Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment

2002/12/07 by Zhenyue Zhang, Zhang, Zhenyue, Hongyuan Zha +1 · 6 citations
Computer Science · #Advanced Vision and Imaging #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #Face and Expression Recognition #I.5.1 #I5.3 #Image Retrieval and Classification Techniques #Machine Learning (cs.LG) #cs.AI #cs.LG

paper · pdf · doi:10.48550/arxiv.cs/0212008

arxiv created 2002/12/07 · openalex publication_date 2002/12/07 · arxiv updated 2016/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

Cited by

• Curvature-aware Manifold Learning
• Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds
• A study of the classification of low-dimensional data with supervised manifold learning
• Manifold Fitting in Ambient Space
• Tangent space estimation for smooth embeddings of Riemannian manifolds
• A Tangent Distance Preserving Dimensionality Reduction Algorithm

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