Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression

TL;DR

This paper proposes a nonlinear sufficient dimension reduction method for distributional data using universal kernels on metric spaces, leveraging Wasserstein and sliced Wasserstein distances, with promising numerical results.

Contribution

The paper introduces a novel kernel-based approach for nonlinear dimension reduction in distribution-on-distribution regression, utilizing Wasserstein and sliced Wasserstein distances.

Findings

Outperforms competing methods on synthetic data

Effective on real-world fertility, mortality, and temperature datasets

Kernel construction ensures rich representation of distributional data

Abstract

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.