Eigenvalues of an alignment matrix in nonlinear manifold learning

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.