从子流形采样的随机几何图上的非参数回归
Nonparametric regression on random geometric graphs sampled from submanifolds
AI总结：
该研究针对协变量位于未知光滑紧子流形的非参数回归问题，基于图拉普拉斯特征基随机基展开的贝叶斯先验，证明了在Hölder光滑性假设下后验收缩率达极小极大最优。
AI中文摘要：
我们研究协变量位于欧氏空间中未知光滑紧子流形上的非参数回归问题。在协变量上定义随机几何图结构后，我们分析了通过图拉普拉斯特征基中的随机基展开设计的贝叶斯先验所产生后验分布的频率论渐近行为。在回归函数和子流形上协变量密度满足Hölder光滑性假设的条件下，我们证明了这类方法的后验收缩率对任意正光滑性指数都是极小极大最优的（至多相差对数因子）。
英文摘要：
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.
