Diffusion Maps

Abstract

In Laplacian eigenmaps method, the DR data is obtained from the eigen-subspace of the Laplace-Beltrami operator on the underlying manifold where the observed data resides. In Chapter 12, it was pointed out that Laplace-Beltrami operator directly links up with the heat diffusion operator by the exponential formula for positive self-adjoint operators. Therefore, they have the same eigenvector set, and the corresponding eigenvalues are linked by the exponential relation too. The relation indicates that the diffusion kernel on the manifold itself can be used in the construction of DR kernel. The diffusion map method (Dmaps) constructs the DR kernel using the diffusion maps. Then the DR data is computed from several leading eigenvectors of the Dmaps DR kernel. The chapter is organized as follows. In Section 14.1, we describe Dmaps method and its mathematical background. In Section 14.2, we present Dmaps algorithms with different types of normalization. The implementation of the various Dmaps algorithms is included in Section 14.3. In Section 14.4, we discuss the applications of Dmaps in the extraction of data features. In Section 14.5, several results of the implementation of Dmaps feature extractors are displayed.