Related papers: Bounding adapted Wasserstein metrics
The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as…
Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities…
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,…
Optimization over the space of probability measures endowed with the Wasserstein-2 geometry is central to modern machine learning and mean-field modeling. However, traditional methods relying on full Wasserstein gradients often suffer from…
Since the introduction of the Sliced Wasserstein distance in the literature, its simplicity and efficiency have made it one of the most interesting surrogate for the Wasserstein distance in image processing and machine learning. However,…
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear…
This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative…
In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
\v{C}ech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $OT_{p}$; however, the extent to which PDs are stable…
It was shown in [6] that the Wasserstein distance is equivalent to the Mean Optimal Sub-Pattern Assignment (MOSPA) measure for empirical probability density functions. A more recent paper [7], extends on it by drawing new connections…
In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We…
Comparing probability distributions is at the crux of many machine learning algorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are two classes of distances between probability distributions that have attracted abundant…
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…
While theoretically appealing, the application of the Wasserstein distance to large-scale machine learning problems has been hampered by its prohibitive computational cost. The sliced Wasserstein distance and its variants improve the…
Random measures provide flexible parameters for Bayesian nonparametric models. Given two different priors for a random measure, we develop a natural framework to investigate the rate at which the corresponding posteriors merge, as the…
Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein…
Comparing two probability measures supported on heterogeneous spaces is an increasingly important problem in machine learning. Such problems arise when comparing for instance two populations of biological cells, each described with its own…
Information is an inherent component of stochastic processes and to measure the distance between different stochastic processes it is not sufficient to consider the distance between their laws. Instead, the information which accumulates…
We investigate here the optimal transportation problem on configuration space for the quadratic cost. It is shown that, as usual, provided that the corresponding Wasserstein is finite, there exists one unique optimal measure and that this…