Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4638812 | Journal of Computational and Applied Mathematics | 2015 | 16 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Qiang Ye, Weifeng Zhi,