Local polynomial regression on unknown manifolds

TL;DR

This paper demonstrates that naive multivariate local polynomial regression can adapt to lower-dimensional manifold structures, achieving optimal convergence rates for nonparametric regression on such manifolds.

Contribution

It reveals that simple local polynomial regression methods can effectively adapt to unknown manifold structures, achieving optimal rates without explicit manifold estimation.

Findings

Achieves optimal convergence rates on manifolds

Adapts to local lower-dimensional structures

Works for functions in Sobolev spaces

Abstract

We reveal the phenomenon that ``naive'' multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression functions belonging to a Sobolev space when the predictor variables live on or close to a lower dimensional manifold.