An improved local tangent space alignment method for manifold learning

Principal component analysis (PCA) is widely used in recently proposed manifold learning algorithms to provide approximate local tangent spaces. However, such approximations provided by PCA may be inaccurate when local neighborhoods of the data manifold do not lie in or close to a linear subspace. Furthermore, the approximated tangent spaces can not fit the change in data distribution density. In this paper, a new method is proposed for providing faithful approximations to the local tangent spaces of a data manifold, which is proved to be more accurate than PCA. With this new method, an improved local tangent space alignment (ILTSA) algorithm is developed, which can efficiently recover the geometric structure of data manifolds even in the case when data are sparse or non-uniformly distributed. Experimental results are presented to illustrate the better performance of ILTSA on both synthetic data and image data.

[1]  Francesco Camastra,et al.  Data dimensionality estimation methods: a survey , 2003, Pattern Recognit..

[2]  H. Zha,et al.  Local smoothing for manifold learning , 2004, CVPR 2004.

[3]  Hongyuan Zha,et al.  Adaptive Manifold Learning , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  I. Jolliffe.  Principal Component Analysis , 2002 .

[5]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[6]  Bo Zhang,et al.  Intrinsic dimension estimation of manifolds by incising balls , 2009, Pattern Recognit..

[7]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[8]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Deli Zhao,et al.  Laplacian PCA and Its Applications , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[10]  Balázs Kégl,et al.  Intrinsic Dimension Estimation Using Packing Numbers , 2002, NIPS.

[11]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[12]  Peter J. Bickel,et al.  Maximum Likelihood Estimation of Intrinsic Dimension , 2004, NIPS.

[13]  H. Zha,et al.  Principal manifolds and nonlinear dimensionality reduction via tangent space alignment , 2004, SIAM J. Sci. Comput..

[14]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.