From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians

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 datadependent weights to some weighted Laplace operator. Our investigation also includes the important case where the data lies on a submanifold of R.