Abstract. 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. We show that under certain conditions the graph Laplacian of a point cloud converges to the Laplace-Beltrami operator on the underlying manifold. Theorem 1 contains the first result showing convergence of a random graph Laplacian to manifold Laplacian in the machine learning context. 1

Abstract. In the machine learning community it is generally believed that graph Laplacians corresponding to a finite sample of data points converge to a continuous Laplace operator if the sample size increases. Even though this assertion serves as a justification for many Laplacianbased algorithms, so far only some aspects of this claim have been rigorously proved. In this paper we close this gap by establishing the strong pointwise consistency of a family of graph Laplacians with data-dependent weights to some weighted Laplace operator. Our investigation also includes the important case where the data lies on a submanifold of R d. 1