A Geometric Study of Wasserstein Spaces: Ultrametrics

Abstract

Abstract. -We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of ℓ 1 . As a consequence, it is connected by 1 p -Hölder arcs, but any α-Hölder arc with α > 1 p must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when X is ultrametric; however thanks to the Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate that needed a strong rectifiability assumption.The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of regular ultrametric spaces, i.e. spaces of the form {1, . . . , k} N with a natural ultrametric.We are also lead to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

Inference for Empirical Wasserstein Distances on Finite Spaces

“…This extends a previous result of Samworth and Johnson (2004) who considered a finite number of point masses on the real line. The Wasserstein distance on trees has, to the best of our knowledge, only been considered in two papers: Kloeckner (2013) studied the geometric properties of the Wasserstein space of measures on a tree and Evans and Matsen (2012) used the Wasserstein distance on phylogenetic trees to compare microbial communities.…”

On isometric embeddings of Wasserstein spaces – the discrete case

“…The most important developments in our considerations have been done by Bertrand and Kloeckner in connection with the Wasserstein metric [2,3,12]. Besides that the Wasserstein distance metrises the weak convergence of probability measures, its importance also lies in its role in geometric investigations of metric spaces, for more information see [14,17,18,19,20] and the references therein.…”

“…We highlight also Kloeckner's results on the isometry group of W 2 (R n ), in particular, isometries of W 2 (R n ) are usually not induced by only one mapping of R n -unlike in the aforementioned results. Moreover, it turned out, which is one of the main results of [12], that W 2 (R) admits exotic isometries that does not even preserve the shape of measures. This result is pretty uncommon and it raises several questions, see Section 8 in [12], and also the last section of this paper.…”

“…Moreover, it turned out, which is one of the main results of [12], that W 2 (R) admits exotic isometries that does not even preserve the shape of measures. This result is pretty uncommon and it raises several questions, see Section 8 in [12], and also the last section of this paper. One can imagine that things could become even more complicated when one drops the surjectivity condition and tries to characterize general isometric embeddings.…”

“…For instance, in [2,4,9,10] and [16] it turned out -as a consequence of the main theorems -that every isometry is automatically affine, i.e. preserves convex combinations of measures; and in [12] that every isometry of W 2 (R) preserves geodesically convex combinations of measures. Here we shall see that for a typical non-surjective isometric embedding of W p (X ) this is not true anymore.…”