  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
observation. 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, a noisy image can be denoised by i) finding the closest
image in the range of the generator or by ii) passing it through an
encoder-generator architecture (known as an autoencoder). However, there is
little theory to justify this success, let alone to predict the denoising
performance as a function of the network parameters. In this paper we consider
the problem of denoising an image from additive Gaussian noise using the two
generator based approaches. In both cases, we assume the image is well
described by a deep neural network with ReLU activations functions, mapping a
$k$-dimensional code to an $n$-dimensional image. In the case of the
autoencoder, we show that the feedforward network reduces noise energy by a
factor of $O(k/n)$. In the case of optimizing over the range of a generative
model, we state and analyze a simple gradient algorithm that minimizes a
non-convex loss function, and provably reduces noise energy by a factor of
$O(k/n)$. We also demonstrate in numerical experiments that this denoising
performance is, indeed, achieved by generative priors learned from data.
