Uniform Convergence of Adaptive Graph-Based Regularization

摘要

The regularization functional induced by the graph Lapla-cian 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 Holder functions. As an intermediate step we derive upper bounds on the covering numbers of Holder functions on compact Riemannian manifolds, which are of independent interest for the theoretical analysis of manifold-based learning methods.