  The deep image prior was recently introduced as a prior for natural images.
It represents images as the output of a convolutional network with random
inputs. For "inference", gradient descent is performed to adjust network
parameters to make the output match observations. This approach yields good
performance on a range of image reconstruction tasks. We show that the deep
image prior is asymptotically equivalent to a stationary Gaussian process prior
in the limit as the number of channels in each layer of the network goes to
infinity, and derive the corresponding kernel. This informs a Bayesian approach
to inference. We show that by conducting posterior inference using stochastic
gradient Langevin we avoid the need for early stopping, which is a drawback of
the current approach, and improve results for denoising and impainting tasks.
We illustrate these intuitions on a number of 1D and 2D signal reconstruction
tasks.
