Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

TLDR

Developing appropriate representations for complex data is a central challenge in machine learning and pattern recognition. The study aims to construct a representation for data that lies on a low‑dimensional manifold embedded in a high‑dimensional space. The authors propose a geometrically motivated algorithm that leverages the graph Laplacian, the Laplace‑Beltrami operator, and the heat equation to represent high‑dimensional data. The resulting algorithm offers a computationally efficient, locality‑preserving nonlinear dimensionality reduction method with a natural link to clustering, and the authors illustrate its potential applications.

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.