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Hessian eigenmaps: locally linear embedding techniques for high-dimensional data.

David L Donoho1, Carrie Grimes

  • 1Department of Statistics, Stanford University, Stanford, CA, USA, 94305-4065.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 2006
PubMed
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We introduce Hessian-based locally linear embedding to recover data parameters on manifolds. This method extends ISOMAP by handling non-convex shapes, enabling broader applications in manifold learning.

Area of Science:

  • Computational geometry
  • Manifold learning
  • Riemannian geometry

Background:

  • Scattered data on manifolds in high-dimensional Euclidean space pose challenges for parameter recovery.
  • Existing methods like ISOMAP have limitations with non-convex manifold structures.
  • Local isometry provides a framework for understanding manifold geometry.

Purpose of the Study:

  • To develop a novel method for recovering the underlying parametrization of scattered data on manifolds.
  • To extend the capabilities of manifold learning algorithms to handle a wider class of geometric situations.
  • To provide a theoretical foundation for recovering isometric coordinates from local geometric properties.

Main Methods:

  • Hessian-based locally linear embedding (HLLE) is proposed, building on concepts of local isometry.

Related Experiment Videos

  • A quadratic form involving the Hessian of functions on the manifold is defined and analyzed.
  • The method utilizes orthogonal coordinates on tangent planes to compute the Hessian.

    • Main Results:

    • The Hessian-based approach identifies a null space related to the manifold's intrinsic dimension (d).
    • Isometric coordinates are recovered up to a linear isometry, revealing the underlying parametrization.
    • The method demonstrates robustness for manifolds that are locally isometric to non-convex Euclidean subsets.

    Conclusions:

    • Hessian-based locally linear embedding offers an advancement over existing manifold learning techniques.
    • The framework successfully recovers intrinsic manifold coordinates by analyzing Hessian properties.
    • This approach expands the applicability of manifold learning to more complex data structures.