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 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 ∑ 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.