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Wasserstein distances provide a powerful framework for comparing data distributions. They can be used to analyze processes over time or to detect inhomogeneities within data. However, simply calculating the Wasserstein distance or analyzing…
We propose a series of metrics between pairs of signals, linear systems or rational spectra, based on optimal transport and linear-systems theory. The metrics operate on the locations of the poles of rational functions and admit very…
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…
Cross-domain alignment between two sets of entities (e.g., objects in an image, words in a sentence) is fundamental to both computer vision and natural language processing. Existing methods mainly focus on designing advanced attention…
Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few.…
Querying uncertain data sets (represented as probability distributions) presents many challenges due to the large amount of data involved and the difficulties comparing uncertainty between distributions. The Earth Mover's Distance (EMD) has…
We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
We propose a new algorithm to approximate the Earth Mover's distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar $L_1$ type minimization. We use a regularization which…
We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous…
As a fundamental problem of natural language processing, it is important to measure the distance between different documents. Among the existing methods, the Word Mover's Distance (WMD) has shown remarkable success in document semantic…
Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem…
Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability…
We propose a methodology for intercomparing climate models and evaluating their performance against benchmarks based on the use of the Wasserstein distance (WD). This distance provides a rigorous way to measure quantitatively the difference…
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…
The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. This work proposes a new canonical interpretation of this distance and provides tools to perform inference on…
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…
We address the estimation problem for general finite mixture models, with a particular focus on the elliptical mixture models (EMMs). Compared to the widely adopted Kullback-Leibler divergence, we show that the Wasserstein distance provides…