I don't have much experience in topology, but I am interested to know if:
• Given a particular problem and associated cost function, how would one deduce what kind of manifold this problem lies on.
I ask this because as far as I know standard gradient descent algorithms implicitly assume we are working on a Euclidean manifold ... but how would I be able to know whether or not I am working on a Euclidean manifold on a problem-by-problem basis?
For example if we can prove that the parameter space for a particular problem lies on a Riemannian manifold, then the natural gradient may prove more fruitful to compute than the standard gradient.
Any help, key-words, pointers in the right direction would be super helpful. What are the key steps to identify this.