From Subspaces to Submanifolds

This paper identifies a broad class of nonlinear dimensionality reduced (NLDR) problems where the exact local isometry between an extrinsically curved data manifold M and a low-dimensional parameterization space can be recovered from a finite set of high-dimensional point sampels. The method, Geodesic Nullsapce Analysis (GNA), rests on two results: First, the exact isometric parameterization of a local point clique on M haas an algebraic reduction to arc-length integrations when the ambient-space embedding of M is locally a product of planar quadrics. Second, the locally isometric global parameterization lies in the left invariant subspace of a linearizing operator that averages the nullspace projectors of the local parameterizations. We show how to use the GNA operator for denosing, dimensionality reduction, and resynthesis of both the original data and of new samples, making such "submanifold" methods an attractive alternative to subspace methods in data analysis.