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

Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression