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In generative modeling, the Wasserstein distance (WD) has emerged as a useful metric to measure the discrepancy between generated and real data distributions. Unfortunately, it is challenging to approximate the WD of high-dimensional…
Optimal transport (OT) and the related Wasserstein metric (W) are powerful and ubiquitous tools for comparing distributions. However, computing pairwise Wasserstein distances rapidly becomes intractable as cohort size grows. An attractive…
This paper serves as a user's guide to sampling strategies for sliced optimal transport. We provide reminders and additional regularity results on the Sliced Wasserstein distance. We detail the construction methods, generation time…
Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and…
In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring…
We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical…
We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces…
Kernel methods underpin many of the most successful approaches in data science and statistics, and they allow representing probability measures as elements of a reproducing kernel Hilbert space without loss of information. Recently, the…
Nowadays stochastic computer simulations with both numeral and distribution inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This paper studies the design and analysis issues of such computer…
We define a modified Wasserstein distance for distribution clustering which inherits many of the properties of the Wasserstein distance but which can be estimated easily and computed quickly. The modified distance is the sum of two terms.…
Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…
This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that…
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…
Wasserstein 1 optimal transport maps provide a natural correspondence between points from two probability distributions, $\mu$ and $\nu$, which is useful in many applications. Available algorithms for computing these maps do not appear to…
Distance measures between graphs are important primitives for a variety of learning tasks. In this work, we describe an unsupervised, optimal transport based approach to define a distance between graphs. Our idea is to derive…
The practical applications of Wasserstein distances (WDs) are constrained by their sample and computational complexities. Sliced-Wasserstein distances (SWDs) provide a workaround by projecting distributions onto one-dimensional subspaces,…
The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions that has exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be…
Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals with non-smooth Riesz kernels show a rich structure as singular measures can become absolutely continuous ones and conversely. In this paper we contribute to the…
Generative Adversarial Networks (GANs) are one of the most practical methods for learning data distributions. A popular GAN formulation is based on the use of Wasserstein distance as a metric between probability distributions.…
Optimal transport (OT) distances are increasingly used as loss functions for statistical inference, notably in the learning of generative models or supervised learning. Yet, the behavior of minimum Wasserstein estimators is poorly…