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

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.

@article{zhang2025_2407.13120,
  title={ HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration },
  author={ Shuchang Zhang and Hui Zhang and Hongxia Wang },
  journal={arXiv preprint arXiv:2407.13120},
  year={ 2025 }
}