In lieu of neural networks and deep learning, what are some approaches to computer vision?
I think you could use some type of probabilistic model to solve this issue. I might be completely wrong, though, as I do not know a lot about probabilistic models.

Some time ago, I made use of Markov chains to generate levels for a game called Mega Man, and I remember that the results were promising, so I looked up "Markov Chains and face recognition" and found the following paper:

http://www.anefian.com/research/nefian98_face.pdf

It deals with the specific problem of face recognition.

Markov Chains also work with Optical Character Recognition, and though I know that comparing a letter to a face is a bit of a stretch, the following paper might give you some insight.

http://www.aclweb.org/anthology/N03-1018

I hope that helps!