Metric space valued Fréchet regression
László Györfi, Pierre Humbert, Batiste Le Bars
TL;DR
The paper addresses non-Euclidean regression by formulating Fréchet and conditional Fréchet means for responses in separable metric spaces and provides computable, universally consistent estimators. It introduces a random-quantization-based estimator for the Fréchet mean with strong consistency under a bounded-loss condition and a support assumption, and extends this to universal, Mahalanobis-free Fréchet regression via a data-driven Voronoi partition and quantization (Proto-NN style). Theoretical results prove almost-sure convergence of the risk to the true Fréchet risk for both the mean and the regression function, with detailed proofs leveraging concentration bounds, vanishing discretization errors, and extension theorems. The approach yields practical, scalable methods for non-Euclidean targets, with applicability to Banach-valued outputs and general metric-space spaces. These contributions provide a unified, computable framework for non-Euclidean Fréchet estimation and regression in broad settings.
Abstract
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.