Rethinking Gradient Step Denoiser: Towards Truly Pseudo-Contractive Operator

Shuchang Zhang, Yaoyun Zeng, Kangkang Deng, Hongxia Wang

Keywords: Plug-and-Play prior, Regularization by Denoising, Pseudo-contractive denoiser, Image restoration

Abstract:

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 operators. 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. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED
via fixed-point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive property and, when integrated into RED-PRO, provides a favorable trade-off between interpretability and empirical performance on inverse problems.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 27569