You're missing the point; if the information content isn't there (i.e. "no clues") and you're not providing it somehow (e.g. by "cheating" and providing more than one image from the Internet), there's literally no way to reconstruct it -- this reeks of the Nyquist sampling theorem.

It's not something that can still be accomplished by someone who isn't a pessimist; it's provably impossible.

That's, like, the entire opposite of the nyquist sampling theorem. Faces are a low dimensional space. Images are a higher dimensional space. https://en.m.wikipedia.org/wiki/Compressed_sensing

There just isn't a learn-able (or "un-learnable" for that matter) function _even_ on the restricted domain of frontal profile images that maps surjectively onto the MUCH larger space of possible 3D face reconstructions. Intuitively, maybe the side of my jaw is deformed in a way that is not visible from the front, for example -- how can any oracle recover this information without seeing the side of my face?

Many people are mistakenly under the impression that certain signal reconstruction techniques like compressed sensing "violate" the Nyquist sampling theorem. Compressed sensing is still under the same umbrella of the Nyquist Sampling Theorem, as is CNN-based reconstruction (the technique used by this paper). I realize my analogy might have been poor; my claim is that there is still unrecoverable information loss.