As novel data collection becomes increasingly common, traditional dimension reduction and data visualization techniques are becoming inadequate to analyze these complex data. A surrogate-assisted sufficient dimension reduction (SDR) method for regression with a general metric-valued response on Euclidean predictors is proposed. The response objects are mapped to a real-valued distance matrix using an appropriate metric and then projected onto a large sample of random unit vectors to obtain scalar-valued surrogate responses. An ensemble estimate of the subspaces for the regression of the surrogate responses versus the predictor is used to estimate the original central space. Under this framework, classical SDR methods such as ordinary least squares and sliced inverse regression are extended. The surrogate-assisted method applies to responses on compact metric spaces including but not limited to Euclidean, distributional, and functional. An extensive simulation experiment demonstrates the superior performance of the proposed surrogate-assisted method on synthetic data compared to existing competing methods where applicable. The analysis of the distributions and functional trajectories of county-level COVID-19 transmission rates in the U.S. as a function of demographic characteristics is also provided. The theoretical justifications are included as well.

翻译：随着新型数据收集方式日益普遍，传统的降维与数据可视化技术已难以胜任对这些复杂数据的分析。本文提出一种基于代理变量的充分降维方法，用于处理以欧氏变量为预测变量、以一般度量空间值作为响应变量的回归问题。首先利用适当度量将响应对象映射为实值距离矩阵，再将其投影至大量随机单位向量上以获得标量代理响应。通过集成估计代理响应与预测变量回归的子空间，进而估计原始中心空间。在此框架下，经典充分降维方法如普通最小二乘法和切片逆回归法得到拓展。该代理变量方法适用于包括欧氏空间、分布空间、函数空间在内的紧致度量空间上的响应变量。广泛模拟实验表明，在合成数据上，所提代理变量方法相较于现有可比方法具有更优性能。同时，本文提供了美国县级COVID-19传播率分布及函数轨迹随人口统计学特征变化的分析，并给出了理论证明。