Metric space valued Fréchet regression

We consider the problem of estimating the Fréchet and conditional Fréchet 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 Fréchet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fréchet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.

💡 Research Summary

The paper addresses the fundamental problem of estimating Fréchet means and conditional Fréchet means when both predictor and response variables take values in arbitrary separable metric spaces. While Euclidean settings enjoy a rich toolbox of estimators, no universal, computationally feasible method existed for general metric spaces. The authors propose two novel estimators based on random quantization (for the output space) and data‑driven partitioning (for the input space), and prove strong universal consistency for both.

First, for the unconditional Fréchet mean (\bar m_{Fr} = \arg\min_{y\in Y} \mathbb{E}