> The CNN is used to come up with the feature space for the pixels (weights of the CNN), and then each pixel is "copy-pasted" from the training set based on the nearest match.
FWIW, what you just described is known as a "Markov process". It is sampling a known conditional probability distribution.
While some interpolation of the data happens because the output represents a mixture of the training images, this is not "interpolation" at the pixel level, it's picking best matches from a search space of image fragments. (And the pixel neighbors are usually synthesized - the best match depends on previous best matches!) This is distinctly different from the kind of nearest neighbor interpolation you'd do when resizing an image.
Note the phrase "nearest neighbor" in this paper has an overloaded double meaning. It is referring both to pixel neighbors and neighbors in the search space of images. The pixel neighbors provide spatial locality within a single image; this is how & why high frequencies are generated from the training set. Nearest neighbor is also referring to the neighborhood matches in the search space, the K nearest neighbors of a given pixel neighborhood are used to generate the next K pixel outputs in the synthesis phase.