Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance - Tanguy, Eloi; Flamary, Rémi; Delon, Julie (Comptes Rendus. Mathématique, 2024)
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 $P_i: \mathbb{R}^d → \mathbb{R}^{d_i}$ (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.