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Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial…
It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the…
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
We show how to sample in parallel from a distribution $\pi$ over $\mathbb R^d$ that satisfies a log-Sobolev inequality and has a smooth log-density, by parallelizing the Langevin (resp. underdamped Langevin) algorithms. We show that our…
Integration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention…
This paper improves the state-of-the-art rate of a first-order algorithm for solving entropy regularized optimal transport. The resulting rate for approximating the optimal transport (OT) has been improved from…
This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_\sigma$, for $\mathcal{N}_\sigma\triangleq\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$, by…
Bayesian inference in high-dimensional discrete-input additive noise models is a fundamental challenge in communication systems, as the support of the required joint a posteriori probability (APP) mass function grows exponentially with the…
Computationally solving multi-marginal optimal transport (MOT) with squared Euclidean costs for $N$ discrete probability measures has recently attracted considerable attention, in part because of the correspondence of its solutions with…
Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations…
In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…
Optimal transportation theory and the related $p$-Wasserstein distance ($W_p$, $p\geq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools has some issues. For instance, it…
Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A…
Sinkhorn algorithm is the de-facto standard approximation algorithm for optimal transport, which has been applied to a variety of applications, including image processing and natural language processing. In theory, the proof of its…
The empirical optimal transport (OT) cost between two probability measures from random data is a fundamental quantity in transport based data analysis. In this work, we derive novel guarantees for its convergence rate when the involved…
We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark…
We investigate the small regularization limit of entropic optimal transport when the cost function is the Euclidean distance in dimensions $d > 1$, and the marginal measures are absolutely continuous with respect to the Lebesgue measure.…
Let $X_1, \ldots , X_n$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\|X_1\| \le \beta$. Then, we show that $\frac{1}{\sqrt{n}}(X_1 + \ldots + X_n)$ converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or…
We show convergence of the gradients of the Schr\"odinger potentials to the Brenier map in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel…