Related papers: DOTmark - A Benchmark for Discrete Optimal Transpo…
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…
We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in…
The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and…
We introduce a new technique, which we call the boundary method, for solving semi-discrete optimal transport problems with a wide range of cost functions. The boundary method reduces the effective dimension of the problem, thus improving…
A common way to quantify the ,,distance'' between measures is via their discrepancy, also known as maximum mean discrepancy (MMD). Discrepancies are related to Sinkhorn divergences $S_\varepsilon$ with appropriate cost functions as…
This paper proposes an efficient numerical optimization approach for solving dynamic optimal transport (DOT) problems on general smooth surfaces, computing both the quadratic Wasserstein distance and the associated transportation path.…
Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent…
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…
Illumination estimation remains a pivotal challenge in image processing, particularly for robotics, where robust environmental perception is essential under varying lighting conditions. Traditional approaches, such as RGB histograms and…
As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this…
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…
With the increasing availability of data objects in the form of probability distributions, there is a growing need for statistical methods tailored to distributional data. Distance measures, especially the pairwise distance matrix between…
In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as…
Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…
Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss…
We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in…
Optimal transport has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. Recently, Bredies et al. (ESAIM:M2AN 54:2351-2382, 2020) and Schmitzer et al. (IEEE T MED…
This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_{2}$). A general form of the Fr\'echet gradient is systematically derived by optimal transportation (OT)…
Classical optimal transport problem seeks a transportation map that preserves the total mass betwenn two probability distributions, requiring their mass to be the same. This may be too restrictive in certain applications such as color or…
Solving large scale Optimal Transport (OT) in machine learning typically relies on sampling measures to obtain a tractable discrete problem. While the discrete solver's accuracy is controllable, the rate of convergence of the discretization…