Comptes Rendus
Research article - Probability theory
Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1121-1129.

This paper deals with the reconstruction of a discrete measure γ Z on d from the knowledge of its pushforward measures P i #γ Z by linear applications P i : d 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 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.

On s’intéresse dans cet article au problème de reconstruction d’une mesure discrète γ Z sur d connaissant ses images par des applications linéaires P i : d 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 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.

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DOI: 10.5802/crmath.601
Classification: 28E99, 15A29
Keywords: Reconstruction, Inverse Problems, Discrete Measures
Mots-clés : Reconstruction, problèmes inverses, mesures discrètes

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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