Discovering Shared Structure in Manifold Learning

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local will suffer from at least four generic problems associated with (1) noise in the data, (2) curvature of the manifold, (3) dimensionality of the manifold, and (4) the presence of many manifolds with little data per manifold. This analysis suggests non-local manifold learning algorithms which attempt to discover shared structure in the tangent planes at different positions. A criterion for such an algorithm is proposed and experiments estimating a tangent plane prediction function are presented. The function has parameters that are shared across space rather than estimated based on the local neighborhood, as in current non-parametric manifold learning algorithms. The results show clearly the advantages of this approach with respect to local manifold learning algorithms.