> contrary to expectations,
ingore the weasel words
> a great deal of image statistics are captured by the structure of a convolutional image generator rather than by any learned capability.
> To show this, we apply untrained ConvNets to the solution of several such problems. Instead of following the common paradigm of training a ConvNet on a large dataset of example images, we fit a generator network to a single degraded image.
How is that _untrained_, if they are training it on an image? Is a generator network different from a ConvNet?
> This is particularly remarkable because no aspect of the network is learned from data;
but they just said "we fit a generator network to a single degraded image"
As others have commented, they appear to be training on all the parts of a single complex image, and using that training to repair local deviations from the global average. This may be better than classical denoising algorithms because the NN can model the image structure better than other approaches, but this doesn't seem like a novel use of NNs.