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In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as…
We study optimal transport between two high-dimensional distributions $\mu,\nu$ in $R^n$ from an algorithmic perspective: given $x \sim \mu$, find a close $y \sim \nu$ in $poly(n)$ time, where $n$ is the dimension of $x,y$. Thus, running…
We propose and axiomatize preferences on a product state space in light of uncertainty regarding the dependency of different payoff-relevant factors. Dependence structures allow to decompose probabilities and allow to pin down behavior…
In Bayesian applications, there is a huge interest in rapid and accurate estimation of the posterior distribution, particularly for high dimensional or hierarchical models. In this article, we propose to use optimization to solve for a…
We study the problem of estimating a function $T$ given independent samples from a distribution $P$ and from the pushforward distribution $T_\sharp P$. This setting is motivated by applications in the sciences, where $T$ represents the…
In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified…
We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure $\lambda$ and an invariant random measure $\mu$ of unit intensity to be finite. We show that for \emph{any} such random measure the…
Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's $\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing…
We establish the validity of asymptotic limits for the general transportation problem between random i.i.d. points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three.…
We consider Kac's random walk on $n$-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on $\mathit{SO}(n)$…
Temperature-dependent transport data, including diffusion coefficients and ionic conductivities, are routinely analysed by fitting empirical models such as the Arrhenius equation. These fitted models yield parameters such as the activation…
We propose three measures of mutual dependence between multiple random vectors. All the measures are zero if and only if the random vectors are mutually independent. The first measure generalizes distance covariance from pairwise dependence…
Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding…
Optimal transport has gained significant attention in recent years due to its effectiveness in deep learning and computer vision. Its descendant metric, the Wasserstein distance, has been particularly successful in measuring distribution…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact support, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $\mu$ and…
We develop a scheme for the computation of the full-counting statistics of transport described by Markovian master equations with an arbitrary time dependence. It is based on a hierarchy of generalized density operators, where the trace of…
The theory of optimal transportation has developed into a powerful and elegant framework for comparing probability distributions, with wide-ranging applications in all areas of science. The fundamental idea of analyzing probabilities by…
For a random variable $X$, we are interested in the blind extraction of its finest mutual independence pattern $\mu ( X )$. We introduce a specific kind of independence that we call dichotomic. If $\Delta ( X )$ stands for the set of all…
Detecting dependence between variables is a crucial issue in statistical science. In this paper, we propose a novel metric called label projection correlation to measure the dependence between numerical and categorical variables. The…