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

Abstract

This paper deals with the reconstruction of a discrete measure γZ on Rd from the knowledge of its pushforward measures Pi #γZ by linear applications Pi : Rd → Rdi (for instance projections onto subspaces). The measure γZ being fixed, assuming that the rows of the matrices Pi are independent realizations of laws which do not give mass to hyperplanes, we show that if Pi di > d, this reconstruction problem has almost certainly a unique solution. This holds for any number of points in γZ . A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.