Predicting outputs that are located in non-Euclidean spaces, such as probability distributions, networks, and symmetric positive-definite matrices, is becoming increasingly important in modern data analysis, particularly when inputs are high-dimensional. We propose DeSI (Deep Single-Index Fréchet Regression), a semiparametric framework for regression with metric space-valued outputs and multivariate inputs that assumes a single-index structure for the conditional Fréchet mean. DeSI estimates an interpretable index direction, which quantifies the relative importance of inputs, using a deep neural network, and performs Fréchet regression along the resulting one-dimensional index in the target metric space. This structure mitigates the curse of dimensionality while retaining interpretability, which stands in contrast to standard deep neural networks. We establish theoretical guarantees for DeSI, including uniform approximation and convergence rates, and demonstrate its strong predictive performance through simulations on distributions, networks, and symmetric positive-definite matrices, as well as an application to compositional mood data from New Jersey.

翻译：在非欧几里得空间（如概率分布、网络和对称正定矩阵）中预测输出在现代数据分析中日益重要，尤其当输入为高维数据时。我们提出DeSI（深度单指标弗雷歇回归），这是一种半参数回归框架，适用于度量空间取值的输出和多变量输入，该框架假设条件弗雷歇均值具有单指标结构。DeSI利用深度神经网络估计可解释的指标方向（用于量化输入的相对重要性），并在目标度量空间中沿生成的一维指标执行弗雷歇回归。这种结构在保持可解释性的同时缓解了维度灾难问题，这与标准深度神经网络形成鲜明对比。我们为DeSI建立了理论保证，包括均匀逼近和收敛速率，并通过在分布、网络和对称正定矩阵上的模拟实验以及新泽西州成分情绪数据的应用，展示了其强大的预测性能。