Eigenvalue bounds for an alignment matrix in manifold learning

Qiang Ye, Weifeng Zhi

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 languageEnglish
Pages (from-to)2944-2962
Number of pages19
JournalLinear Algebra and Its Applications
Issue number8
StatePublished - 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).


  • 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


Dive into the research topics of 'Eigenvalue bounds for an alignment matrix in manifold learning'. Together they form a unique fingerprint.

Cite this