Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance

Eloi Tanguy; Rémi Flamary; Julie Delon

doi:10.5802/crmath.601

Article de recherche - Probabilités

Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance
[Reconstruction de mesures discrètes à partir de projections. Conséquences sur la distance de Sliced Wasserstein empirique]

Eloi Tanguy ¹ ; Rémi Flamary ² ; Julie Delon ¹

¹ Université Paris Cité, CNRS, MAP5, F-75006 Paris, France
² CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1121-1129.

Résumé
Abstract

On s’intéresse dans cet article au problème de reconstruction d’une mesure discrète $γ_{Z}$ sur $ℝ^{d}$ connaissant ses images par des applications linéaires $P_{i} : ℝ^{d} \to ℝ^{d_{i}}$ (par exemple des projections sur des sous-espaces). La mesure $γ_{Z}$ étant fixée, en supposant que les lignes des matrices $P_{i}$ sont des réalisations indépendantes de lois ne donnant pas de masse aux hyperplans, on montre que si $\sum_{i} d_{i} > d$ , ce problème de reconstruction a presque sûrement une unique solution, et ceci quelque soit le nombre de points dans $γ_{Z}$ . Ce résultat permet de démontrer une propriété de séparabilité presque sûre pour la distance de Sliced–Wasserstein empirique.

This paper deals with the reconstruction of a discrete measure $γ_{Z}$ on $ℝ^{d}$ from the knowledge of its pushforward measures $P_{i} # γ_{Z}$ by linear applications $P_{i} : ℝ^{d} \to ℝ^{d_{i}}$ (for instance projections onto subspaces). The measure $γ_{Z}$ being fixed, assuming that the rows of the matrices $P_{i}$ are independent realizations of laws which do not give mass to hyperplanes, we show that if $\sum_{i} d_{i} > d$ , this reconstruction problem has almost certainly a unique solution. This holds for any number of points in $γ_{Z}$ . A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.

Reçu le : 2023-04-25
Révisé le : 2023-10-17
Accepté le : 2023-12-18
Publié le : 2024-11-05

Zbl

DOI : 10.5802/crmath.601

Classification : 28E99, 15A29
Keywords: Reconstruction, Inverse Problems, Discrete Measures
Mots-clés : Reconstruction, problèmes inverses, mesures discrètes

Affiliations des auteurs :

Eloi Tanguy ¹ ; Rémi Flamary ² ; Julie Delon ¹

¹ Université Paris Cité, CNRS, MAP5, F-75006 Paris, France
² CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Eloi Tanguy and R\'emi Flamary and Julie Delon},
     title = {Reconstructing discrete measures from projections. {Consequences} on the empirical {Sliced} {Wasserstein} {Distance}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1121--1129},
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Eloi Tanguy; Rémi Flamary; Julie Delon. Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1121-1129. doi : 10.5802/crmath.601. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.601/

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