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.

翻译：我们研究了高斯过程（GP）先验下的后验收缩率如何依赖于预测变量的本征维度和回归函数的平滑度。一个悬而未决的问题是：一个不包含预测变量本征低维结构知识的通用GP先验，是否能为一大类此类结构获得自适应速率。我们证明情况确实如此，建立了后验收缩率在数据域$X$的覆盖数（闵可夫斯基维数）方面自适应于本征维度$\varrho$的条件，并证明了在再生核希尔伯特空间（RKHS）在${X}$上具有近似阶$s$的假设下，最优后验收缩率为$O(n^{-s/(2s +\varrho)})$（忽略对数因子）。当${X}$是一个$\varrho$维紧致光滑流形时，我们通过对流形上核近似的新颖分析，研究了RKHS对任意正数$s$的流形上本征定义的$s$阶H\"older函数的逼近，从而得到最优的自适应后验收缩率。我们提出了一种基于核亲和度与$k$近邻统计量的核带宽经验贝叶斯先验，无需事先知道本征维度。所提出的贝叶斯回归方法的效率在各种数值实验中得到了验证。