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This paper focuses on a similarity measure, known as the Wasserstein distance, with which to compare images. The Wasserstein distance results from a partial differential equation (PDE) formulation of Monge's optimal transport problem. We…
We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates…
The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this…
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…
Issued from Optimal Transport, the Wasserstein distance has gained importance in Machine Learning due to its appealing geometrical properties and the increasing availability of efficient approximations. In this work, we consider the problem…
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on…
This paper presents a probabilistic generalization of the Generalized Optimal Sub-Pattern Assignment (GOSPA) metric, termed P-GOSPA. The GOSPA metric has been widely used to evaluate the distance between finite sets, particularly in…
We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal…
A novel framework for density estimation under expectation constraints is proposed. The framework minimizes the Wasserstein distance between the estimated density and a prior, subject to the constraints that the expected value of a set of…
This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…
In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\varrho$. The system's response $f$ pushes forward $\varrho$ to a new measure $f\circ \varrho$ which we…
Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures $\mu$ and $\nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $\mu$ and $\nu$, which we call…
Optimal transportation theory and the related $p$-Wasserstein distance ($W_p$, $p\geq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools has some issues. For instance, it…
This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on…
In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular,…
We develop a general framework for statistical inference with the 1-Wasserstein distance. Recently, the Wasserstein distance has attracted considerable attention and has been widely applied to various machine learning tasks because of its…
A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to…
Leveraging the Wasserstein distance -- a summation of sample-wise transport distances in data space -- is advantageous in many applications for measuring support differences between two underlying density functions. However, when supports…
The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'{e}chet or…