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.

翻译：非欧几里得响应（例如概率分布、网络、对称正定矩阵和组合）的回归在现代应用中变得越来越重要。本文提出深度Fréchet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Fréchet风险来近似给定预测变量时响应的条件Fréchet均值（即条件期望在度量空间中的类比）。该框架具有高度灵活性，可适应多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新进展推广到一般的度量空间值响应，而无需做出模型假设或依赖局部平滑。对合成分布值和网络值响应的实证研究，以及预测就业职业构成的实际应用表明，DFNNs始终优于现有方法。