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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.601/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |