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

Hein, M; Audibert, J; von Luxburg, U

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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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https://link.springer.com/content/pdf/10.1007%2F11503415_32.pdf
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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: https://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.