Hessian Eigenmaps: new locally linear embedding techniques for high-dimensional data

We describe a method to recover the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based Locally Linear Embedding (HLLE), derives from a conceptual framework of Local Isometry in which the manifold M , viewed as a Riemannian submanifold of the ambient Euclidean space Rn, is locally isometric to an open, connected subset Θ of Euclidean space Rd. Since Θ does not have to be convex, this framework is able to handle a significantly wider class of situations than the original Isomap algorithm. The theoretical framework revolves around a quadratic formH(f) = ∫ M ||Hf (m)|| 2 Fdm defined on functions f : M 7→ R. Here Hf denotes the Hessian of f , and H(f) averages the Frobenius norm of the Hessian over M . To define the Hessian, we use orthogonal coordinates on the tangent planes of M . The key observation is that, if M truly is locally isometric to an open connected subset of Rd, then H(f) has a (d+1)-dimensional nullspace, consisting of the constant functions and a d-dimensional space of functions spanned by the original isometric coordinates. Hence, the isometric coordinates can be recovered up to a linear isometry. Our method may be viewed as a modification of the Locally Linear Embedding and our theoretical framework as a modification of the Laplacian Eigenmaps framework, where we substitute a quadratic form based on the Hessian in place of one based on the Laplacian.