HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration
<|reference_start|>HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration: Preconditioned Proximal Point (PPP) algorithms provide a unified framework for splitting methods in image restoration. Recent advancements with RED (Regularization by Denoising) and PnP (Plug-and-Play) priors have achieved state-of-the-art performance in this domain, emphasizing the need for a meaningful particular solution. However, degenerate PPP algorithms typically exhibit weak convergence in infinite-dimensional Hilbert space, leading to uncertain solutions. To address this issue, we propose the Halpern-type Preconditioned Proximal Point (HPPP) algorithm, which leverages the strong convergence properties of Halpern iteration to achieve a particular solution. Based on the implicit regularization defined by gradient RED, we further introduce the Gradient REgularization by Denoising via HPPP called GraRED-HP3 algorithm. The HPPP algorithm is shown to have the regularity converging to a particular solution by a toy example. Additionally, experiments in image deblurring and inpainting validate the effectiveness of GraRED-HP3, showing it surpasses classical methods such as Chambolle-Pock (CP), PPP, RED, and RED-PRO.<|reference_end|>
@article{zhang2024hppp:,
    title={HPPP: Halpern-type Preconditioned Proximal Point Algorithms and
  Applications to Image Restoration},
    author={Shuchang Zhang, Hui Zhang, and Hongxia Wang},
    journal={arXiv preprint arXiv:2407.13120},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.13120},
    primaryClass={cs.CV math.OC}
}