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

arXiv 2407.13120 ↗First posted 18 Jul 2024cs.CV math.OCcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 34 of its 67 references that are in the map. First posted on arXiv 18 Jul 2024.

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34 of 67 references.

References · 34

  1. 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,358
  2. 2020Plug-and-Play Image Restoration with Deep Denoiser Priorcited by 470
  3. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 307
  4. 2020Deep Unfolding Network for Image Super-Resolutioncited by 254
  5. 2016Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applicationscited by 246
  6. 2019Plug-and-Play Methods Provably Converge with Properly Trained Denoiserscited by 198
  7. 2015Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restorationcited by 195
  8. 2021Gradient Step Denoiser for convergent Plug-and-Playcited by 108
  9. 2018Regularization by Denoising: Clarifications and New Interpretationscited by 103
  10. 2024Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priorscited by 97
  11. 2022Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imagingcited by 96
  12. 2022Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularizationcited by 80
22 more
  1. 2015Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolationcited by 78
  2. 2019Plug and play methods for magnetic resonance imaging (long version)cited by 75
  3. 2017A First Order Method for Solving Convex Bi-Level Optimization Problemscited by 67
  4. 2020Learning Maximally Monotone Operators for Image Recoverycited by 62
  5. 2020Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalitiescited by 61
  6. 2017Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibriumcited by 58
  7. 2020Regularization by Denoising via Fixed-Point Projection (RED-PRO)cited by 55
  8. 2018A characterization of proximity operatorscited by 32
  9. 2019Generalized monotone operators and their averaged resolventscited by 27
  10. 2022Exact Optimal Accelerated Complexity for Fixed-Point Iterationscited by 25
  11. 2022From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problemscited by 20
  12. 2023Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iterationcited by 19
  13. 2023Provably Convergent Plug-and-Play Quasi-Newton Methodscited by 15
  14. 2024Learning pseudo-contractive denoisers for inverse problemscited by 6
  15. 2021Degenerate Preconditioned Proximal Point algorithmscited by 3
  16. 2019The Douglas--Rachford Algorithm Converges Only Weaklycited by 2
  17. 2019Warped Proximal Iterations for Monotone Inclusionscited by 1
  18. 2019A strongly convergent Krasnosel'skivi-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators in Hilbert spacescited by 1
  19. 2021A generalized forward-backward splitting operator: Degenerate analysis and applicationscited by 1
  20. 2022Strong Convergence of Forward-Reflected-Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Controlcited by 1
  21. 2023On the Bredies-Chenchene-Lorenz-Naldi algorithmcited by 1
  22. 2024Accelerating preconditioned ADMM via degenerate proximal point mappingscited by 1

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