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).

Funding

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).

Funders	Funder number
National Science Foundation (NSF)	DMS-0915062
Directorate for Mathematical and Physical Sciences	0915062

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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author = "Qiang Ye and Weifeng Zhi",

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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).

PY - 2012/4/15

Y1 - 2012/4/15

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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DO - 10.1016/j.laa.2011.07.007

M3 - Article

AN - SCOPUS:84858081634

SN - 0024-3795

VL - 436

SP - 2944

EP - 2962

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 8

ER -

Eigenvalue bounds for an alignment matrix in manifold learning

Abstract

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Funding

Keywords

ASJC Scopus subject areas

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