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

Eloi Tanguy; Rémi Flamary; Julie Delon

doi:10.5802/crmath.601

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

Titre traduit de la contribution: Reconstruction de mesures discrètes à partir de projections. Conséquences sur la distance de Sliced Wasserstein empirique

Eloi Tanguy
, Rémi Flamary
, Julie Delon

CNRS

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

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 ^P_i d_i > 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.

Titre traduit de la contribution	Reconstruction de mesures discrètes à partir de projections. Conséquences sur la distance de Sliced Wasserstein empirique
langue originale	Anglais
Pages (de - à)	1121-1129
Nombre de pages	9
journal	Comptes Rendus Mathematique
Volume	362
Les DOIs	https://doi.org/10.5802/crmath.601
état	Publié - 1 janv. 2024

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10.5802/crmath.601

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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.",

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AB - 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.

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KW - Inverse Problems

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

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