  We introduce a new approach to nonlinear sufficient dimension reduction in
cases where both the predictor and the response are distributional data,
modeled as members of a metric space. Our key step is to build universal
kernels (cc-universal) on the metric spaces, which results in reproducing
kernel Hilbert spaces for the predictor and response that are rich enough to
characterize the conditional independence that determines sufficient dimension
reduction. For univariate distributions, we construct the universal kernel
using the Wasserstein distance, while for multivariate distributions, we resort
to the sliced Wasserstein distance. The sliced Wasserstein distance ensures
that the metric space possesses similar topological properties to the
Wasserstein space while also offering significant computation benefits.
Numerical results based on synthetic data show that our method outperforms
possible competing methods. The method is also applied to several data sets,
including fertility and mortality data and Calgary temperature data.
