Related papers: Quantization and martingale couplings
Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one,…
We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate transitions according to a martingale…
Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability…
Given two probability measures $\mu$ and $\nu$ in "convex order" on $\R^d$, we study the profile of one-step martingale plans $\pi$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all…
Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…
We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex…
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various…
We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we…
Coupling probability measures lies at the core of many problems in statistics and machine learning, from domain adaptation to transfer learning and causal inference. Yet, even when restricted to deterministic transports, such couplings are…
Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using…
It is well known that martingale transport plans between marginals $\mu\neq\nu$ are never given by Monge maps -- with the understanding that the map is over the first marginal $\mu$, or forward in time. Here, we change the perspective, with…
A classical result of Strassen asserts that given probabilities $\mu, \nu$ on the real line which are in convex order, there exists a \emph{martingale coupling} with these marginals, i.e.\ a random vector $(X_1,X_2)$ such that $X_1\sim \mu,…
Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment…
We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their…
A logconcave likelihood is as important to proper statistical inference as a convex cost function is important to variational optimization. Quantization is often disregarded when writing likelihood models, ignoring the limitations of the…
Approximating a convex function by a polyhedral function that has a limited number of facets is a fundamental problem with applications in various fields, from mitigating the curse of dimensionality in optimal control to bi-level…
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…
Optimal transport (OT) is a popular tool in machine learning to compare probability measures geometrically, but it comes with substantial computational burden. Linear programming algorithms for computing OT distances scale cubically in the…
Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure.…
This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The…