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

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Recently, the degenerate preconditioned proximal point (PPP) method provided a unified and flexible framework for designing and analyzing operator-splitting algorithms, such as the Douglas–Rachford (DR) splitting. 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 the 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.