Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

作者：Mikhail Belkin, Partha Niyogi · 发表于：Neural Computation · 年份：2003 · DOI：10.1162/089976603321780317 · 被引用次数：7807 · 研究领域：Face and Expression Recognition、Topological and Geometric Data Analysis、Neural Networks and Applications

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.