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1. Quantifying the Empirical Wasserstein Distance to a Set of Measures: Beating the Curse of Dimensionality

   Si, Nian; Blanchet, Jose; Ghosh, Soumyadip; Squillante, Mark (April 2020, Quantifying the Empirical Wasserstein Distance to a Set of Measures: Beating the Curse of Dimensionality)

   null (Ed.)

   We consider the problem of estimating the Wasserstein distance between the empirical measure and a set of probability measures whose expectations over a class of functions (hypothesis class) are constrained. If this class is sufficiently rich to characterize a particular distribution (e.g., all Lipschitz functions), then our formulation recovers the Wasserstein distance to such a distribution. We establish a strong duality result that generalizes the celebrated Kantorovich-Rubinstein duality. We also show that our formulation can be used to beat the curse of dimensionality, which is well known to affect the rates of statistical convergence of the empirical Wasserstein distance. In particular, examples of infinite-dimensional hypothesis classes are presented, informed by a complex correlation structure, for which it is shown that the empirical Wasserstein distance to such classes converges to zero at the standard parametric rate. Our formulation provides insights that help clarify why, despite the curse of dimensionality, the Wasserstein distance enjoys favorable empirical performance across a wide range of statistical applications. 

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2. Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

   https://doi.org/10.1007/s00205-021-01631-w  

   Esposito, Antonio; Patacchini, Francesco S.; Schlichting, André; Slepčev, Dejan (May 2021, Archive for Rational Mechanics and Analysis)

   null (Ed.)

   Abstract We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL $$^2$$ 2 IE). We develop the existence theory for the solutions of the NL $$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL $$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result. 

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3. Nonlocal Wasserstein distance: metric and asymptotic properties

   https://doi.org/10.1007/s00526-023-02576-6  

   Slepčev, Dejan; Warren, Andrew (September 2023, Calculus of Variations and Partial Differential Equations)

   Abstract The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. Here we consider a fundamental modification of the framework where the paths are solutions of nonlocal (jump) continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We characterize the basic properties of the distance and obtain sharp conditions on the (jump) kernel specifying the nonlocal transport that determine whether the topology metrized is the weak or the strong topology. A key result of the paper are the quantitative comparisons between the nonlocal and local Wasserstein distance. 

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4. Accelerated convergence in Hit-and-Run Monte Carlo and a coordinate-free randomized Kaczmarz algorithm

   https://doi.org/10.1214/25-EJP1418  

   Bou-Rabee, Nawaf; Eberle, Andreas; Oberdörster, Stefan (October 2025, Electronic Journal of Probability)

   Hit-and-Run is a coordinate-free Gibbs sampler, yet the quantitative advantages of its coordinate-free property remain largely unexplored beyond empirical studies. In this paper, we prove sharp estimates for the Wasserstein contraction of Hit-and-Run in Gaussian target measures via coupling methods and conclude mixing time bounds. Our results uncover accelerated convergence rates in certain settings. Furthermore, we extend these insights to a coordinate-free variant of the randomized Kaczmarz algorithm, an iterative method for linear systems, and demonstrate analogous convergence rates. These findings offer new insights into the advantages and limitations of coordinate-free methods for both sampling and optimization. 

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5. A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

   https://doi.org/10.1007/s10915-025-02816-1  

   Wang, Zhongjian; Xin, Jack; Zhang, Zhiwen (March 2025, Journal of Scientific Computing)

   Abstract We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three-dimensional (3D) space. In our algorithm, the KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by a spectral method. Instead of using heat kernels that cause history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green’s function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high-gradient part of solutions. Despite the lack of analytical knowledge (such as a self-similar ansatz) of a blowup, the SIPF algorithm provides a low-cost approach to studying the emergence of finite-time blowup in 3D space using only dozens of Fourier modes and by varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and the formation of a finite time singularity with ease. 

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