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.