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.
|Number of pages||16|
|Journal||Journal of Computational and Applied Mathematics|
|State||Published - Apr 15 2015|
Bibliographical noteFunding Information:
The research was supported in part by National Science Foundation under grants DMS-1317424 and DMS-1318633 .
© 2014 Elsevier B.V. All rights reserved.
- Dimensionality reduction
- Hessian matrix
- Null space
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics