Issue |
ESAIM: COCV
Volume 29, 2023
|
|
---|---|---|
Article Number | 19 | |
Number of page(s) | 36 | |
DOI | https://doi.org/10.1051/cocv/2022088 | |
Published online | 07 March 2023 |
Hellinger–Kantorovich barycenter between Dirac measures
1
Campus Institute Data Science, Göttingen University, Germany
2
Department of Computer Science, University of Verona, Italy
* Corresponding author: schmitzer@cs.uni-goettingen.de
Received:
15
August
2022
Accepted:
16
December
2022
The Hellinger-Kantorovich (HK) distance is an unbalanced extension of the Wasserstein-2 distance. It was shown recently that the HK barycenter exhibits a much more complex behaviour than the Wasserstein barycenter. Motivated by this observation we study the HK barycenter in more detail for the case where the input measures are an uncountable collection of Dirac measures, in particular the dependency on the length scale parameter of HK, the question whether the HK barycenter is discrete or continuous and the relation between the expected and the empirical barycenter. The analytical results are complemented with numerical experiments that demonstrate that the HK barycenter can provide a coarse-to-fine representation of an input pointcloud or measure.
Mathematics Subject Classification: 49K35 / 49M29 / 49Q22
Key words: optimal transport / unbalanced optimal transport / barycenter / data analysis
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.