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 map a noisy image to a near noise-free image. The underlying principle is simple: images are well described by priors that map a low-dimensional latent representations to image. Based on a prior, a noisy image can be denoised by finding a close image in the range of the prior. Since deep networks trained on large set of images have empirically been shown to be good priors, they enable effective denoisers. However, there is little theory to justify this success, let alone to predict the denoising performance. In this paper we consider the problem of denoising an image from additive Gaussian noise with variance $\sigma^2$, 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 provide an iterative algorithm minimizing a non-convex loss that provably removes noise energy by a fraction $\sigma^2 k/n$. We also demonstrate in numerical experiments that this denoising performance is, indeed, achieved by generative priors learned from data.