Calculus of Variations and Geometric Measure Theory

P. Pegon - M. Petrache

Optimal quantization with branched optimal transport distances

created by pegon on 15 Sep 2023
modified on 08 Feb 2024



Inserted: 15 sep 2023
Last Updated: 8 feb 2024

Year: 2024

ArXiv: 2309.08677 PDF

39 pages + bibliography


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 semi-discrete optimal transport with Wasserstein distance, the interfaces between cells associated with neighboring atoms have Voronoï 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 behaviour 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 $N$ Hölder regularity of the landscape function, a branched transport analog to Kantorovich potentials in classical optimal transport.

Keywords: Gamma-convergence, Optimal transport, Branched transport, quantization, convergence of measures, clustering, optimal partitions, uniform point configurations