Reconstructing discrete measures from projections. Consequences on the empirical Sliced Wasserstein Distance
This paper deals with the reconstruction of a discrete measure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> </mml:math> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> from the knowledge of its pushforward measures <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>#</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> </mml:mrow> </mml:math> by linear applications <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msup> </mml:mrow> </mml:math> (for instance projections onto subspaces). The measure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> </mml:math> being fixed, assuming that the rows of the matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are independent realizations of laws which do not give mass to hyperplanes, we show that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∑</mml:mo> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> , this reconstruction problem has almost certainly a unique solution. This holds for any number of points in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> </mml:math> . A direct consequence of this result is an almost-sure separability property on the empirical Sliced Wasserstein distance.
Authors
- Julie Delon (ORCID: https://orcid.org/0000-0002-7182-7537)
- Rémi Flamary (ORCID: https://orcid.org/0000-0002-4212-6627)
- Eloi Tanguy
Institutions
- École Polytechnique (FR)
- Université Paris Cité (FR)
- Département mathématiques, informatique, sciences de la donnée et technologies du numérique (FR)
- Centre de Mathématiques Appliquées de l'École polytechnique (FR)
- Mathématiques Appliquées à Paris 5 (FR)
Publication Details
- Journal
- Comptes Rendus Mathématique
- Published
- 2024-11-05
- DOI
- https://doi.org/10.5802/crmath.601
- Primary Topic
- Topological and Geometric Data Analysis
- Type
- article
- Field-Weighted Citation Impact
- 0.00