Eigenvalue bounds for an alignment matrix in manifold learning

Qiang Ye; Weifeng Zhi

doi:10.1016/j.laa.2011.07.007

Eigenvalue bounds for an alignment matrix in manifold learning

Qiang Ye, Weifeng Zhi

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

4 Scopus citations

Abstract

This paper presents a spectral analysis for an alignment matrix that arises in reconstruction of a global coordinate system from local coordinate systems through alignment in manifold learning. Some characterizations of its eigenvalues and its null space as well as a lower bound for the smallest positive eigenvalue are given, which generalize earlier results of Li et al. (2007) [4] to include a more general situation that arises in alignments of local sections of different dimensions. Our results provide a theoretical understanding of the Local Tangent Space Alignment (LTSA) method (Zhang and Zha (2004) [12]) for nonlinear manifold learning and address some computational issues related to the method.

Original language	English
Pages (from-to)	2944-2962
Number of pages	19
Journal	Linear Algebra and Its Applications
Volume	436
Issue number	8
DOIs	https://doi.org/10.1016/j.laa.2011.07.007
State	Published - Apr 15 2012

Bibliographical note

Funding Information:
Supported in part by the National Science Foundation under Grant DMS-0915062. Corresponding author. E-mail addresses: [email protected] (Q. Ye), [email protected] (W. Zhi).

Keywords

Alignment matrix
Dimensionality reduction
Eigenvalue bound
Manifold learning
Smallest nonzero eigenvalue

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2011.07.007

Cite this

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title = "Eigenvalue bounds for an alignment matrix in manifold learning",

abstract = "This paper presents a spectral analysis for an alignment matrix that arises in reconstruction of a global coordinate system from local coordinate systems through alignment in manifold learning. Some characterizations of its eigenvalues and its null space as well as a lower bound for the smallest positive eigenvalue are given, which generalize earlier results of Li et al. (2007) [4] to include a more general situation that arises in alignments of local sections of different dimensions. Our results provide a theoretical understanding of the Local Tangent Space Alignment (LTSA) method (Zhang and Zha (2004) [12]) for nonlinear manifold learning and address some computational issues related to the method.",

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AU - Ye, Qiang

AU - Zhi, Weifeng

N1 - Funding Information: Supported in part by the National Science Foundation under Grant DMS-0915062. Corresponding author. E-mail addresses: [email protected] (Q. Ye), [email protected] (W. Zhi).

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N2 - This paper presents a spectral analysis for an alignment matrix that arises in reconstruction of a global coordinate system from local coordinate systems through alignment in manifold learning. Some characterizations of its eigenvalues and its null space as well as a lower bound for the smallest positive eigenvalue are given, which generalize earlier results of Li et al. (2007) [4] to include a more general situation that arises in alignments of local sections of different dimensions. Our results provide a theoretical understanding of the Local Tangent Space Alignment (LTSA) method (Zhang and Zha (2004) [12]) for nonlinear manifold learning and address some computational issues related to the method.

AB - This paper presents a spectral analysis for an alignment matrix that arises in reconstruction of a global coordinate system from local coordinate systems through alignment in manifold learning. Some characterizations of its eigenvalues and its null space as well as a lower bound for the smallest positive eigenvalue are given, which generalize earlier results of Li et al. (2007) [4] to include a more general situation that arises in alignments of local sections of different dimensions. Our results provide a theoretical understanding of the Local Tangent Space Alignment (LTSA) method (Zhang and Zha (2004) [12]) for nonlinear manifold learning and address some computational issues related to the method.

KW - Alignment matrix

KW - Dimensionality reduction

KW - Eigenvalue bound

KW - Manifold learning

KW - Smallest nonzero eigenvalue

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

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Eigenvalue bounds for an alignment matrix in manifold learning

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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