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OPTIMAL QUANTIZATION WITH BRANCHED OPTIMAL TRANSPORT DISTANCES.

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Title: OPTIMAL QUANTIZATION WITH BRANCHED OPTIMAL TRANSPORT DISTANCES.
Authors: PEGON, PAUL1 pegon@ceremade.dauphine.fr, PETRACHE, MIRCEA2 mpetrache@mat.uc.cl
Source: SIAM Journal on Mathematical Analysis. 2025, Vol. 57 Issue 4, p3649-3694. 46p.
Subject Terms: *SIGNAL quantization, *VORONOI polygons, *DISTRIBUTION (Probability theory)
Abstract: We consider the problem of optimal approximation of a target measure by an atomic measure with N atoms in branched optimal transport distance. This is a new branched transport version of optimal quantization problems. New difficulties arise, since in classical semidiscrete optimal transport with Wasserstein distance, the interfaces between cells associated with neighboring atoms have Voronoi structure and satisfy an explicit description. This description is missing for our problem, in which the cell interfaces are thought to have fractal boundary. We study the asymptotic behavior of optimal quantizers for absolutely continuous measures as the number N of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish uniformity bounds of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is d-Ahlfors regular. A crucial technical tool is the uniform in N Holder regularity of the landscape function, a branched transport analogue to Kantorovich potentials in classical optimal transport. [ABSTRACT FROM AUTHOR]
Database: Academic Search Index
Description
Abstract:We consider the problem of optimal approximation of a target measure by an atomic measure with N atoms in branched optimal transport distance. This is a new branched transport version of optimal quantization problems. New difficulties arise, since in classical semidiscrete optimal transport with Wasserstein distance, the interfaces between cells associated with neighboring atoms have Voronoi structure and satisfy an explicit description. This description is missing for our problem, in which the cell interfaces are thought to have fractal boundary. We study the asymptotic behavior of optimal quantizers for absolutely continuous measures as the number N of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish uniformity bounds of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is d-Ahlfors regular. A crucial technical tool is the uniform in N Holder regularity of the landscape function, a branched transport analogue to Kantorovich potentials in classical optimal transport. [ABSTRACT FROM AUTHOR]
ISSN:00361410
DOI:10.1137/24M1650193