# Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric

arXiv: https://arxiv.org/abs/2101.09039 · HTML: https://mlsanity.com/paper/2101.09039/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 69 references that are in the map.
- First posted on arXiv 22 Jan 2021.

## Builds on (most cited first, 9 of 9)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Approximate Bayesian computation with the Wasserstein distance](https://mlsanity.com/paper/1905.03747/) (2019)
- [Multi-marginal Wasserstein GAN](https://mlsanity.com/paper/1911.00888/) (2019)
- [Wasserstein Autoregressive Models for Density Time Series](https://mlsanity.com/paper/2006.12640/) (2020)
- [Backward Nested Descriptors Asymptotics with Inference on Stem Cell Differentiation](https://mlsanity.com/paper/1609.00814/) (2016)
- [Bayesian Quantile Regression Using Random B-spline Series Prior](https://mlsanity.com/paper/1609.02950/) (2016)

## Built on by (most cited first, 2 of 2)

- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Barycentric model aggregation in the Wasserstein space of distributions and a variational approach to consistency](https://mlsanity.com/paper/2507.11719/) (2025)

## Abstract (arXiv)

We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, exploiting a constrained B-spline approximation. As a byproduct of our approach, we are also able to derive faster routines for previous work on PCA for distributions. By means of simulation studies, we compare our approaches to previously proposed methods, showing that our projected PCA has similar performance for a fraction of the computational cost and that the projected regression is extremely flexible even under misspecification. Several theoretical properties of the models are investigated and asymptotic consistency is proven. Two real world applications to Covid-19 mortality in the US and wind speed forecasting are discussed.