Metric space valued Fr{\'e}chet regression

2026-02-05

AI Summary

This work addresses the challenge of universally and efficiently estimating Fréchet means and their conditional counterparts in separable metric spaces, where existing methods often lack generality or computational feasibility. The authors propose a Fréchet mean estimator based on random quantization and develop a data-driven partitioning strategy to construct a conditional Fréchet mean estimator applicable to general metric spaces and Banach space-valued outputs. They establish, for the first time, universal consistency of the Fréchet mean estimator in arbitrary separable metric spaces and prove universal consistency of the conditional estimator in Banach spaces. This framework provides a theoretically sound and computationally tractable approach to regression analysis in non-Euclidean settings.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

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

Problem

• Fréchet mean
• conditional Fréchet mean
• metric space
• universal consistency
• separable metric space

Innovation

• Fréchet regression
• metric space
• random quantization
• universal consistency
• conditional Fréchet mean