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

Eloi Tanguy
, Rémi Flamary
, Julie Delon

Centre national de la recherche scientifique

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Translated title of the contribution	Reconstruction de mesures discrètes à partir de projections. Conséquences sur la distance de Sliced Wasserstein empirique
Original language	English
Pages (from-to)	1121-1129
Number of pages	9
Journal	Comptes Rendus Mathematique
Volume	362
DOIs	https://doi.org/10.5802/crmath.601
Publication status	Published - 1 Jan 2024

Keywords

Discrete Measures
Inverse Problems
Reconstruction

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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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AU - Tanguy, Eloi

AU - Flamary, Rémi

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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

KW - Reconstruction

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JF - Comptes Rendus Mathematique

ER -

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

Abstract

Keywords

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