Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance

This paper deals with the reconstruction of a discrete measure γ Z on R d from the knowledge of its pushforward measures P i #γ Z by linear applications P i : R d → R di (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 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