Related papers: Optimal transport of measures via autonomous vecto…
The ability to measure similarity between documents enables intelligent summarization and analysis of large corpora. Past distances between documents suffer from either an inability to incorporate semantic similarities between words or from…
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…
Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…
In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval,…
During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott-Sturm-Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in…
Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 approximate…
We study first- and second-order linear transport equations, as well as ODE and SDE flows, with velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In…
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are…
In the classical Monge-Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Thus, it does not allow for congestion effects. Using the…
We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb R^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption…
Mode-based model-reduction is used to reduce the degrees of freedom of high dimensional systems, often by describing the system state by a linear combination of spatial modes. Transport dominated phenomena, ubiquitous in technical and…
Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g. decision…
The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr\"odinger map. We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$…
We generalize known results on transport equations associated to a Lipschitz field $\mathbf{F}$ on some subspace of $\mathbb{R}^N$ endowed with some general space measure $\mu$. We provide a new definition of both the transport operator and…
In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the…
Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is…
This study examines the time complexities of the unbalanced optimal transport problems from an algorithmic perspective for the first time. We reveal which problems in unbalanced optimal transport can/cannot be solved efficiently.…
We present a 2-step optimal transport approach that performs a mapping from a source distribution to a target distribution. Here, the target has the particularity to present new classes not present in the source domain. The first step of…
We show that in a locally compact complete $CAT(0)$ space satisfying positive angles property and a disintegration regularity for its canonical Hausdorff measure, there exists a unique optimal transport map that push-forwards a given…
We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization…