Sure, try this: http://math.ucr.edu/home/baez/information/.

In a nutshell, you take probability distributions and project them onto topological manifolds, such that the distribution consists of points on the manifold. You can find more by searching for key words like "information geometry" or "differential geometry machine learning".

there's also the stuff that ayasdi and carlsson were/are doing but i never really saw the point of that (e.g. just compute connected components instead persistent homology).