相关论文: Maximizers of nonlocal interactions of Wasserstein…
The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial…
For a large class of supermodular integrands, we establish conditions under which balls are the unique (up to translations) maximizers of the Riesz-type functionals with constraints.
We consider sets in $\mathbb R^N$ which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel $g:\mathbb R^N\setminus\{0\}\to \mathbb R^+$. We establish some general existence and…
Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.
Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities…
In this paper we maximize a class of functionals under certain constraints. We find sufficient and necessary conditions for these maximizers to exist and be unique. Moreover, we characterize them and discuss the optimality of our results by…
An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space $\mathbb{R}^d$ of specified Lebesgue measures, balls centered at the origin are maximizers of certain functionals defined by…
A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among…
In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We…
Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein…
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in…
We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we…
In this paper, we study a maximization and a minimization problem associated with a Poisson boundary value problem. Optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
We propose a family of relaxations of the optimal transport problem which regularize the problem by introducing an additional minimization step over a small region around one of the underlying transporting measures. The type of…
In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics, is described by a transport equation with non-local velocities and is subject to…
Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss…
We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric…
We present a novel multiscale framework for analyzing sequences of probability measures in Wasserstein spaces over Euclidean domains. Exploiting the intrinsic geometry of optimal transport, we construct a multiscale transform applicable to…
As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this…