Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment
<|reference_start|>Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment: Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of the neighborhood connection matrix. We present a careful error analysis of our algorithm and show that the reconstruction errors are of second-order accuracy. We illustrate our algorithm using curves and surfaces both in   2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face images with various pose and lighting conditions. We also address several theoretical and algorithmic issues for further research and improvements.<|reference_end|>
@article{zhang2002principal,
    title={Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent
  Space Alignment},
    author={Zhenyue Zhang and Hongyuan Zha},
    journal={arXiv preprint arXiv:cs/0212008},
    year={2002},
    archivePrefix={arXiv},
    eprint={cs/0212008},
    primaryClass={cs.LG cs.AI}
}