Local Fr'echet Regression via RKHS embedding and Its Applications to Data Analysis on Manifolds

TL;DR

This paper introduces a novel nonparametric regression method called Local Fréchet Regression (LFR) for data in metric spaces, extending it via RKHS embedding to handle manifold data and providing asymptotic theory and confidence regions.

Contribution

It extends LFR to Hilbert spaces, develops an RKHS-based estimator for metric space data, and applies the method to manifold data with confidence region construction.

Findings

Derived the asymptotic distribution of LFR in Hilbert spaces.

Proposed an RKHS-based estimator for metric space data.

Constructed confidence regions for data on manifolds.

Abstract

Local Fr'echet Regression (LFR) is a nonparametric regression method for settings in which the explanatory variable lies in a Euclidean space and the response variable lies in a metric space. It is used to estimate smooth trajectories in general metric spaces from noisy observations of random objects taking values in such spaces. Since metric spaces form a broad class of spaces that often lack algebraic structures such as addition or scalar multiplication characteristics typical of vector spaces the asymptotic theory for conventional random variables cannot be directly applied. As a result, deriving the asymptotic distribution of the LFR estimator is challenging. In this paper, we first extend nonparametric regression models for real-valued responses to Hilbert spaces and derive the asymptotic distribution of the LFR estimator in a Hilbert space setting. Furthermore, we propose a new estimator based on the LFR estimator in a reproducing kernel Hilbert space (RKHS), by mapping data from a general metric space into an RKHS. Finally, we consider applications of the proposed method to data lying on manifolds and construct confidence regions in metric spaces based on the derived asymptotic distribution.