# Autoregressive Optimal Transport Models

arXiv: https://arxiv.org/abs/2105.05439 · HTML: https://mlsanity.com/paper/2105.05439/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2021.

## Built on by (most cited first, 5 of 5)

- [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)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Wasserstein multivariate auto-regressive models for modeling distributional time series](https://mlsanity.com/paper/2207.05442/) (2022)
- [Wasserstein Exponential Smoothing for Distributional Time Series Forecasting](https://mlsanity.com/paper/2606.05560/) (2026)

## Abstract (arXiv)

Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu and Shao (2004), using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.