Building Firmly Nonexpansive Convolutional Neural Networks

Building nonexpansive Convolutional Neural Networks (CNNs) is a challenging problem that has recently gained a lot of attention from the image processing community. In particular, it appears to be the key to obtain convergent Plugand-Play algorithms. This problem, which relies on an accurate control of the the Lipschitz constant of the convolutional layers, has also been investigated for Generative Adversarial Networks to improve robustness to adversarial perturbations. However, to the best of our knowledge, no efficient method has been developed yet to build nonexpansive CNNs. In this paper, we develop an optimization algorithm that can be incorporated in the training of a network to ensure the nonexpansiveness of its convolutional layers. This is shown to allow us to build firmly nonexpansive CNNs. We apply the proposed approach to train a CNN for an image denoising task and show its effectiveness through simulations.