# Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression

arXiv: https://arxiv.org/abs/2207.04613 · HTML: https://mlsanity.com/paper/2207.04613/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 55 references that are in the map.
- First posted on arXiv 11 Jul 2022.

## Builds on (most cited first, 11 of 11)

- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Sufficient Forecasting Using Factor Models](https://mlsanity.com/paper/1505.07414/) (2015)
- [Inverse Moment Methods for Sufficient Forecasting using High-Dimensional Predictors](https://mlsanity.com/paper/1705.00395/) (2017)
- [Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions](https://mlsanity.com/paper/2107.09218/) (2021)
- [Dimension Reduction for Fr'echet Regression](https://mlsanity.com/paper/2110.00467/) (2021)

## Built on by (most cited first, 8 of 8)

- [FAStEN: An Efficient Adaptive Method for Feature Selection and Estimation in High-Dimensional Functional Regressions](https://mlsanity.com/paper/2303.14801/) (2023)
- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)
- [Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression](https://mlsanity.com/paper/2609.23789/) (2026)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [On relative universality, regression operator, and conditional independence](https://mlsanity.com/paper/2504.11044/) (2025)
- [Structure-Preserving Nonlinear Sufficient Dimension Reduction for Tensors](https://mlsanity.com/paper/2512.20057/) (2025)
- [Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects](https://mlsanity.com/paper/2607.28981/) (2026)

## Abstract (arXiv)

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.