  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.
