标题： 基于加权拉普拉斯特征映射的非参数回归自适应和非自适应的最小最大速率 显示英文标题

标题： Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression

摘要： 我们展示了基于加权Laplacian-Eigenmap的非参数回归方法族在真回归函数属于Sobolev空间且采样密度上下有界的条件下，自适应和非自适应的最小最大收敛率。 该自适应方法基于对Lepski方法的扩展，并针对光滑性参数（$s\in\mathbb{N}_{+}$）和确定Sobolev空间约束的范数参数（$M>0$）进行了扩展。 我们的结果推广了\cite{green2021minimax}中为特定归一化图Laplacian建立的非自适应结果，将其扩展到实践中使用的广泛类别加权Laplacian矩阵，包括未归一化的Laplacian和随机游走Laplacian。 显示英文摘要

摘要： We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods, when the true regression function belongs to a Sobolev space and the sampling density is bounded from above and below. The adaptation methodology is based on extensions of Lepski's method and is over both the smoothness parameter ($s\in\mathbb{N}_{+}$) and the norm parameter ($M>0$) determining the constraints on the Sobolev space. Our results extend the non-adaptive result in \cite{green2021minimax}, established for a specific normalized graph Laplacian, to a wide class of weighted Laplacian matrices used in practice, including the unnormalized Laplacian and random walk Laplacian.