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

• URL: http://arxiv.org/abs/2407.13120v2
• Date: Sun, 21 Jul 2024 12:13:03 GMT
• Title: HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration
• Authors: Shuchang Zhang, Hui Zhang, Hongxia Wang,
• Abstract summary: Preconditioned Proximal Point (PPP) algorithms provide a unified framework for splitting methods in image restoration. PPP algorithms typically degenerate in infinite-dimensional convergence leading to uncertain solutions. We propose the Halpern-type HPPP algorithm, which leverages the strong convergence properties to achieve a particular solution.
• Score: 7.614347936574962
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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