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

Abstract: This paper deals with the reconstruction of a discrete measure

g a m m a_{Z}

on

m a t h b b R^{d}

from the knowledge of its pushforward measures

by linear applications

P_{i} : m a t h b b R^{d} i g h t a r r o w m a t h b b R^{d_{i}}

(for instance projections onto subspaces). The measure

g a m m a_{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

s u m_{i} d_{i} > d

, this reconstruction problem has almost certainly a unique solution. This holds for any number of points in

g a m m a_{Z}

. A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.

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