  Deep convolutional neural networks trained on large datsets have emerged as
an intriguing alternative for compressing images and solving inverse problems
such as denoising and compressive sensing. However, it has only recently been
realized that even without training, convolutional networks can function as
concise image models, and thus regularize inverse problems. In this paper, we
provide further evidence for this finding by studying variations of
convolutional neural networks that map few weight parameters to an image. The
networks we consider only consist of convolutional operations, with either
fixed or parameterized filters followed by ReLU non-linearities. We demonstrate
that with both fixed and parameterized convolutional filters those networks
enable representing images with few coefficients. What is more, the
underparameterization enables regularization of inverse problems, in particular
recovering an image from few observations. We show that, similar to standard
compressive sensing guarantees, on the order of the number of model parameters
many measurements suffice for recovering an image from compressive
measurements. Finally, we demonstrate that signal recovery with a un-trained
convolutional network outperforms standard l1 and total variation minimization
for magnetic resonance imaging (MRI).
