Snowflake universality  of Wasserstein spaces

Andoni, Alexandr; Naor, Assaf; Neiman, Ofer

doi:10.24033/asens.2363

Snowflake universality of Wasserstein spaces
[Universalité des espaces de Wasserstein à floconnage près]

Andoni, Alexandr ; Naor, Assaf ; Neiman, Ofer

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 657-700.

Résumé
Abstract

Pour $p \in (1, \infty)$ notons $𝒫_{p} (ℝ^{3})$ l'espace métrique des mesures de probabilité $p$ -intégrables sur $ℝ^{3}$ , muni de la $p$ -métrique de Wasserstein $𝖶_{p}$ . Nous montrons que pour tout $ε > 0$ , tout $θ \in (0, 1 / p]$ et tout espace métrique fini $(X, d_{X})$ , l'espace métrique $(X, d_{X}^{θ})$ se plonge dans $𝒫_{p} (ℝ^{3})$ avec distortion au plus $1 + ε$ . Nous montrons que cela est optimal quand $p \in (1, 2]$ au sens où l'exposant $1 / p$ ne peut pas être augmenté. En fait pour $n \in ℕ$ assez grand il existe un espace métrique à $n$ points $(X_{n}, d_{n})$ tel que pour tout $α \in (1 / p, 1]$ tout plongement de l'espace métrique $(X_{n}, d_{n}^{α})$ dans $𝒫_{p} (ℝ^{3})$ a une distortion au moins égale à un multiple par une constante de ${(log n)}^{α - 1 / p}$ . Ces résultats impliquent qu'il existe un espace d'Alexandrov de courbure positive, à savoir $𝒫_{2} (ℝ^{3})$ , vis- à-vis duquel il n'existe pas de suite de graphes expanseurs de degré borné. Il en résulte aussi que $𝒫_{2} (ℝ^{3})$ n'admet pas de plongement uniforme, grossier ou quasisymétrique dans un espace de Banach de type non trivial. Nous discutons le lien avec plusieurs questions ouvertes depuis longtemps en géométrie des espaces métriques, dont la caractérisation des sous-ensembles des espaces d'Alexandrov, l'existence d'expandeurs, le problème d'universalité pour $𝒫_{2} (ℝ^{k})$ , et le problème de dichotomie pour le cotype métrique.

For $p \in (1, \infty)$ let $𝒫_{p} (ℝ^{3})$ denote the metric space of all $p$ -integrable Borel probability measures on $ℝ^{3}$ , equipped with the Wasserstein $p$ metric $𝖶_{p}$ . We prove that for every $ε > 0$ , every $θ \in (0, 1 / p]$ and every finite metric space $(X, d_{X})$ , the metric space $(X, d_{X}^{θ})$ embeds into $𝒫_{p} (ℝ^{3})$ with distortion at most $1 + ε$ . We show that this is sharp when $p \in (1, 2]$ in the sense that the exponent $1 / p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n \in ℕ$ there exists an $n$ -point metric space $(X_{n}, d_{n})$ such that for every $α \in (1 / p, 1]$ any embedding of the metric space $(X_{n}, d_{n}^{α})$ into $𝒫_{p} (ℝ^{3})$ incurs distortion that is at least a constant multiple of ${(log n)}^{α - 1 / p}$ . These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $𝒫_{2} (ℝ^{3})$ , with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $𝒫_{2} (ℝ^{3})$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $𝒫_{2} (ℝ^{k})$ , and the metric cotype dichotomy problem.

Publié le : 2018-11-12

DOI : 10.24033/asens.2363

Classification : 46B85, 53C23, 46E27.
Keywords: Metric embeddings, Wasserstein spaces, Alexandrov spaces, Snowflakes of metric spaces, nonlinear spectral gaps, metric cotype, Markov type.
Mot clés : Plongements d'espaces métriques, espaces de Wasserstein, espaces d'Alexandrov, floconnage d'espaces métriques, trou spectral non linéaire, cotype métrique, type de Markov.

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     author = {Andoni, Alexandr and Naor, Assaf and Neiman, Ofer},
     title = {Snowflake universality  of {Wasserstein} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {657--700},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {3},
     year = {2018},
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     mrnumber = {3831034},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2363/}
}

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Andoni, Alexandr; Naor, Assaf; Neiman, Ofer. Snowflake universality  of Wasserstein spaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 3, pp. 657-700. doi : 10.24033/asens.2363. http://www.numdam.org/articles/10.24033/asens.2363/

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