Hier finden Sie wissenschaftliche Publikationen aus den Fraunhofer-Instituten.

From graphs to manifolds - weak and strong pointwise consistency of graph Laplacians

: Hein, M.; Audibert, J.-Y.; Luxburg, U. von


Auer, P.:
Learning theory. 18th Annual Conference on Learning Theory, COLT 2005 : Bertinoro, Italy, June 27 - 30, 2005; Proceedings
Berlin: Springer, 2005 (Lecture Notes in Artificial Intelligence 3559)
ISBN: 3-540-26556-2
ISBN: 978-3-540-26556-6
Conference on Learning Theory (COLT) <18, 2005, Bertinoro/Italy>
Conference Paper
Fraunhofer IPSI; 2007

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.