We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.

翻译：我们考虑当协变量位于欧几里得空间中一个未知光滑紧致子流形上的非参数回归问题。通过在协变量上定义随机几何图结构，我们分析了由在图拉普拉斯特征基中通过随机基展开设计的贝叶斯先验所产生的后验分布的渐近频率学派性质。在回归函数和协变量在子流形上密度的赫尔德光滑性假设下，我们证明了此类方法的后验收缩率对于任意正光滑性指标是最小化极大最优的（在对数因子范围内）。