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 R-d from the knowledge of its pushforward measures P-i #gamma(Z) by linear applications P-i : R-d -> R-di (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 Sigma(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.