Discrete Hessian Eigenmaps method for dimensionality reduction

Qiang Ye, Weifeng Zhi

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


For a given set of data points lying on a low-dimensional manifold embedded in a high-dimensional space, the dimensionality reduction is to recover a low-dimensional parametrization from the data set. The recently developed Hessian Eigenmaps method is a mathematically rigorous method that also sets a theoretical framework for the nonlinear dimensionality reduction problem. In this paper, we develop a discrete version of the Hessian Eigenmaps method and present an analysis, giving conditions under which the method works as intended. As an application, a procedure to modify the standard constructions of k-nearest neighborhoods is presented to ensure that Hessian LLE can recover the original coordinates up to an affine transformation.

Original languageEnglish
Pages (from-to)197-212
Number of pages16
JournalJournal of Computational and Applied Mathematics
StatePublished - Apr 15 2015

Bibliographical note

Funding Information:
The research was supported in part by National Science Foundation under grants DMS-1317424 and DMS-1318633 .

Publisher Copyright:
© 2014 Elsevier B.V. All rights reserved.


  • Dimensionality reduction
  • Eigenmaps
  • Hessian
  • Hessian matrix
  • Null space

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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