# Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions

arXiv: https://arxiv.org/abs/2107.09218 · HTML: https://mlsanity.com/paper/2107.09218/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2021.

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

- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [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)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [An Optimal Transport Approach for Network Regression](https://mlsanity.com/paper/2406.12204/) (2024)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Embedding Empirical Distributions for Computing Optimal Transport Maps](https://mlsanity.com/paper/2504.17740/) (2025)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)

## Abstract (arXiv)

Increasingly complex data analysis tasks motivate the study of the dependency of distributions of multivariate continuous random variables on scalar or vector predictors. Statistical regression models for distributional responses so far have primarily been investigated for the case of one-dimensional response distributions. We investigate here the case of multivariate response distributions while adopting the 2-Wasserstein metric in the distribution space. The challenge is that unlike the situation in the univariate case, the optimal transports that correspond to geodesics in the space of distributions with the 2-Wasserstein metric do not have an explicit representation for multivariate distributions. We show that under some regularity assumptions the conditional Wasserstein barycenters constructed for a geodesic in the Euclidean predictor space form a corresponding geodesic in the Wasserstein distribution space and demonstrate how the notion of conditional barycenters can be harnessed to interpolate as well as extrapolate multivariate distributions. The utility of distributional inter- and extrapolation is explored in simulations and examples. We study both global parametric-like and local smoothing-like models to implement conditional Wasserstein barycenters and establish asymptotic convergence properties for the corresponding estimates. For algorithmic implementation we make use of a Sinkhorn entropy-penalized algorithm. Conditional Wasserstein barycenters and distribution extrapolation are illustrated with applications in climate science and studies of aging.