Is it reasonable to suggest a PhD thesis topic in an interdisciplinary area in which none of the faculty members specialize? I think a similar question was answered in PhD Research under guide/advisor of a different department by JeffE. I'm asking a slightly different question. Yet if you find it identical to the above one, you can report it to be 'duplicate.'

I'm interested in both pure math and machine learning, and I chose to apply to math PhD exclusively due to my background. However, recently I've been into (1) developing an algorithm related to proof verification and others for application to mathematics using the state-of-the-art deep learning techniques and (2) deep learning proper without much of sophisticated math involved. I'm certain most or all professors in math department I'm applying to (or anywhere else) aren't working in such an approach as (1), since that's more of a job of professors in CS department. 


Though the answer may be case-by-case, is a topic as (1) appropriate for my PhD thesis? I understand that I can have a co-adviser from CS dept., but is this topic likely to get accepted from math department? 
If so, by any chance if my interest totally shifts to (2) and has not much to do with mathematics at later stage of my PhD, do I still have to work on something related to math like (1)? (I assume the answer is Yes.)
In fact, one of the programs I got accepted to is quite lenient in letting a math PhD student into its ML PhD program. Is going to this program the only reasonable path for the case that my interest totally shifts to (2)?
Well, no answer yet? I'll give it a try, even though I'm in CS and I'm residing in Europe. But maybe it helps a bit:

To adress your questions 1-3:


In CS it would - in math it depends on the will of the supervisor. 
Depends on your PhD program and supervisor. It might depend who is sponsoring the PhD program as well. 
I don't know enough about the US system to answer this one - but you asked for an other option and there is one: 
Just quit the program you are in and look for an other one in a CS department! Sounds a bit disruptive, but if this is your research interest, you should follow it!