Towards a Theoretical Foundation for Laplacian-Based Manifold Methods

In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacian-based manifold methods. These methods utilize the graph Laplacian associated to a data set for a variety of applications in semi-supervised learning, clustering, data representation. We show that under certain conditions the graph Laplacian of a point cloud of data samples converges to the Laplace-Beltrami operator on the underlying manifold. Theorem 3.1 contains the first result showing convergence of a random graph Laplacian to the manifold Laplacian in the context of machine learning.