Uniform convergence of adaptive graph-based regularization

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 Hölder functions. As an intermediate step we derive upper bounds on the covering numbers of Hölder functions on compact Riemannian manifolds, which are of independent interest for the theoretical analysis of manifold-based learning methods. © Springer-Verlag Berlin Heidelberg 2006.