Metric space valued Fr\'echet regression

TL;DR

This paper introduces universal, consistent estimators for the Fréchet and conditional Fréchet means in separable metric spaces, extending regression methods beyond Euclidean settings.

Contribution

It develops practical, universally consistent estimators for Fréchet means in general metric spaces and for conditional Fréchet means in Banach spaces, using random quantization and data-driven partitioning.

Findings

Proposed a computable estimator for the Fréchet mean with universal consistency.

Introduced a conditional Fréchet mean estimator with proven consistency in Banach spaces.

Established theoretical guarantees for the estimators across various metric spaces.

Abstract

We consider the problem of estimating the Fr\'echet and conditional Fr\'echet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fr\'echet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fr\'echet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.