Related papers: Gaussian Optimal Transport Beyond Brenier's Theore…
We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Swiech (1998). We prove results on the existence, uniqueness and…
By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem…
We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may flow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to…
In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients…
We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified…
A generalized unbalanced optimal transport distance ${\rm WB}_{\Lambda}$ on matrix-valued measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] \`{a} la Benamou-Brenier, which extends the Kantorovich-Bures and the…
The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when…
We introduce the Gaussian transform (GT), an optimal transport inspired iterative method for denoising and enhancing latent structures in datasets. Under the hood, GT generates a new distance function (GT distance) on a given dataset by…
The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:X\times…
We propose Gaussian optimal transport for Image style transfer in an Encoder/Decoder framework. Optimal transport for Gaussian measures has closed forms Monge mappings from source to target distributions. Moreover interpolates between a…
We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target…
Given two n-dimensional measures $\mu$ and $\nu$ on Polish spaces, we propose an optimal transportation's formulation, inspired by classical Kan-torovitch's formulation in the scalar case. In particular, we established a strong duality…
We completely characterise the optimal solutions for the three-marginal optimal transport problem - introduced in [K. Bolbotowski, G. Bouchitt\'e, Kantorovich-Rubinstein duality theory for the Hessian, 2024, preprint], and whose relaxation…
We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $\Lambda$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures…
We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace $X(\Omega)$ of first order distribution. A particular…
In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous.…
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations…
We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich…