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

TL;DR

This paper introduces the HPPP algorithm, combining Halpern's iteration with preconditioned proximal point methods, to achieve strong convergence and acceleration, and applies it to image restoration with denoiser priors, demonstrating improved performance.

Contribution

The paper proposes a novel Halpern-type preconditioned proximal point algorithm that enhances convergence and acceleration, and applies it to image restoration with denoiser priors, addressing limitations of existing PPP methods.

Findings

HPPP achieves strong convergence in Hilbert spaces.

The combined HPPP and denoiser prior method improves image restoration quality.

Numerical experiments validate the effectiveness of the proposed algorithms.

Abstract

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.