Related papers: Statistical Aspects of Wasserstein Distances
Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…
This brief note aims to introduce the recent paradigm of distributional robustness in the field of shape and topology optimization. Acknowledging that the probability law of uncertain physical data is rarely known beyond a rough…
Understanding the space of probability measures on a metric space equipped with a Wasserstein distance is one of the fundamental questions in mathematical analysis. The Wasserstein metric has received a lot of attention in the machine…
The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked…
We consider a data-driven robust hypothesis test where the optimal test will minimize the worst-case performance regarding distributions that are close to the empirical distributions with respect to the Wasserstein distance. This leads to a…
The adapted Wasserstein distance is a metric for quantifying distributional uncertainty and assessing the sensitivity of stochastic optimization problems on time series data. A computationally efficient alternative to it, is provided by the…
Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different…
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…
We study the problem of quantifying how far an empirical distribution deviates from Gaussianity under the framework of optimal transport. By exploiting the cone geometry of the relative translation invariant quadratic Wasserstein space, we…
We formulate and solve a regression problem with time-stamped distributional data. Distributions are considered as points in the Wasserstein space of probability measures, metrized by the 2-Wasserstein metric, and may represent images,…
In this paper, we establish sharp upper and lower bounds on the convergence rate of the empirical measures of point processes under the Wasserstein distance. To this end, we first introduce a new metric on the space of counting measures…
Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional…
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants…
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…
We study the problem of estimating a sequence of evolving probability distributions from historical data, where the underlying distribution changes over time in a nonstationary and nonparametric manner. To capture gradual changes, we…
We propose a fundamental metric for measuring the distance between two distributions. This metric, referred to as the decision-focused (DF) divergence, is tailored to stochastic linear optimization problems in which the objective…
Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from…
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…
Distribution data refers to a data set where each sample is represented as a probability distribution, a subject area receiving burgeoning interest in the field of statistics. Although several studies have developed…
The sliced Wasserstein metric compares probability measures on $\mathbb{R}^d$ by taking averages of the Wasserstein distances between projections of the measures to lines. The distance has found a range of applications in statistics and…