Abstract
One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
Keywords
Dimensionality reductionNonlinear dimensionality reductionLaplace operatorManifold (fluid mechanics)LocalityDiffusion mapCluster analysisManifold alignmentRepresentation (politics)MathematicsGraphLaplacian matrixArtificial intelligenceConnection (principal bundle)IsomapComputer scienceTheoretical computer scienceMathematical analysisGeometry
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Publication Info
- Year
- 2003
- Type
- article
- Volume
- 15
- Issue
- 6
- Pages
- 1373-1396
- Citations
- 7514
- Access
- Closed
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Cite This
Mikhail Belkin,
Partha Niyogi
(2003).
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation.
Neural Computation
, 15
(6)
, 1373-1396.
https://doi.org/10.1162/089976603321780317
Identifiers
- DOI
- 10.1162/089976603321780317