Statistical Aspects of Wasserstein Distances

By Victor M. Panaretos, Yoav Zemel. Annual Review of Statistics and Its Application, 2018

980 citations recorded by the source corpus.

Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catalyze core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: They can be used to derive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of perturbation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and they have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein distances and optimal transportation, and a succinct overview of some of their many statistical aspects.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/3a0d81ff7917b16156e11fe812182af82edeaa85

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

Bayesian Learning with Wasserstein Barycenters (2018, background): “This was discovered by Panaretos and Zemel in [41, 33].”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “…distances have a long history and continue to attract interest from diverse fields in statistics, machine learning, and computer science, in particular image analysis; see for instance the monographs and reviews by Santambrogio (2015), Peyré and Cuturi (2019), and Panaretos and Zemel (2019).”
Distribution-on-Distribution Regression via Optimal Transport Maps (2021, background): “In the case of probability distributions, the Wasserstein metric [13, 14] has been exhibited as a canonical choice [12], primarily because it captures deformations, which are typically the main form of variation for probability distributions.”
YOLOv10: Real-Time End-to-End Object Detection (2024, background): “The supervision gap between two branches can thus be derived by the 1-Wasserstein distance [46] of different classification objectives, i.e ., We can observe that the gap decreases as t o 2 m,i increases, i.e ., i ranks higher within Ω .”
Plugin estimation of smooth optimal transport maps (2021, background): “Optimal transport maps play a central role in the theory of optimal transport (Rachev and Rüschendorf, 1998; Villani, 2003; Santambrogio, 2015), and form an increasingly important methodology in statistics and machine learning (Kolouri et al., 2017; Panaretos and Zemel, 2019).”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, background): “Most of them are related to Wasserstein distance (see Panaretos and Zemel (2019)) and somewhat disconnected from the problems considered here.”
DSTAGNN: Dynamic Spatial-Temporal Aware Graph Neural Network for Traffic Flow Forecasting (2022, methodology): “…the dynamic spatial dependency among nodes by capturing the correlation attributes between the probability distributions of each node, using e.g. the Wasserstein Distance (Panaretos & Zemel, 2019) which measures the minimum e ff ort taken to reconfigure a probability distribution to another.”
The Wasserstein space of stochastic processes (2021, background): “[60, 14]) as well as in pure mathematics (e.”
Recovering manifold structure in LLM responses through a joint Euclidean mirror (2026, unknown): “For details see the note on the curse of dimensionality in Panaretos and Zemel (2019), Section 3.3.”