  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.
