Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance
This paper deals with the reconstruction of a discrete measure $\gamma _Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#\gamma _Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (for instance projections onto subspaces). The measure $\gamma _Z$ bein...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.601/ |
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| author | Tanguy, Eloi Flamary, Rémi Delon, Julie |
| author_facet | Tanguy, Eloi Flamary, Rémi Delon, Julie |
| author_sort | Tanguy, Eloi |
| collection | DOAJ |
| description | This paper deals with the reconstruction of a discrete measure $\gamma _Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#\gamma _Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (for instance projections onto subspaces). The measure $\gamma _Z$ 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 $\gamma _Z$. A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance. |
| format | Article |
| id | doaj-art-8a3888d3c39c4c1ea93fa63c4f59e48a |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-8a3888d3c39c4c1ea93fa63c4f59e48a2025-02-07T11:23:31ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G101121112910.5802/crmath.60110.5802/crmath.601Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein DistanceTanguy, Eloi0Flamary, Rémi1Delon, Julie2Université Paris Cité, CNRS, MAP5, F-75006 Paris, FranceCMAP, CNRS, École Polytechnique, Institut Polytechnique de ParisUniversité Paris Cité, CNRS, MAP5, F-75006 Paris, FranceThis paper deals with the reconstruction of a discrete measure $\gamma _Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#\gamma _Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (for instance projections onto subspaces). The measure $\gamma _Z$ 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 $\gamma _Z$. A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.601/ReconstructionInverse ProblemsDiscrete Measures |
| spellingShingle | Tanguy, Eloi Flamary, Rémi Delon, Julie Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance Comptes Rendus. Mathématique Reconstruction Inverse Problems Discrete Measures |
| title | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| title_full | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| title_fullStr | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| title_full_unstemmed | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| title_short | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| title_sort | reconstructing discrete measures from projections consequences on the empirical sliced wasserstein distance |
| topic | Reconstruction Inverse Problems Discrete Measures |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.601/ |
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