Non-Isometric Manifold Learning: Analysis and an Algorithm

In this work we take a novel view of nonlinear manifold learning. Usually, manifold learning is formulated in terms of finding an embedding or "unrolling" of a manifold into a lower dimensional space. Instead, we treat it as the problem of learning a representation of a nonlinear, possibly non-isometric manifold that allows for the manipulation of novel points. Central to this view of manifold learning is the concept of generalization beyond the training data. Drawing on concepts from supervised learning, we establish a framework for studying the problems of model assessment, model complexity, and model selection for manifold learning. We present an extension of a recent algorithm, Locally Smooth Manifold Learning (L S M L), and show it has good generalization properties. L S M L learns a representation of a manifold or family of related manifolds and can be used for computing geodesic distances, finding the projection of a point onto a manifold, recovering a manifold from points corrupted by noise, generating novel points on a manifold, and more.