Expected path length on random manifolds

Manifold learning is one of the cornerstones of unsupervised learning. Classical methods such as Isomap [31], Locally linear embeddings [29], Laplacian eigenmaps [4] and more [30, 11] all seek a low dimensional embedding of high dimensional data that preserves prespecified aspects of data. Probabilistic methods often view the data manifold as governed by a latent variable along with a generative model that describes how the latent manifold is to be embedded in the data space. The common theme is the quest for a low dimensional representation that faithfully captures the data. Ideally, we want an operational representation, that is we want to be able to make mathematically meaningful calculations with respect to the learned representation. It has been argued [17] that a good representation should at least support the following: - Interpolation: given two points, a natural unique interpolating curve that follows the manifold should exist.