How to identify manifolds for an optimisation problem

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.

• What distinction are you making between "Euclidean" and Riemannian manifolds? – whuber Dec 19 '18 at 16:04
• In my limited experience a Euclidean surface is one which is flat(ish), and globally consists of parallel lines, and triangles which have internal angles summing to 180. A riemannian surface, would be like a sphere, and is specifically only ever locally euclidean. – pche8701 Dec 19 '18 at 18:03
• That indeed is a special case of a Riemannian manifold. Typically, Riemannian manifolds arise as submanifolds of Euclidean space due to equality constraints imposed on the problem. I hope that characterization shows how very broad and general the situation is. – whuber Dec 19 '18 at 18:30
• Have you tried this one: arxiv.org/pdf/1412.1193 ? – Karel Macek Dec 20 '18 at 7:30