Abstract
We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in two-dimensional/three-dimensional (2D/3D) Euclidean spaces and in higher-dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.
Keywords
Tangent spaceMathematicsNonlinear dimensionality reductionManifold (fluid mechanics)Dimensionality reductionDiffusion mapEuclidean spaceManifold alignmentParameterized complexityTangent coneTangentTangent vectorCurse of dimensionalityAlgorithmIsomapPseudo-Riemannian manifoldNonlinear systemMathematical analysisGeometryComputer scienceCurvatureArtificial intelligence
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Publication Info
- Year
- 2004
- Type
- article
- Volume
- 26
- Issue
- 1
- Pages
- 313-338
- Citations
- 1466
- Access
- Closed
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Cite This
Zhenyue Zhang,
Hongyuan Zha
(2004).
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment.
SIAM Journal on Scientific Computing
, 26
(1)
, 313-338.
https://doi.org/10.1137/s1064827502419154
Identifiers
- DOI
- 10.1137/s1064827502419154