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In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In…
We consider the global well-posedness of weak energy conservative solution to a general quasilinear wave equation through variational principle, where the solution may form finite time cusp singularity, when energy concentrates. As a main…
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators…
In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical…
We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and…
This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural…
Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm.…
Optimal transport has emerged as a fundamental methodology with applications spanning multiple research areas in recent years. However, the convergence rate of the empirical estimator to its population counterpart suffers from the curse of…
We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived…
Adversarial examples have pointed out Deep Neural Networks vulnerability to small local noise. It has been shown that constraining their Lipschitz constant should enhance robustness, but make them harder to learn with classical loss…
Optimal transport has found numerous applications across data science, many of which require differentiating the optimal transport map with respect to the underlying probability densities in the Fr\'echet sense. In this work, we show that…
We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information…
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean…
This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger…
Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach…
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the…
We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the…
We develop a general approach to prove global regularity estimates for quadratic optimal transport using the entropic regularisation of the problem and the Prekopa-Leindler inequality.
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…