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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…
We study estimation problems in safety-critical applications with streaming data. Since estimation problems can be posed as optimization problems in the probability space, we devise a stochastic projected Wasserstein gradient flow that…
In this paper, we apply the framework of optimal transport to the formulation of optimal design problems. By considering the Wasserstein space as a set of design variables, we associate each probability measure with a shape configuration of…
This paper discusses a class of combinatorial optimization problems with uncertain costs in the objective function. It is assumed that a sample of the cost realizations is available, which defines an empirical probability distribution for…
Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year…
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or…
In 2012, Pflug and Pichler proved, under regularity assumptions, that the value function in Multistage Stochastic Programming (MSP) is Lipschitz continuous w.r.t. the Nested Distance, which is a distance between scenario trees (or discrete…
While the existing stochastic control theory is well equipped to handle dynamical systems with stochastic uncertainties, a paradigm shift using distance measure based decision making is required for the effective further exploration of the…
Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional…
Optimal transport and the Wasserstein distance $\mathcal{W}_p$ have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely…
The adapted Wasserstein distance controls the calibration errors of optimal values in various stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. However, statistical aspects of the adapted…
Wasserstein distributionally robust optimization offers a framework for model fitting in machine learning under potential shifts in the data distribution. We study a regularized variant of this problem in which entropic smoothing produces a…
We propose moment relaxations for data-driven Wasserstein distributionally robust optimization problems. Conditions are identified to ensure asymptotic consistency of such relaxations for both single-stage and two-stage problems, together…
We consider a distributionally robust second-order stochastic dominance constrained optimization problem. We require the dominance constraints hold with respect to all probability distributions in a Wasserstein ball centered at the…
Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately,…
This paper investigates the robust optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity. We consider a class of discrete-time optimal control problems where the controller \emph{jointly}…
Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely…
Personalized recommender systems are playing an increasingly important role as more content and services become available and users struggle to identify what might interest them. Although matrix factorization and deep learning based methods…
Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance--an easily computable variant of the Wasserstein distance.…
Computing the infinity Wasserstein distance and retrieving projections of a probability measure onto a closed subset of probability measures are critical sub-problems in various applied fields. However, the practical applicability of these…