Annual Review of Statistics and Its Application· 2018Q1· Review
Statistical Aspects of Wasserstein Distances
- 802citations
- Q1SCImago
- 2018year
Short summary
Wasserstein distances, inspired by optimal mass transportation, offer a versatile statistical tool for measuring differences between probability distributions by incorporating domain geometry and quantifying perturbations.
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Key points
- Wasserstein distances measure the minimal effort to transform one probability distribution into another.
- They naturally incorporate the geometry of the domain of the distributions.
- They are useful for quantifying perturbations and bounding distributions.
- Wasserstein distances have applications in statistical theory, inferential methodology, and are becoming objects of inference themselves.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Annual Review of Statistics and Its Application, 2018 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics