Minimax Optimal Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps

Zhaoyang Shi, Krishna Balasubramanian, Wolfgang Polonik

Keywords: Nonsmooth Nonparametric Regression, Minimax Optimal, Fractional Laplacian Eigenmaps, Principal Component Regression

Abstract: We develop minimax optimal estimators for nonparametric regression methods when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. This function class is a Hilbert space lying between the space of square-integrable functions and the first-order Sobolev space consisting of differentiable functions. It contains fractional power functions, piecewise constant or piecewise polynomial functions and bump function as canonical examples. We construct an estimator based on performing Principal Component Regression using Fractional Laplacian Eigenmaps and show that the in-sample mean-squared estimation error of this estimator is of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the order parameter and $n$ is the number of observations. We next prove a minimax lower bound of the same order, thereby establishing that no other estimator can improve upon the proposed estimator, up to context factors. We also provide preliminary empirical results validating the practical performance of the developed estimators.

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Submission Number: 326