Metric reconstruction via optimal transport

Let X be a sample of points from a metric space M. How do we recover the geometric and topological properties of M from X? One way is to build a metric thickening space P(X;r) of all probability measures on X whose "size" is at most r, where size could be measured as an L^p variance or as an L^p diameter, for 1 <= p <= infty. Can a Morse theory be developed to describe how P(X;r) changes as r increases? Particular cases of interest are when X is finite (which I will mention) or when X is a Riemannian manifold (which I will focus on).