This paper deals with the reconstruction of a discrete measure on from the knowledge of its pushforward measures by linear applications (for instance projections onto subspaces). The measure being fixed, assuming that the rows of the matrices are independent realizations of laws which do not give mass to hyperplanes, we show that if , this reconstruction problem has almost certainly a unique solution. This holds for any number of points in . 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 sur connaissant ses images par des applications linéaires (par exemple des projections sur des sous-espaces). La mesure étant fixée, en supposant que les lignes des matrices sont des réalisations indépendantes de lois ne donnant pas de masse aux hyperplans, on montre que si , ce problème de reconstruction a presque sûrement une unique solution, et ceci quelque soit le nombre de points dans . Ce résultat permet de démontrer une propriété de séparabilité presque sûre pour la distance de Sliced–Wasserstein empirique.
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Keywords: Reconstruction, Inverse Problems, Discrete Measures
Mots-clés : Reconstruction, problèmes inverses, mesures discrètes
Eloi Tanguy 1; Rémi Flamary 2; Julie Delon 1
@article{CRMATH_2024__362_G10_1121_0, 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}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.601}, language = {en}, }
TY - JOUR AU - Eloi Tanguy AU - Rémi Flamary AU - Julie Delon TI - Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance JO - Comptes Rendus. Mathématique PY - 2024 SP - 1121 EP - 1129 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.601 LA - en ID - CRMATH_2024__362_G10_1121_0 ER -
%0 Journal Article %A Eloi Tanguy %A Rémi Flamary %A Julie Delon %T Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance %J Comptes Rendus. Mathématique %D 2024 %P 1121-1129 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.601 %G en %F CRMATH_2024__362_G10_1121_0
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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