From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians
Matthias Hein, Jean-Yves Audibert and Ulrike Von Luxburg
In: COLT 2005, 27 - 30 Jun 2005, Bertinoro, Italy.

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 Laplacian-based 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 Euclidean space.

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