Metric space valued Fr{é}chet regression

Abstract: We consider the problem of estimating the Fr{é}chet and conditional Fr{é}chet 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{é}chet 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{é}chet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.