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

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Hein,  M
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Hein, M., Audibert, J., & von Luxburg, U. (2005). From Graphs to Manifolds: Weak and Strong Pointwise Consistency of Graph Laplacians. In P. Auer, & R. Meir (Eds.), Learning Theory: 18th Annual Conference on Learning Theory, COLT 2005, Bertinoro, Italy, June 27-30, 2005 (pp. 470-485). Berlin, Germany: Springer.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-D6D7-0
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 R^d.