# Tangential Wasserstein Projections

arXiv: https://arxiv.org/abs/2207.14727 · HTML: https://mlsanity.com/paper/2207.14727/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 95 references that are in the map.
- First posted on arXiv 29 Jul 2022.

## Builds on (most cited first, 26 of 26)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [BDD100K: A Diverse Driving Dataset for Heterogeneous Multitask Learning](https://mlsanity.com/paper/1805.04687/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [On Robustness of Principal Component Regression](https://mlsanity.com/paper/1902.10920/) (2019)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data](https://mlsanity.com/paper/1507.07218/) (2015)
- [Fr'echet Means and Procrustes Analysis in Wasserstein Space](https://mlsanity.com/paper/1701.06876/) (2017)
- [A fast approach to optimal transport: The back-and-forth method](https://mlsanity.com/paper/1905.12154/) (2019)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Distributional synthetic controls](https://mlsanity.com/paper/2001.06118/) (2020)
- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Log-PCA versus Geodesic PCA of histograms in the Wasserstein space](https://mlsanity.com/paper/1708.08143/) (2017)
- [A Design-Based Perspective on Synthetic Control Methods](https://mlsanity.com/paper/2101.09398/) (2021)
- [A novel notion of barycenter for probability distributions based on optimal weak mass transport](https://mlsanity.com/paper/2102.13380/) (2021)
- [Synthetic Control As Online Linear Regression](https://mlsanity.com/paper/2202.08426/) (2022)
- [Geodesic PCA in the Wasserstein space](https://mlsanity.com/paper/1307.7721/) (2013)

## Built on by (most cited first, 4 of 4)

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications](https://mlsanity.com/paper/2410.23602/) (2024)
- [Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs](https://mlsanity.com/paper/2603.26940/) (2026)
- [Conditional Random Ordered Transport Spaces](https://mlsanity.com/paper/2606.08113/) (2026)

## Abstract (arXiv)

We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and provides a unique solution in regular settings. The idea is to work on regular tangent cones of the Wasserstein space using generalized geodesics. Its structure and computational properties make the method applicable in a variety of settings, from causal inference to the analysis of object data. An application to estimating causal effects yields a generalization of the notion of synthetic controls to multivariate data with individual-level heterogeneity, as well as a way to estimate optimal weights jointly over all time periods.