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HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration

Shuchang Zhang, Hui Zhang, Hongxia Wang

TL;DR

This work introduces Halpern-type Preconditioned Proximal Point (HPPP) algorithms to overcome weak convergence and lack of acceleration in degenerate PPP, achieving strong convergence to an $\mathcal{M}$-projection and sublinear $\mathcal{O}(1/k)$ rates on fixed-point residuals. Building on HPPP, the GraRED-HP$^3$ method pairs Halpern-based PPP with Plug-and-Play priors to yield an accelerated PnP approach for image restoration, including deblurring and inpainting, with competitive or superior PSNR performance and faster residual decay than prior RED/DRS-based methods. Theoretical results establish strong convergence under maximal monotone assumptions and Lipschitz resolvent conditions, plus explicit convergence-rate bounds, while experiments on toy, infinite-dimensional, and real IR tasks demonstrate robustness to initialization and state-of-the-art restoration quality when using advanced denoisers. The framework unifies preconditioning, Halpern regularization, and denoiser priors, providing a principled route to fast, reliable fixed-point computations in high-dimensional imaging applications.

Abstract

Recently, the degenerate preconditioned proximal point (PPP) method provides a unified and flexible framework for designing and analyzing operator-splitting algorithms such as Douglas-Rachford (DR). However, the degenerate PPP method exhibits weak convergence in the infinite-dimensional Hilbert space and lacks accelerated variants. To address these issues, we propose a Halpern-type PPP (HPPP) algorithm, which leverages the strong convergence and acceleration properties of Halpern's iteration method. Moreover, we propose a novel algorithm for image restoration by combining HPPP with denoiser priors such as Plug-and-Play (PnP) prior, which can be viewed as an accelerated PnP method. Finally, numerical experiments including several toy examples and image restoration validate the effectiveness of our proposed algorithms.

HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration

TL;DR

This work introduces Halpern-type Preconditioned Proximal Point (HPPP) algorithms to overcome weak convergence and lack of acceleration in degenerate PPP, achieving strong convergence to an -projection and sublinear rates on fixed-point residuals. Building on HPPP, the GraRED-HP method pairs Halpern-based PPP with Plug-and-Play priors to yield an accelerated PnP approach for image restoration, including deblurring and inpainting, with competitive or superior PSNR performance and faster residual decay than prior RED/DRS-based methods. Theoretical results establish strong convergence under maximal monotone assumptions and Lipschitz resolvent conditions, plus explicit convergence-rate bounds, while experiments on toy, infinite-dimensional, and real IR tasks demonstrate robustness to initialization and state-of-the-art restoration quality when using advanced denoisers. The framework unifies preconditioning, Halpern regularization, and denoiser priors, providing a principled route to fast, reliable fixed-point computations in high-dimensional imaging applications.

Abstract

Recently, the degenerate preconditioned proximal point (PPP) method provides a unified and flexible framework for designing and analyzing operator-splitting algorithms such as Douglas-Rachford (DR). However, the degenerate PPP method exhibits weak convergence in the infinite-dimensional Hilbert space and lacks accelerated variants. To address these issues, we propose a Halpern-type PPP (HPPP) algorithm, which leverages the strong convergence and acceleration properties of Halpern's iteration method. Moreover, we propose a novel algorithm for image restoration by combining HPPP with denoiser priors such as Plug-and-Play (PnP) prior, which can be viewed as an accelerated PnP method. Finally, numerical experiments including several toy examples and image restoration validate the effectiveness of our proposed algorithms.
Paper Structure (22 sections, 13 theorems, 61 equations, 19 figures, 8 tables, 5 algorithms)

This paper contains 22 sections, 13 theorems, 61 equations, 19 figures, 8 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preconditioned proximal point
  4. PnP prior and RED
  5. PnP prior
  6. RED prior
  7. Implicit Gradient RED
  8. Halpern-type preconditioned proximal point (HPPP)
  9. Convergence analysis
  10. Convergence rate
  11. GraRED-HP$^3$
  12. Acceleration for PnP methods
  13. Compared with RED-PRO
  14. Experiments
  15. A toy example
  16. ...and 7 more sections

Key Result

Lemma 3

\newlabellemma: MFNE0 Let $\mathcal{A}:\mathcal{H}\to 2^\mathcal{H}$ be an operator with $\mathrm{zer} \mathcal{A} \neq \emptyset$, and let $\mathcal{M}$ be an admissible preconditioner such that $\mathcal{M}^{-1}\mathcal{A}$ is $\mathcal{M}$-monotone. Then $\mathcal{T}$ is $\mathcal{M}$-FNE, i.e.

Figures (19)

  • Figure 1: The image of $f(x)+g(x)$.
  • Figure 2: $\mathcal{M}$-projection onto the saddle-point set.
  • Figure 3: Trajectory of $\mathbf{u}^k = (x^k,y^k)$ generated by HPPP or PPP.
  • Figure 4: Trajectories, distance to solution, and two residuals of iterates for the 2D toy example. Here $\gamma = 1/0.95, \theta =15^\circ, N=100$.
  • Figure 5: Distance to solution
  • ...and 14 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2: $\mathcal{M}$-monotonicity
  • Lemma 3: Bredies2022
  • Definition 4: beck2017first
  • Proposition 5: moreau1965proximiteGribonval2020
  • Lemma 6
  • Proof 1
  • Lemma 7
  • Theorem 1
  • Proof 2
  • ...and 19 more