Rethinking Gradient Step Denoiser: Towards Truly Pseudo-Contractive Operator
Shuchang Zhang, Yaoyun Zeng, Kangkang Deng, Hongxia Wang
Learning pseudo-contractive denoisers is a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail to guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and pseudo-contractive denoisers. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Additionally, we theoretically propose another novel construction method for truly pseudo-contractive denoisers. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments validate the pseudo-contractive property and effectiveness of the learned truly pseudo-contractive denoiser, highlighting its ability to balance interpretability and performance in inverse problems.