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

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

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von Luxburg,  U
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Zitation

Hein, M., Audibert, J., & von Luxburg, U. (2005). From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians. In Conference on Learning Theory (pp. 470-485).


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0013-D6D7-0
Zusammenfassung
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.