Principal manifolds and nonlinear dimensionality reduction via tangent space alignment

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Principal manifolds and nonlinear dimensionality reduction via tangent space alignment (2004)

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by Zhenyue Zhang , Hongyuan Zha

Venue:	SIAM JOURNAL ON SCIENTIFIC COMPUTING
Citations:	260 - 15 self

BibTeX

@ARTICLE{Zhang04principalmanifolds,
    author = {Zhenyue Zhang and Hongyuan Zha},
    title = {Principal manifolds and nonlinear dimensionality reduction via tangent space alignment },
    journal = {SIAM JOURNAL ON SCIENTIFIC COMPUTING},
    year = {2004},
    pages = {313--338}
}

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Abstract

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.

Keyphrases

tangent space alignment zhenyue zhang hongyuan zha data point principal manifold nonlinear dimensionality reduction tangent space second-order accuracy data visualization problem affine subspace internal global coordinate local geometry algorithmic issue unsupervised learning unorganized data point 64-by-64 pixel face image great variety dimensional euclidean space nonlinear dimension reduction careful error analysis partial eigendecomposition manifold learning neighborhood connection matrix reconstruction error new algorithm various pose

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