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

Zhaoyang Shi; Krishnakumar Balasubramanian; Wolfgang Polonik

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arXiv · 2311.00140

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

Zhaoyang Shi·Krishnakumar Balasubramanian·Wolfgang Polonik

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Abstract

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.

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Zhaoyang Shi, Krishnakumar Balasubramanian, Wolfgang Polonik. 2023-10-31. Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression. https://arxiv.org/abs/2311.00140

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