Communications in Mathematical Sciences

Volume 5 (2007)

Number 2

Eigenvalues of an alignment matrix in nonlinear manifold learning

Pages: 313 – 329



Chi-Kwong Li

Ren-Cang Li

Qiang Ye


The alignment algorithm of Zhang and Zha is an effective method recently proposed for nonlinear manifold learning (or dimensionality reduction). By first computing local coordinates of a data set, it constructs an alignment matrix from which a global coordinate is obtained from its null space. In practice, the local coordinates can only be constructed approximately and so is the alignment matrix. This together with roundoff errors requires that we compute the the eigenspace associated with a few smallest eigenvalues of an approximate alignment matrix. For this purpose, it is important to know the first nonzero eigenvalue of the alignment matrix or a lower bound in order to computationally separate the null space. This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctly compute the desired null space of the approximate alignment matrix.


Smallest nonzero eigenvalues; alignment matrix; overlapped partition; nonlinear manifold learning; dimensionality reduction

2010 Mathematics Subject Classification

15A18, 65F15

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