Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912542163468288 |
|---|---|
| author | Tanguy, Eloi Flamary, Rémi Delon, Julie |
| author_facet | Tanguy, Eloi Flamary, Rémi Delon, Julie |
| contents | This paper deals with the reconstruction of a discrete measure $γ_Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#γ_Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12029 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance Tanguy, Eloi Flamary, Rémi Delon, Julie Probability This paper deals with the reconstruction of a discrete measure $γ_Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#γ_Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (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. |
| title | Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance |
| topic | Probability |
| url | https://arxiv.org/abs/2304.12029 |