With the rapid development of data collection techniques, complex data objects that are not in the Euclidean space are frequently encountered in new statistical applications. Fr\'echet regression model (Peterson & M\"uller 2019) provides a promising framework for regression analysis with metric space-valued responses. In this paper, we introduce a flexible sufficient dimension reduction (SDR) method for Fr\'echet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors and to provide a visual inspection tool for Fr\'echet regression. Our approach is flexible enough to turn any existing SDR method for Euclidean (X,Y) into one for Euclidean X and metric space-valued Y. The basic idea is to first map the metric-space valued random object $Y$ to a real-valued random variable $f(Y)$ using a class of functions, and then perform classical SDR to the transformed data. If the class of functions is sufficiently rich, then we are guaranteed to uncover the Fr\'echet SDR space. We showed that such a class, which we call an ensemble, can be generated by a universal kernel. We established the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the proposed methods is illustrated through simulation studies for several commonly encountered metric spaces that include Wasserstein space, the space of symmetric positive definite matrices, and the sphere. We illustrated the data visualization aspect of our method by exploring the human mortality distribution data across countries and by studying the distribution of hematoma density.

翻译：随着数据收集技术的迅速发展,在新的统计应用中经常遇到不在欧几里特空间的复杂数据对象。Fr\'echet回归模型(Peterson & M\""uller 2019)为回归分析提供了充满希望的框架,并提供了空间价值的量化反应。在本文件中,我们为Fr\'echet回归引入了一种灵活而充分的维度减少方法,以实现两个目的:减轻高维预测器造成的维度诅咒,并为Fr\'echche回归提供一个视觉检查工具。我们的方法足够灵活,可以将Euclidean(X,Y)的任何现有特别提款权方法转化为Euclidean X 和多度空间价值Y的回归模型。我们的基本想法是,首先用一种功能类别来绘制价值为Y$美元至实际价值的随机值$f(Y)美元(SDR)的方法,然后对变化的数据进行经典特别提法。如果功能类别足够丰富,那么我们就能发现Fr\'echet TRich 空间空间空间空间空间空间空间空间空间空间空间。我们展示了这样一个分类,我们称之为精确度的精确度的精确度的精确度分布,我们称之为一个分类,我们称之为一个平整级的精确度的精确度的精确度分析方法,我们通过一个用来研究, 我们用平比度的平比度分析方法, 我们通过一系列的精确度分析研究提出了一种平比度的精确度的精确度的精确度分析方法, 的计算方法,我们所拟议的标准的计算法, 通过一系列的精确度的精确度分析方法,我们用一个解释的计算。我们通过一系列的精确度分析方法, 通过一系列的精确度的计算法的计算法的计算法的精确度的精确度的计算, 通过一系列的计算方法,我们所研判法的计算方法,我们所研判法的计算方法, 通过一系列的计算方法,我们所研。