Discrete Hessian Eigenmaps method for dimensionality reduction

Qiang Ye; Weifeng Zhi

doi:10.1016/j.cam.2014.09.011

Discrete Hessian Eigenmaps method for dimensionality reduction

Qiang Ye, Weifeng Zhi

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

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

Original language	English
Pages (from-to)	197-212
Number of pages	16
Journal	Journal of Computational and Applied Mathematics
Volume	278
DOIs	https://doi.org/10.1016/j.cam.2014.09.011
State	Published - Apr 15 2015

Bibliographical note

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

Funding

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

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1318633, 1317424, DMS-1318633, DMS-1317424
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China

Keywords

Dimensionality reduction
Eigenmaps
Hessian
Hessian matrix
Null space

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2014.09.011

Cite this

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KW - Hessian

KW - Hessian matrix

KW - Null space

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JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

ER -

Discrete Hessian Eigenmaps method for dimensionality reduction

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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