Related papers: Minimum cross-entropy distributions on Wasserstein…
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on…
For $\ell\colon \mathbb{R}^d \to [0,\infty)$ we consider the sequence of probability measures $\left(\mu_n\right)_{n \in \mathbb{N}}$, where $\mu_n$ is determined by a density that is proportional to $\exp(-n\ell)$. We allow for infinitely…
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…
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…
Given two discrete random variables $X$ and $Y$, with probability distributions ${\bf p} =(p_1, \ldots , p_n)$ and ${\bf q}=(q_1, \ldots , q_m)$, respectively, denote by ${\cal C}({\bf p}, {\bf q})$ the set of all couplings of ${\bf p}$ and…
Given two discrete random variables $X$ and $Y,$ with probability distributions ${\bf p}=(p_1, \ldots , p_n)$ and ${\bf q}=(q_1, \ldots , q_m)$, respectively, denote by ${\cal C}({\bf p}, {\bf q})$ the set of all couplings of ${\bf p}$ and…
We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic…
The subject of this paper is the estimation of a probability measure on ${\mathbb R}^d$ from data observed with an additive noise, under the Wasserstein metric of order $p$ (with $p\geq 1$). We assume that the distribution of the errors is…
We initiate a study of the following problem: Given a continuous domain $\Omega$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $\mu$ on $\Omega$, find the probability density over $\Omega$ whose…
Estimating the density of a distribution from samples is a fundamental problem in statistics. In many practical settings, the Wasserstein distance is an appropriate error metric for density estimation. For example, when estimating…
Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used $K$-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the…
This brief note aims to introduce the recent paradigm of distributional robustness in the field of shape and topology optimization. Acknowledging that the probability law of uncertain physical data is rarely known beyond a rough…
We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We…
We consider an extension of $\epsilon$-entropy to a KL-divergence based complexity measure for randomized density estimation methods. Based on this extension, we develop a general information-theoretical inequality that measures the…
The Wasserstein distance has emerged as a key metric to quantify distances between probability distributions, with applications in various fields, including machine learning, control theory, decision theory, and biological systems.…
We obtain an estimate for the expected subspace robust Wasserstein distance between any probability measure on the unit ball of a separable Hilbert space, and its empirical distribution from $n$ i.i.d. samples.
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…
We consider distributionally robust optimization problems where the uncertainty is modeled via a structured Wasserstein ambiguity set. Specifically, the ambiguity is restricted to product measures $P^{\otimes N}$, where $P$ lies within a…
The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the…
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…