Discrete Hessian Eigenmaps method for dimensionality reduction

• Qiang Ye ^[1] ; Weifeng Zhi ^[2]
  1. [1] University of Kentucky

     University of Kentucky

     Estados Unidos

     
  2. [2] University of California
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 278, Nº 1 (15 April 2015), 2015, págs. 197-212
• Idioma: inglés
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• Resumen

  • 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 kk-nearest neighborhoods is presented to ensure that Hessian LLE can recover the original coordinates up to an affine transformation.