Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data

We describe a method for recovering the underlying parametrization of scattered data (m(i)) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based locally linear embedding, derives from a conceptual framework of local isometry in which the manifold M, viewed as a Riemannian submanifold of the ambient Euclidean Space R(n), is locally isometric to an open, connected subset Θ of Euclidean space R(d). Because 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 form H(f) = ∫(M)||H(f)(m)||²(F)dm defined on functions f: M--> R. Here Hf…

• NS
  National Science Foundation

  Award: 0140698
• YU
  Yale University
• UO
  University of Pennsylvania
• HU
  Harvard University
• DA
  Defense Advanced Research Projects Agency
• AR
  Advanced Research Projects Agency