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Erschienen in:
2005 | OriginalPaper | Buchkapitel
From Graphs to Manifolds – Weak and Strong Pointwise Consistency of Graph Laplacians
verfasst von : Matthias Hein, Jean-Yves Audibert, Ulrike von Luxburg
Erschienen in: Learning Theory
Verlag: Springer Berlin Heidelberg
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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
${\mathbb R}^{d}$
.