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.