HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restoration
This work addresses convergence issues in optimization algorithms for image restoration, offering an incremental improvement with potential benefits in computational imaging.
The paper tackled the weak convergence and lack of acceleration in degenerate preconditioned proximal point methods by proposing a Halpern-type variant (HPPP) that ensures strong convergence and acceleration, and applied it to image restoration with denoiser priors, achieving improved performance in numerical experiments.
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.