Related papers: Box distance and observable distance via optimal t…
Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several…
A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial…
In 1999, M. Gromov introduced the box distance function $\sikaku$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding (cf. \cite[Lemma 5.10]{Colding}), we estimate $\sikaku(\mathbb{S}^n,\mathbb{S}^m)$ and…
One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and…
In this article, we define the transport dimension of probability measures on $\mathbb{R}^m$ using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski…
We lay out the phenomenological behavior of event-shape observables evaluated by solving optimal transport problems between collider events and reference geometries -- which we name 'manifold distances' -- to provide guidance regarding…
Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure…
Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between…
We generalize the box and observable distances to those between metric measure spaces with group actions, and prove some fundamental properties. As an application, we obtain an example of a sequence of lens spaces with unbounded dimension…
We give a detailed proof to Gromov's statement that precompact sets of metric measure spaces are bounded with respect to the box distance and the Lipschitz order.
It is shown that for any two compact metric spaces there exists an "optimal" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric…
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels,…
In this paper, we investigate compact ultrametric measure spaces which form a subset $\mathcal{U}^w$ of the collection of all metric measure spaces $\mathcal{M}^w$. Similar as for the ultrametric Gromov-Hausdorff distance on the collection…
A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff…
We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization…
We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by…
We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last…
Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from…
This paper is devoted to the study of the Monge-Kantorovich theory of optimal mass transport and its applications, in the special case of one-dimensional and circular distributions. More precisely, we study the Monge-Kantorovich distances…
Optimal transport is a geometrically intuitive, robust and flexible metric for sample comparison in data analysis and machine learning. Its formal Riemannian structure allows for a local linearization via a tangent space approximation. This…