DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses

2025-10-19

AI Summary

This paper addresses regression problems with non-Euclidean response variables—such as probability distributions, networks, and symmetric positive-definite matrices—lying in general metric spaces. We propose a deep Fréchet neural network framework that directly estimates the conditional Fréchet mean given covariates by end-to-end minimization of the Fréchet risk, marking the first integration of deep neural networks into nonparametric Fréchet mean estimation for metric-space-valued responses. Theoretically, we establish a universal approximation theorem for Fréchet regression without assuming parametric models or local smoothness, thereby extending the theoretical foundations of neural networks to non-Euclidean domains. Practically, the framework accommodates arbitrary metric structures and high-dimensional covariate inputs. Empirical evaluations on synthetic distributional data, network-valued responses, and real-world occupational composition prediction tasks demonstrate substantial performance gains over state-of-the-art methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses -- which are considered as random objects in a metric space -- from Euclidean predictors. Our method leverages the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses without making model assumptions or relying on local smoothing. Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.

Problem

• Predicting non-Euclidean responses from Euclidean predictors
• Learning conditional Fréchet means in metric spaces
• Accommodating diverse metrics and high-dimensional predictors

Innovation

• Deep Fréchet networks for non-Euclidean response prediction
• Minimizing Fréchet risk to approximate conditional means
• Universal approximation theorem for metric-space-valued responses