  Manifold learning seeks a low dimensional representation that faithfully
captures the essence of data. Current methods can successfully learn such
representations, but do not provide a meaningful set of operations that are
associated with the representation. Working towards operational representation
learning, we endow the latent space of a large class of generative models with
a random Riemannian metric, which provides us with elementary operators. As
computational tools are unavailable for random Riemannian manifolds, we study
deterministic approximations and derive tight error bounds on expected
distances.
