Eigenvalue bounds for an alignment matrix in manifold learning

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.