基于加权拉普拉斯特征映射的非参数回归的自适应与非自适应极小极大速率
Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression
AI总结：
本文针对基于加权拉普拉斯特征映射的非参数回归，在索伯列夫空间设定下建立了自适应与非自适应极小极大收敛速率，将已有结果推广到包括非归一化和随机游走拉普拉斯在内的广泛加权拉普拉斯矩阵。
AI中文摘要：
当真实回归函数属于索伯列夫空间且采样密度上下有界时，我们证明了一类基于加权拉普拉斯特征映射的非参数回归方法的自适应与非自适应极小极大收敛速率。自适应方法基于Lepski方法的扩展，同时针对光滑度参数（$s\in\mathbb{N}_{+}$）和决定索伯列夫空间约束的范数参数（$M>0$）进行自适应。我们的结果将文献\cite{green2021minimax}中针对特定归一化图拉普拉斯算子建立的非自适应结果，推广到实际中广泛使用的一类加权拉普拉斯矩阵，包括非归一化拉普拉斯算子和随机游走拉普拉斯算子。
英文摘要：
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.
