Georgii Riabov
A representation for the Kantorovich--Rubinstein distance between probability measures on an abstract Wiener space in terms of the extended stochastic integral (or, divergence) operator is obtained.
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We prove a version for random measures of the celebrated Kantorovich-Rubinstein duality theorem and we give an application to the coupling of random variables which extends and unifies known results.
This work is devoted to studying complex dynamical systems under non-Gaussian fluctuations. We first estimate the Kantorovich-Rubinstein distance for solutions of non-local Fokker-Planck equations associated with stochastic differential equations with non-Gaussian Levy noise. This is then applied to establish weak convergence of the corresponding probability distributions. Furthermore, this leads to smooth approximation for non-local Fokker-Planck equations, as illustrated in an example.
Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich
本文研究了广义Kantorovich-Rubinstein对偶性，展示了Levy-Prokhorov距离和凸集分布标准度量在该对偶性下的应用。
Local Convergence near Equilibria for Distribution-Dependent SDEs in a Generalized Kantorovich--Rubinstein Metric
本文通过线性化非线性 Markov 半群，将分布依赖 SDE 解在平稳分布附近的局部指数收敛归结为线性算子半群的收敛，并给出稳定性判据与颗粒介质方程实例。
On the Use of the Kantorovich-Rubinstein Distance for Dimensionality Reduction
本文研究利用 Kantorovich-Rubinstein 距离作为分类样本复杂度描述符，证明类间该距离较大时存在性能良好的 1-Lipschitz 分类器，并讨论其作为描述符的局限性。
A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error …
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N}{f(x_k)} \right| \leq \left\| \nabla f \right\|_{L^{\infty}} \cdot W_1\left( \frac{1}{N} \sum_{k=1}^{N}{δ_{x_k}} , dx\right),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Mover's) Distance. We prove another such inequality with a smaller norm on $\nabla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e…
Relative Optimal Transport
本文提出相对最优传输理论，基于特定子集作为质量库，比较不同总变分的测度。研究了相对Kantorovich-Rubinstein范数、Wasserstein距离及对偶性，并证明了Riesz-Markov-Kakutani定理的相对版本。
In this paper, we apply the Stein's method in the context of point processes, namely when the target measure is the distribution of a finite Poisson point process. We show that the so-called Kantorovich-Rubinstein distance between such a measure and another finite point process is bounded by the $L^1$-distance between their respective Papangelou intensities. Then, we deduce some convergence rates for sequences of point processes approaching a Poisson or a Cox point process.
We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the in-degree and out-degree is finite, we show that the distance grows logarithmically in the size of the graph. In contrast with the undirected case, this can happen even when the variance of the degrees is infinite. The main tool in the analysis is a new coupling between a breadth-first graph exploration process and a suitable branching process based on the Kantoro…
Sharp inequalities between Zolotarev and Wasserstein distances in $\mathrm{P}_2(\mathbb{R}^d)$
基于新的Kantorovich-Rubinstein对偶原理，本文将将将Rio不等式最优扩展至任意维度，并为具有指定方差的中心化概率下Zolotarev距离与Wasserstein距离的比值提出最优上界。
In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,μ)$, where $H$ is Cameron-Martin space and $μ$ is the Gaussian measure. We study the regularity of optimal transport maps with a quadratic cost function assuming that both initial and target measures have a strictly positive Radon-Nikodym density with respect to $μ$. Under conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivative of so-called Monge-Brenier maps, or Monge potentials. We show Sobolev regularity of the…
Estimates of transport distance in the central limit theorem
本文研究了中心极限定理中运输距离的估计，提出了更精确的边界值结果，并扩展到多变量情况。
Disintegration theorem for multifunctions, with applications to empirical Wasserstein distances and average-case statistical bounds
本文将经典分解定理推广到多值函数情形，提出渐近可分解性概念，并据此在Wasserstein空间中改进大数定律的高概率量化速率，同时给出仅依赖局部Lipschitz常数经验平均的平均情形统计误差界。
Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems
本文提出一种运输-熵框架，用于研究无限积空间上的高斯集中性不等式，揭示了运输成本与熵之间的关系，并在有限体积内证明了积分概率度量与耦合函数的等价性。
We prove that the linear heat flow in a RCD(K,\infty) metric measure space (X,d,m) satisfies a contraction property with respect to every L^p-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a precise estimate for the optimal W_\infty-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the RCD(K,\infty) lower Ricci bound and the corresponding Bakry-Émery condition for the canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension to the non-smooth metric measur…
Estimates of the total variation distance between laws of Sobolev mappings on Gaussian spaces
研究高斯空间上\(\mathbb{R}^k\)值Sobolev映射律间总变差距离，利用Malliavin行列式小球界，通过Kantorovich - Rubinstein距离等估计，基于分数正则性估计证明，为特定随机向量分布提供新估计及更高指数。
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