Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

2003/05/22 by Mikhail Belkin, Partha Niyogi · 436 citations
Computer Science · #Face and Expression Recognition #Neural Networks and Applications #Topological and Geometric Data Analysis

paper · doi:10.1162/089976603321780317

openalex publication_date 2003/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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.

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