Uniform Convergence of Adaptive Graph-Based Regularization

Hein, M

doi:10.1007/11776420_7

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Uniform Convergence of Adaptive Graph-Based Regularization

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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. (2006). Uniform Convergence of Adaptive Graph-Based Regularization. In G. Lugosi, & H. Simon (Eds.), Learning Theory: 19th Annual Conference on Learning Theory, COLT 2006, Pittsburgh, PA, USA, June 22-25, 2006 (pp. 50-64). Berlin, Germany: Springer.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-D05B-D

Abstract

The regularization functional induced by the graph Laplacian of a random
neighborhood graph based on the data is adaptive in two ways. First it adapts to an underlying
manifold structure and second to the density of the data-generating probability measure.
We identify in this paper the limit of the regularizer and show
uniform convergence over the space of Hoelder functions. As an intermediate
step we derive upper bounds on the covering numbers of Hoelder functions on
compact Riemannian manifolds, which are of independent interest
for the theoretical analysis of manifold-based learning methods.