A Geometric Study of Wasserstein Spaces: Hadamard Spaces

Bertrand, Jérôme; Kloeckner, Benoît

doi:10.1142/s1793525312500227

Cited by 23 publications

(20 citation statements)

References 18 publications

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“…For Corollary 1.2, recall the fact that a convex subset of ℓ 1 is geodesic (thus connected by Lipschitz arcs) and that for any metric space (Y, d), d β defines a distance without any non-constant Lipschitz curve whenever β ∈ (0, 1) (which is folklore, see e.g. Lemma 5.4 in [BK12], for a simple proof).…”

Section: ℓ 1 Coordinatesmentioning

confidence: 99%

“…In a series of papers we try to understand what kind of geometric information on Wp (X) can be obtained from given geometric information on X. We considered for example isometry groups and embeddability questions when X is a Euclidean space [Klo10] or, with Jérôme Bertrand, a Hadamard space [BK12], and the size of Wp (X) when X is (close to be) a compact manifold [Klo12].…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Study of Wasserstein Spaces: Ultrametrics

Kloeckner

2014

Mathematika

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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.

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Section: ℓ 1 Coordinatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Geometric Study of Wasserstein Spaces: Ultrametrics

Kloeckner

2014

Mathematika

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“…We recall that these measures are probability measures on the cone c∂X over the geodesic boundary of X. But by Proposition 5.2 of [BK12], these measures are in fact concentrated on ∂X, viewed as a subset of c∂X. In particular, W 2 (X) is already far from satisfying the visibility condition.…”

Section: A First Necessary Condition: Antipodalitymentioning

confidence: 99%

“…The asymptotic formula (Theorem 4.2 of [BK12]) gives us a first necessary condition valid for any Hadamard space. Let us say that two points ζ, ξ ∈ ∂X are antipodal if they are linked by a geodesic, that two sets A − , A + ⊂ ∂X are antipodal if all pairs (ζ, ξ) ∈ A − × A + are antipodal, and that two measures ν − , ν + on ∂X are antipodal when they are concentrated on antipodal sets.…”

Section: A First Necessary Condition: Antipodalitymentioning

confidence: 99%

A Geometric Study of Wasserstein Spaces: An Addendum on the Boundary

Bertrand

Kloeckner

2013

Lecture Notes in Computer Science

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We extend the geometric study of the Wasserstein space W2(X) of a simply connected, negatively curved metric space X by investigating which pairs of boundary points can be linked by a geodesic, when X is a tree.Let X be a Hadamard space, by which we mean that X is a complete globally CAT(0), locally compact metric space. Mainly, X is a space where triangles are "thin": points on the opposite side to a vertex are closer to the vertex than they would be in the Euclidean plane. This assumption can also be interpreted as X having non-positive curvature, in a setting more general than manifolds; it has a lot of consequences (the distance is convex, X is contractible, it admits a natural boundary and an associated compactification, . . . ) An important example of Hadamard space, on which we shall focus in this paper, is simply an infinite tree.The set of Borel probability measures of X having finite second moment can be endowed with a natural distance defined using optimal transportation, giving birth to the Wasserstein space W 2 (X). It is well-known that W 2 (X) does not have non-positive curvature even when X is a tree.This note is an addendum to [BK12], where we defined and studied the boundary of W 2 (X). We refer to that article and references therein for the background both on Hadamard space and optimal transportation, as well as for notations. Note that a previous (long) version of [BK12] contained the present content, but has been split after remarks of a referee.Let us quickly sum up the content of [BK12]. The boundary of X can be defined by looking at geodesic rays, and identifying rays that stay at bounded distance one to another ("asymptote" relation). We showed that there is a natural boundary ∂ W 2 (X) of the Wasserstein space that is both close to the traditional boundary of Hadamard spaces (a boundary point can be defined as an asymptote class of rays) and relevant to optimal transportation (a boundary point can be seen as a measure on the cone over ∂X, encoding the asymptotic direction and speed distribution of the mass along a ray). This boundary can

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“…It is known that the metric space (X , ρ) is a R-tree space (see Kirk 2007, p 197), and R-tree spaces, as mentioned above, are Hadamard spaces (e.g. Bertrand and Kloeckner 2012). For x = 0 and t ∈ R ++ , letB(x, t) be the open ball with center x and radius t in the radial metric, and B(x, t) the Euclidean open ball with the same center and radius.…”

mentioning

confidence: 99%