标题： DFNN：一种用于学习度量空间值响应的深度弗雷歇神经网络框架 显示英文标题

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

摘要： 非欧几里得响应的回归——例如概率分布、网络、对称正定矩阵和组成——在现代应用中变得越来越重要。 在本文中，我们提出了深度弗雷歇神经网络（DFNNs），这是一种端到端的深度学习框架，用于从欧几里得预测变量预测非欧几里得响应——这些响应被视为度量空间中的随机对象。 我们的方法利用深度神经网络（DNNs）的表征学习能力，通过最小化弗雷歇风险来完成给定预测变量的响应条件弗雷歇均值的近似任务，这是条件期望的度量空间类比。 该框架高度灵活，可以适应各种度量和高维预测变量。 我们建立了DFNNs的通用逼近定理，将神经网络逼近理论的最先进水平推进到一般度量空间值响应，而无需做出模型假设或依赖局部平滑。 在合成分布和网络值响应的实证研究以及一个现实世界的应用——预测就业职业组成——表明，DFNNs始终优于现有方法。 显示英文摘要

摘要： 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 Fr\'echet 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 Fr\'echet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fr\'echet 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.