Deconvolution is pretty hard since it's basically inverting a very poorly-conditioned matrix - that's also assuming you know the kernel. The solution is general

Deconvolution is pretty hard since it's basically inverting a very poorly-conditioned matrix - that's also assuming you know the kernel. The solution is generally to use some kind of prior on the recovered image. While a normal compressive sensing technique would use, say, some sort of wavelet sparsity as a prior, this paper's approach is effectively imposing a far stricter prior - that the face is one of 530 individuals. If you can model face transformations arbitrarily well (pretrained deep model) then that prior is going to make the face recovery much easier. Of course they're only saying which face, not recovering the actual image, but close enough. That's one of the things I'm most interested in about deep learning - priors can be far better than simple linear transformations + norms. This means potentially recovering useful information from far noisier/smaller data sources than current compressive sensing techniques are capable of.