The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. In this paper, we focus on the Gromov-Wasserstein distance with a ground cost defined as the squared Euclidean distance and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also solution of the linear Gromov-Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov-Wasserstein distance between Gaussian distributions.
This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when $N\to\infty$, to the standard 2-Wasserstein distance W_2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.
We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\varepsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability measures on $X_n$ and endow it with a discrete Wasserstein distance $W_n$ as introduced independently by Chow et al, Maas, and Mielke for general finite Markov chains. We show that as long as $\varepsilon= \varepsilon_n$ decays towards zero slower than an explicit rate depending …
We construct a coupling between the random walk composed of Lévy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaus…
We develop a new method for showing that a given sequence of random variables verifies an appropriate law of the iterated logarithm. Our tools involve the use of general estimates on multidimensional Wasserstein distances, that are in turn based on recently developed inequalities involving Stein matrices and transport distances. Our main application consists in the proof of the exact law of the iterated logarithm for the Hermite variations of a fractional Brownian motion in the critical case.
We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov's original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov's panc…
The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures…
In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In particular, we give a detailed proof that the Lott-Villani-Sturm notion of generalized Ricci bounds agree with the classical notion on smooth manifolds. We also give the proof that generalized Ricci bounds are preserved under Gromov-Hausdorff convergence. In particular, we examine in detail the space of probability measures over the interval, $P(X)$ equipped wit…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sink…
We study the Wasserstein distance $W_2$ for Gaussian samples. We establish the exact rate of convergence $\sqrt{\log\log n/n}$ of the expected value of the $W_2$ distance between the empirical and true $c.d.f.$'s for the normal distribution. We also show that the rate of weak convergence is unexpectedly $1/\sqrt{n}$ in the case of two correlated Gaussian samples.
