Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps

Laplacian based nonparametric regression is a widely used approach in machine learning that leverages the Laplacian Eigenmaps algorithm to perform regression tasks without relying on explicit parametric models. The nonparametric nature of the approach makes it flexible and adaptable to data generating process without imposing strict assumptions about the functional form of the relationship between the response and the covariates. Existing theoretical studies of this approach are restricted to establishing minimax rates of convergence and adaptivity properties when the true regression function lies in Sobolev spaces; see Section 1.1 for details. Such spaces are inherently smooth in nature and exclude important function classes in nonparametric statistics, such as piecewise constant or polynomial functions, bump functions and other such nonsmooth function classes. In this work, using the framework of fractional Laplacians, we propose a novel approach called Principal Component Regression using Fractional Laplacian Eigenmaps (PCR-FLE) for nonsmooth and nonparametric regression. The PCR-FLE algorithm generalizes the PCR-LE algorithm by Green et al. (2023) and the PCR-WLE algorithm by Shi et al. (2024), and is designed to naturally handle the case when the true regression function lies in an L