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

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... 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 (LSML), and show it has good generalization properties. LSML 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.