(I apologise in advance for a naive and/or misplaced question)
I don’t have a math background, rather robotics and self-taught ML, but I am trying to learn more and understand some abstract concepts or at least get a clearer intuition (i.e. forgive my poor terminology use).
I am reading into (Riemannian) manifolds and obtaining geodesics as distance metrics on manifolds.
What still confuses me is the following:
Most examples explaining (Riemannian) manifolds involve either low-dimensional and ideal examples (e.g. circles and spheres) or very abstract equations. I got some intuition from this blogpost (http://bjlkeng.github.io/posts/manifolds/) on how to get the metric, but I am not sure how can this be done on some arbitrary manifold.
To be more specific: How to define a manifold based on some data samples? As I understand, I first need to fit some (probability) function to the data I have, and this function needs to be smooth/differentiable so I can get the tangent space. From there I can estimate geodesics/metric etc.
Moreover, how can I assure that the fitted function actually gives me a manifold?