Regularization by Denoising: Clarifications and New Interpretations (1806.02296v4)

Abstract: Regularization by Denoising (RED), as recently proposed by Romano, Elad, and Milanfar, is powerful image-recovery framework that aims to minimize an explicit regularization objective constructed from a plug-in image-denoising function. Experimental evidence suggests that the RED algorithms are state-of-the-art. We claim, however, that explicit regularization does not explain the RED algorithms. In particular, we show that many of the expressions in the paper by Romano et al. hold only when the denoiser has a symmetric Jacobian, and we demonstrate that such symmetry does not occur with practical denoisers such as non-local means, BM3D, TNRD, and DnCNN. To explain the RED algorithms, we propose a new framework called Score-Matching by Denoising (SMD), which aims to match a "score" (i.e., the gradient of a log-prior). We then show tight connections between SMD, kernel density estimation, and constrained minimum mean-squared error denoising. Furthermore, we interpret the RED algorithms from Romano et al. and propose new algorithms with acceleration and convergence guarantees. Finally, we show that the RED algorithms seek a consensus equilibrium solution, which facilitates a comparison to plug-and-play ADMM.

• The paper challenges the traditional RED view by showing its explicit regularization holds only for denoisers with local homogeneity and symmetric Jacobians.
• It introduces the Score-Matching by Denoising (SMD) framework, reinterpreting image recovery as a score-matching problem rather than variational regularization.
• Novel algorithmic enhancements, including accelerated RED-ADMM and RED-FP variants, significantly reduce computational overhead and improve convergence rates.

Regularization by Denoising: Clarifications and New Interpretations

The paper "Regularization by Denoising: Clarifications and New Interpretations" by Edward T. Reehorst and Philip Schniter embarks on a detailed exploration of the Regularization by Denoising (RED) framework, a methodology proposed by Romano, Elad, and Milanfar. The RED framework utilizes advanced image-denoising techniques to solve image recovery problems, claiming state-of-the-art experimental performance. This paper revisits the theoretical underpinnings of RED, critiques its current explanations, and proposes novel interpretations.

The authors challenge the prevailing explanation that RED effectively minimizes an explicit regularization objective through a plug-in denoising function. Through rigorous analysis, it is demonstrated that the claim holds only if the employed denoiser exhibits local homogeneity and possesses a symmetric Jacobian. This is a significant theoretical claim as practical denoisers like non-local means, BM3D, TNRD, and DnCNN often lack this symmetry, raising questions about the validity of the existing theoretical grounding for RED.

Furthermore, the paper introduces an alternative framework termed Score-Matching by Denoising (SMD). The SMD framework interprets RED algorithms through the approximation of a score, specifically, the gradient of a log-prior probability. This interpretation provides fresh insights by linking RED algorithms to kernel density estimation and constrained minimum mean-squared error (MMSE) denoising. Crucially, this perspective allows us to view image recovery as a problem of score-matching rather than directly minimizing a variational objective, thereby bypassing the limitations of explicit regularization.

The numerical analysis presented supports the theoretical claims, providing evidence that many practical denoisers do not meet the conditions required for the original interpretation of RED. The authors show that deviations in local homogeneity and Jacobian symmetry can adversely affect the accuracy of RED gradient expressions.

On the algorithmic front, the paper proposes several novel enhancements to existing RED algorithms. It puts forward variants of RED-ADMM and RED-FP algorithms, introducing acceleration techniques through proximal gradient methods to improve convergence rates significantly. These algorithmic adaptations come with practical implications, particularly in reducing computational overhead and accelerating convergence without sacrificing the quality of the image recovery.

In terms of implications, the findings of this paper hold considerable repercussions for the ongoing development of denoising-based regularizers in machine learning and signal processing. By shifting the focus towards score-matching, the authors suggest a new vantage point for the optimization community, which could guide the development of more robust algorithms. The insights into Jacobian properties provide nuance for researchers developing or employing advanced denoisers, emphasizing the importance of understanding the structural properties of such methods.

Looking forward, this paper paves a path for future research within the domain of artificial intelligence and image processing. The intersection of score-matching and denoising presents a fertile ground for exploring sophisticated models that do not conform to traditional regularization paradigms. Through these clarifications and innovations, the scope for novel applications of denoising in ambiguous and high-dimensional problems is immense.

In summary, Reehorst and Schniter's work stands as a foundational critique and contribution to the RED framework, advancing both theoretical understanding and practical algorithmic strategies in image processing and beyond.