Related papers: Computing Kantorovich-Wasserstein Distances on $d$…

Wasserstein GANs with Gradient Penalty (WGAN-GP) are a very popular method for training generative models to produce high quality synthetic data. While WGAN-GP were initially developed to calculate the Wasserstein 1 distance between…

We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from $\tbinom{N+\ell-1}{\ell-1}$ to $\ell\cdot(N+1)$, where $\ell$ is…

In this paper we give an $\widetilde{O}((nm)^{2/3}\log C)$ time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by $C$. For planar multigraphs, this…

In this paper we provide a method of finding possible numbers of shortest paths between two points in a space of compact sets in Euclidean space with Hausdorff distance. We also prove that there cannot be some of the numbers of shortest…

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. These problems consist of minimizing a linear objective function over a high-dimensional polytope, here…

A result of Hohloch links the theory of integer partitions with the Monge formulation of the optimal transport problem, giving the optimal transport map between (Young diagrams of) integer partitions and their corresponding symmetric…

This note addresses computational difficulty of the Gromov-Wasserstein distance frequently mentioned in the literature. We provide details on the structure of the Gromov-Wasserstein distance optimization problem that show its non-convex…

This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal…

Let $X$ be a Polish space, $\mathcal{P}(X)$ be the set of Borel probability measures on $X$, and $T\colon X\to X$ be a homeomorphism. We prove that for the simplex $\mathrm{Dom} \subseteq \mathcal{P}(X)$ of all $T$-invariant measures, the…

Analyzing relationships between objects is a pivotal problem within data science. In this context, Dimensionality reduction (DR) techniques are employed to generate smaller and more manageable data representations. This paper proposes a new…

We propose a new minimum-distance estimator for linear random coefficient models. This estimator integrates the recently advanced sliced Wasserstein distance with the nearest neighbor methods, both of which enhance computational efficiency.…

Consider a finite directed graph without cycles in which the arrows are weighted. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context…

Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions -- such as the Wasserstein distance -- but also for its…

Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance…

We propose a balanced coarsening scheme for multilevel hypergraph partitioning. In addition, an initial partitioning algorithm is designed to improve the quality of k-way hypergraph partitioning. By assigning vertex weights through the LPT…

This short note gives a proof of the triangle inequality based on the Kantorovich duality formula for the Wasserstein distances of exponent $p\in[1,+\infty)$ in the case of a general Polish space. In particular it avoids the "glueing of…

By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L^p Wasserstein distances between probability measures, we define the corresponding spectral distances d_p on the set of all graphs.…

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…

The Gromov--Hausdorff distance measures the difference in shape between metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions…

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…