  We study how the posterior contraction rate under a Gaussian process (GP)
prior depends on the intrinsic dimension of the predictors and smoothness of
the regression function. An open question is whether a generic GP prior that
does not incorporate knowledge of the intrinsic lower-dimensional structure of
the predictors can attain an adaptive rate for a broad class of such
structures. We show that this is indeed the case, establishing conditions under
which the posterior contraction rates become adaptive to the intrinsic
dimension $\varrho$ in terms of the covering number of the data domain $X$ (the
Minkowski dimension), and prove the optimal posterior contraction rate
$O(n^{-s/(2s +\varrho)})$, up to a logarithmic factor, assuming an
approximation order $s$ of the reproducing kernel Hilbert space (RKHS) on
${X}$. When ${X}$ is a $\varrho$-dimensional compact smooth manifold, we study
RKHS approximations to intrinsically defined $s$-order H\"older functions on
the manifold for any positive $s$ by a novel analysis of kernel approximations
on manifolds, leading to the optimal adaptive posterior contraction rate. We
propose an empirical Bayes prior on the kernel bandwidth using kernel affinity
and $k$-nearest neighbor statistics, eliminating the need for prior knowledge
of the intrinsic dimension. The efficiency of the proposed Bayesian regression
approach is demonstrated on various numerical experiments.
