving non-unique solutions or extreme sensitivity to noise. Diffusion models offer a powerful probabilistic approach, framing the inverse problem as sampling from the posterior distribution of the unknown signal given noisy measurements. However, traditional diffusion-based methods often encounter limitations such as the difficulty in accurately approximating the conditional score function and the computational burden of high-dimensional inference. The proposed diffusion optimal control approach addresses these issues by reframing the inverse problem as a discrete optimal control episode, enabling efficient and stable solutions. By leveraging the iterative Linear Quadratic Regulator (iLQR) algorithm, this method sidesteps the need for computationally expensive score function approximation and allows for flexible handling of complex forward operators. The theoretical foundation of this approach is rooted in optimal control theory, providing a rigorous framework for analyzing and solving inverse problems. Empirical results showcase significant improvements in image reconstruction tasks, demonstrating the efficacy and robustness of this novel methodology.

Future Work
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Future research directions stemming from this work on diffusion optimal control for inverse problems could explore several promising avenues. Extending the framework to handle more complex forward models beyond those considered in the paper (e.g., involving non-differentiable or stochastic components) is a key area. Investigating alternative control strategies beyond iLQR, such as model predictive control or reinforcement learning approaches, could potentially improve efficiency or robustness. Analyzing the theoretical properties of the method under less restrictive assumptions (e.g., weaker noise models or approximations of the score function) could further enhance its understanding and general applicability. Incorporating learned priors into the optimization could help improve the quality of reconstructions. Finally, applying the method to diverse real-world applications in areas like medical imaging, remote sensing, and materials science, could showcase its practical benefits and drive further refinement.