# Distribution-on-Distribution Regression via Optimal Transport Maps

arXiv: https://arxiv.org/abs/2104.09418 · HTML: https://mlsanity.com/paper/2104.09418/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2021.

## Built on by (most cited first, 16 of 16)

- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [Medoid splits for efficient random forests in metric spaces](https://mlsanity.com/paper/2306.17031/) (2023)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [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)
- [Wasserstein Transfer Learning](https://mlsanity.com/paper/2505.17404/) (2025)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Measure-to-measure Regression with Transformers](https://mlsanity.com/paper/2605.28075/) (2026)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Wasserstein multivariate auto-regressive models for modeling distributional time series](https://mlsanity.com/paper/2207.05442/) (2022)
- [Heterogeneous Image-based Classification Using Distributional Data Analysis](https://mlsanity.com/paper/2403.07126/) (2024)
- [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)
- [Gaussian Sheaf Neural Networks](https://mlsanity.com/paper/2605.21435/) (2026)
- [Wasserstein Exponential Smoothing for Distributional Time Series Forecasting](https://mlsanity.com/paper/2606.05560/) (2026)

## Abstract (arXiv)

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval Omegasubsetℝ. Our regression model is based on the theory of optimal transportation and links the conditional Fr'echet mean of the response distribution to the covariate distribution via an optimal transport map. We define a Fr'echet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observation of the regression pairs. Computation of the estimator is shown to reduce to an isotonic regression problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data.