Adaptive and non-adaptive minimax rates for weighted Laplacian-Eigenmap based nonparametric regression

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 Green et al., (2023), 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.