Deep Denoising: Rate-Optimal Recovery of Structured Signals with a Deep Prior

Deep neural networks provide state-of-the-art performance for image denoising, where the goal is to recover a near noise-free image from a noisy image. The underlying principle is that neural networks trained on large datasets have empirically been shown to be able to generate natural images well from a low-dimensional latent representation of the image. Given such a generator network, or prior, a noisy image can be denoised by finding the closest image in the range of the prior. However, there is little theory to justify this success, let alone to predict the denoising performance as a function of the networks parameters. In this paper we consider the problem of denoising an image from additive Gaussian noise, assuming the image is well described by a deep neural network with ReLu activations functions, mapping a k-dimensional latent space to an n-dimensional image. We state and analyze a simple gradient-descent-like iterative algorithm that minimizes a non-convex loss function, and provably removes a fraction of (1 - O(k/n)) of the noise energy. We also demonstrate in numerical experiments that this denoising performance is, indeed, achieved by generative priors learned from data.