Many modern applications involve predicting structured, non-Euclidean outputs such as probability distributions, networks, and symmetric positive-definite matrices. These outputs are naturally modeled as elements of general metric spaces, where classical regression techniques that rely on vector space structure no longer apply. We introduce E2M (End-to-End Metric regression), a deep learning framework for predicting metric space-valued outputs. E2M performs prediction via weighted Fréchet means over training outputs, where the weights are learned by a neural network conditioned on the input. This construction provides a principled mechanism for geometry-aware prediction that avoids surrogate embeddings and restrictive parametric assumptions, while fully preserving the intrinsic geometry of the output space. We establish theoretical guarantees, including a universal approximation theorem that characterizes the expressive capacity of the model and a convergence analysis of the entropy-regularized training objective. Through extensive simulations involving probability distributions, networks, and symmetric positive-definite matrices, we show that E2M consistently achieves state-of-the-art performance, with its advantages becoming more pronounced at larger sample sizes. Applications to human mortality distributions and New York City taxi networks further demonstrate the flexibility and practical utility of this framework.

翻译：许多现代应用涉及预测结构化、非欧几里得输出，例如概率分布、网络和对称正定矩阵。这些输出自然被建模为一般度量空间的元素，而依赖向量空间结构的经典回归技术在此不再适用。我们提出E2M（端到端度量回归），这是一个用于预测度量空间值输出的深度学习框架。E2M通过对训练输出的加权弗雷歇均值进行预测，其中权重由基于输入条件化的神经网络学习得到。这种构造为几何感知预测提供了原则性机制，避免了替代嵌入和限制性参数假设，同时完整保留了输出空间的内在几何结构。我们建立了理论保证，包括刻画模型表达能力的一般逼近定理，以及熵正则化训练目标的收敛性分析。通过在概率分布、网络和对称正定矩阵上的广泛模拟，我们证明E2M持续达到最先进的性能，且其优势在更大样本量下更为显著。在人类死亡率分布和纽约市出租车网络中的应用进一步展示了该框架的灵活性和实用价值。