Nonparametric Regression in Nonstandard Spaces

TL;DR

This paper extends nonparametric regression techniques to metric space responses, introducing Fréchet and geodesic regression methods, analyzing their convergence, and comparing their performance through simulations on the sphere.

Contribution

It adapts classical nonparametric regression estimators to metric spaces using Fréchet and geodesic approaches, providing new methods and theoretical analysis.

Findings

Derived convergence rates for the proposed estimators

Compared performance of methods via simulation on the sphere

Introduced new nonparametric regression procedures for metric data

Abstract

A nonparametric regression setting is considered with a real-valued covariate and responses from a metric space. One may approach this setting via Fr\'echet regression, where the value of the regression function at each point is estimated via a Fr\'echet mean calculated from an estimated objective function. A second approach is geodesic regression, which builds upon fitting geodesics to observations by a least squares method. These approaches are applied to transform two of the most important nonparametric regression estimators in statistics to the metric setting -- the local linear regression estimator and the orthogonal series projection estimator. The resulting procedures consist of known estimators as well as new methods. We investigate their rates of convergence in a general setting and compare their performance in a simulation study on the sphere.