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Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex Optimization with Applications to Image Restoration

Zhongming Wu, Chaoyan Huang, Tieyong Zeng

TL;DR

This work develops an extrapolated three-operator Davis–Yin splitting framework for nonconvex optimization and extends it to Plug-and-Play denoiser settings, providing theoretical convergence guarantees under Kurdyka–Łojasiewicz conditions. The authors unify extrapolated forward–backward and Douglas–Rachford schemes within a single DYS-based approach and introduce gradient-step denoisers that act as proximal mappings of nonconvex functionals, yielding two extrapolated PnP-DYS methods with provable convergence. They establish descent properties, sublinear convergence, and KL-based global convergence for both smooth and nonsmooth variants, and substantiate the methods with extensive image deblurring and super-resolution experiments showing accelerated convergence and strong restoration quality. The practical impact lies in delivering convergent, acceleration-enabled PnP optimization tools that leverage learned denoisers for high-quality image restoration tasks while providing rigorous guarantees. The combination of a solid nonconvex analysis framework with state-of-the-art denoisers demonstrates both theoretical and empirical viability of extrapolated PnP-DYS in large-scale imaging problems.

Abstract

This paper investigates the convergence properties and applications of the three-operator splitting method, also known as Davis-Yin splitting (DYS) method, integrated with extrapolation and Plug-and-Play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possible nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward-backward splitting and extrapolated Douglas-Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka-Łojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization prior is replaced by a gradient step-based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image super-resolution problems, where our results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.

Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex Optimization with Applications to Image Restoration

TL;DR

This work develops an extrapolated three-operator Davis–Yin splitting framework for nonconvex optimization and extends it to Plug-and-Play denoiser settings, providing theoretical convergence guarantees under Kurdyka–Łojasiewicz conditions. The authors unify extrapolated forward–backward and Douglas–Rachford schemes within a single DYS-based approach and introduce gradient-step denoisers that act as proximal mappings of nonconvex functionals, yielding two extrapolated PnP-DYS methods with provable convergence. They establish descent properties, sublinear convergence, and KL-based global convergence for both smooth and nonsmooth variants, and substantiate the methods with extensive image deblurring and super-resolution experiments showing accelerated convergence and strong restoration quality. The practical impact lies in delivering convergent, acceleration-enabled PnP optimization tools that leverage learned denoisers for high-quality image restoration tasks while providing rigorous guarantees. The combination of a solid nonconvex analysis framework with state-of-the-art denoisers demonstrates both theoretical and empirical viability of extrapolated PnP-DYS in large-scale imaging problems.

Abstract

This paper investigates the convergence properties and applications of the three-operator splitting method, also known as Davis-Yin splitting (DYS) method, integrated with extrapolation and Plug-and-Play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possible nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward-backward splitting and extrapolated Douglas-Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka-Łojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization prior is replaced by a gradient step-based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image super-resolution problems, where our results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.
Paper Structure (21 sections, 11 theorems, 77 equations, 12 figures, 5 tables, 3 algorithms)

This paper contains 21 sections, 11 theorems, 77 equations, 12 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Organization
  4. Notation
  5. Preliminaries
  6. Extrapolated DYS method with convergence analysis
  7. The extrapolated DYS method
  8. Convergence analysis
  9. Extrapolated PnP-DYS methods
  10. When $f$ is smooth with Lipschitz continuous gradient
  11. When $f$ is nonsmooth
  12. Numerical experiments
  13. Effect of extrapolation
  14. Parameter analysis
  15. Image deblurring and super-resolution
  16. ...and 6 more sections

Key Result

Lemma 2.3

\newlabelLem2.10 bolte2014alternating (Uniformized KL property) Let $f:\mathbb{R}^{n}\rightarrow(-\infty,+\infty]$ be a proper and lower semicontinuous function and $\Omega$ be a compact set. Assume that $f$ is a constant on $\Omega$ and satisfies the KL property at each point of $\Omega$. Then, th for all $\bar{\bf x}\in \Omega$ and each ${\bf x}$ satisfying ${\rm dist}({\bf x},\Omega)<\varsigma$

Figures (12)

  • Figure 1: Effect of $\alpha$ in \ref{['alg:buildtree']} for solving DeTik model on 'butterfly' with Ker1 and noise level $2.55$. Increasing the extrapolation parameter $\alpha$ speeds-up the convergence of the algorithm. This increased convergence speed does not alter the quality of the proposed restoration.
  • Figure 1: Average results (PSNR(dB)) of TVTik and DeTik for image deblurring with 10 different blur kernels and 3 noise levels on Set3C, Set14, Kodak24, and Set17 datasets.
  • Figure 2: Influence of the parameter $\gamma_\nu=\frac{\nu^2}{\gamma}$ for deblurring with DeTik model. First column: average PSNR along with the $\gamma_\nu$. The other parameters are fixed. Remaining columns: visual results for deblurring 'leaves' with various $\gamma_\nu$.
  • Figure 2: Average results (PSNR(dB)) of TVTik and DeTik for image super-resolution with 2 scales ($\times 2$ and $\times 3$), 10 different blur kernels and 3 noise levels on Set5, CBSD68, and Urban100 datasets.
  • Figure 3: Influence of the parameter $\sigma_\nu$ for deblurring with DeTik model. First column: average PSNR along with the $\sigma_\nu$. The other parameters are fixed. Remaining columns: visual results for deblurring 'leaves' with various $\sigma_\nu$.
  • ...and 7 more figures

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • Proof 1
  • Lemma 3.6
  • ...and 15 more