Related papers: Coupling Theory, Optimal Transport, and Strassen's…
We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space $(\Omega,\mathcal{F})$, we consider pairs $(E,\mathcal{G})$ where $E$ is…
The dual representation of the martingale optimal transport problem in the Skorokhod space of multi dimensional cadlag processes is proved. The dual is a minimization problem with constraints involving stochastic integrals and is similar to…
We solve constrained optimal transport problems in which the marginal laws are given by the laws of solutions of stochastic differential equations (SDEs). We consider SDEs with irregular coefficients, making only minimal regularity…
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly…
Optimal transport is a powerful framework for computing distances between probability distributions. We unify the two main approaches to optimal transport, namely Monge-Kantorovitch and Sinkhorn-Cuturi, into what we define as Tsallis…
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.…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
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…
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…
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on…
Optimal transport maps define a one-to-one correspondence between probability distributions, and as such have grown popular for machine learning applications. However, these maps are generally defined on empirical observations and cannot be…
Linear transfers between probability distributions were introduced in [5,6] in order to extend the theory of optimal mass transportation while preserving the important duality established by Kantorovich. It is shown here that $\{0,…
We introduce and study the permanence properties of the class of linear transfers between probability measures. This class contains all cost minimizing mass transports, but also martingale mass transports, the Schrodinger bridge associated…
This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows…
The theory of Monge-Kantorovich Optimal Mass Transport (OMT) has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in…
Distributionally robust optimization has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e., if an adversary…
This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of…
Let $X,Y$ be two finite sets of points having $\#X = m$ and $\#Y = n$ points with $\mu = (1/m) \sum_{i=1}^{m} \delta_{x_i}$ and $\nu = (1/n) \sum_{j=1}^{n} \delta_{y_j}$ being the associated uniform probability measures. A result of…
We derive limit distributions for certain empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the…
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and…