It relates in the following way. Suppose you have some framework for calculation which depends on matrix multiplications. You use that framework to develop some approximation for matrix multiplications which are faster. Then to make that framework faster, you substitute that back into the framework. That is a sort of bootstrapping which user qsort has related to compiler bootstrapping.

The goal in compiler boostrapping is to reach a fixed point, in a single iteration. When the compiler binary compiles itself, what should come out is a faithful replica of that compiler binary. Having done compiler bootstrapping, I know about hair-pulling problems that occur when you don't have a fixed point. E.g. a bug in the optimization causes the compiler to subtly miscompile itself. The resulting compiler then severely miscompiles itself; e.g. fails to recompile itself entirely, or produces something that no longer runs at all. (And that's actually a lucky case; a compiler compiling itself is basically a dog food test. When that fails, you've caught a bug early. What keeps you awake at night is the compiler compiling itself perfectly fine, but rarely miscompiling other code in the wild.)

Anyway, this sort of malfunction is exactly what will happen if you take a working, self-hosted compiler and then replace some important chunk of it with some hokey machine-learning approximation. It's unlikely you can do that without breaking the fixed point.

I don't see how you can not have fixed point issues in the matrix scenario. Maybe it will stabilize if you iterate on it. You take the machine learning framework with approximate matrix multiplications in it and run the process for figuring out those matrix multiplications. You may get some different values in it, which you can substitute back and try again. Will that converge? If it does, how do you know the whole thing is still valid? (What definition of valid applies?)