Metric space valued Fr\'echet regression

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.