Problems and methods of reconstruction of hidden geometric structure of the data from distance matrices

We will consider a classical problem on how to reconstruct the metric measure space from the information on distances between points from a very large subsets (covering densely the space in the limit), as well as how to reconstruct its embedding in a Euclidean or Hilbert space (this is quite natural, for instance, in the case when the space studied is a smooth Riemannian manifold).