Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fr\'ech\'et dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.

翻译：本文提出了针对一般度量空间值响应与欧几里得预测变量之间回归的数据可视化与降维方法。现有的Fr\'ech\'et降维方法要求响应度量空间能够连续嵌入希尔伯特空间，这对度量类型和核函数选择构成了限制。我们通过提出一种避免使用核函数的欧几里得嵌入技术来放宽这一假设。在此框架下，经典降维方法如普通最小二乘法和切片逆回归得到了推广。大量模拟实验表明，在适用场景下，所提方法在合成数据上的性能优于现有方法。同时研究了影响美国COVID-19传播分布的因素分析，以及健康人群BMI与脑结构连接性关联的真实数据分析。