# Wasserstein Regression

arXiv: https://arxiv.org/abs/2006.09660 · HTML: https://mlsanity.com/paper/2006.09660/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2020.

## Built on by (most cited first, 30 of 35)

- [Wasserstein K-means for clustering probability distributions](https://mlsanity.com/paper/2209.06975/) (2022)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Dimension Reduction for Fr'echet Regression](https://mlsanity.com/paper/2110.00467/) (2021)
- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances](https://mlsanity.com/paper/2509.20508/) (2025)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations](https://mlsanity.com/paper/2307.12508/) (2023)
- [Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric](https://mlsanity.com/paper/2101.09039/) (2021)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)
- [Interpretable Causal Inference for Analyzing Wearable, Sensor, and Distributional Data](https://mlsanity.com/paper/2312.10569/) (2023)
- [Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2501.14635/) (2025)
- [Wasserstein Transfer Learning](https://mlsanity.com/paper/2505.17404/) (2025)
- [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)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Energy Trees: Regression and Classification With Structured and Mixed-Type Covariates](https://mlsanity.com/paper/2207.04430/) (2022)
- [Wasserstein multivariate auto-regressive models for modeling distributional time series](https://mlsanity.com/paper/2207.05442/) (2022)
- [Semi-supervised Fr'echet Regression](https://mlsanity.com/paper/2404.10444/) (2024)
- [A Metric-based Principal Curve Approach for Learning One-dimensional Manifold](https://mlsanity.com/paper/2405.12390/) (2024)
- [Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age](https://mlsanity.com/paper/2501.06868/) (2025)
- [Differentially Private Geodesic Regression](https://mlsanity.com/paper/2504.11304/) (2025)
- [Embedding Empirical Distributions for Computing Optimal Transport Maps](https://mlsanity.com/paper/2504.17740/) (2025)
- [Consistency and inconsistency in k-means clustering](https://mlsanity.com/paper/2507.06226/) (2025)
- [Barycentric model aggregation in the Wasserstein space of distributions and a variational approach to consistency](https://mlsanity.com/paper/2507.11719/) (2025)
- [Toward an Asymptotic Efficiency Theory on Regular Parameter Manifolds](https://mlsanity.com/paper/2510.13703/) (2025)
- [Wasserstein Regression as a Variational Approximation of Probabilistic Trajectories through the Bernstein Basis](https://mlsanity.com/paper/2510.26607/) (2025)
- [Bures-Wasserstein Importance-Weighted Evidence Lower Bound: Exposition and Applications](https://mlsanity.com/paper/2602.04272/) (2026)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data](https://mlsanity.com/paper/2604.07671/) (2026)

## Abstract (arXiv)

The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we utilize the geometry of tangent bundles of the space of random measures endowed with the Wasserstein metric for mapping distributions to tangent spaces. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. In simulations, it performs better than an alternative transformation approach where one maps distributions to a Hilbert space through the log quantile density transformation and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression operator and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributional time series of house prices.