  We consider the theory of regression on a manifold using reproducing kernel
Hilbert space methods. Manifold models arise in a wide variety of modern
machine learning problems, and our goal is to help understand the effectiveness
of various implicit and explicit dimensionality-reduction methods that exploit
manifold structure. Our first key contribution is to establish a novel
nonasymptotic version of the Weyl law from differential geometry. From this we
are able to show that certain spaces of smooth functions on a manifold are
effectively finite-dimensional, with a complexity that scales according to the
manifold dimension rather than any ambient data dimension. Finally, we show
that given (potentially noisy) function values taken uniformly at random over a
manifold, a kernel regression estimator (derived from the spectral
decomposition of the manifold) yields minimax-optimal error bounds that are
controlled by the effective dimension.
