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.

翻译：我们证明了当真实回归函数属于Sobolev空间且采样密度有上下界时，一类基于加权拉普拉斯特征映射的非参数回归方法的适应性与非适应性极小化收敛速率。该自适应方法基于Lepski方法的扩展，覆盖了决定Sobolev空间约束的光滑参数（$s\in\mathbb{N}_{+}$）和范数参数（$M>0$）。本文结果将\cite{green2021minimax}中针对特定归一化图拉普拉斯建立的非适应性结论，推广至实践中广泛应用的一类加权拉普拉斯矩阵，包括非归一化拉普拉斯与随机游走拉普拉斯。