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 $P_i\mathrm{\#}{\gamma }_Z$ by linear applications $P_i:{\mathbb{R}}^d\to {\mathbb{R}}^{d_i}$ (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.