Entropy-transport distances between unbalanced metric measure spaces. (English) Zbl 1503.53086
Inspired by the work [M. Liero et al., Invent. Math. 211, No. 3, 969–1117 (2018; Zbl 1412.49089)] in the theory of optimal entropy-transport problems and by the construction of \(\textbf{D}\)-distance in [K.-T. Sturm, Acta Math. 196, No. 1, 65–131 (2006; Zbl 1105.53035)], the authors introduce a new class of complete and separable distances between metric measure spaces (mm-spaces) by replacing the Wasserstein distance with an entropy-transport distance. The authors first define a regular entropy-transport distance and then the Sturm-entropy-transport distance (SET-distance), which actually origins from the definition of Gromov-Hausdorff distance, between two mm-spaces (which do not necessarily have the same total measures).
The authors show their main result in Theorem 2, proving that every SET-distance defines a complete and separable metric structure on the set of equivalence classes of mm-spaces X. Furthermore, such metric structures on X also turn out to be length (resp., geodesic) when the underlying entropy-transport distance is length (resp., geodesic).
It is proved that the weak measured-Gromov convergence coincides with the convergence induced by any SET-distances. The authors define the strong measured-Gromov convergence (which was also defined independently in [J. Deng, SIGMA, Symmetry Integrability Geom. Methods Appl. 17, Paper 013, 20 p. (2021; Zbl 1466.53049)] in Remark 2.3) and then prove that this convergence coincides with the convergence induced by the distance \(\textbf{PL}_p\) for very \(p\geq 1\).
In the last section, the authors discover an upper bound of the conic Gromov-Wasserstein distance in terms of the SET-distance.
The authors show their main result in Theorem 2, proving that every SET-distance defines a complete and separable metric structure on the set of equivalence classes of mm-spaces X. Furthermore, such metric structures on X also turn out to be length (resp., geodesic) when the underlying entropy-transport distance is length (resp., geodesic).
It is proved that the weak measured-Gromov convergence coincides with the convergence induced by any SET-distances. The authors define the strong measured-Gromov convergence (which was also defined independently in [J. Deng, SIGMA, Symmetry Integrability Geom. Methods Appl. 17, Paper 013, 20 p. (2021; Zbl 1466.53049)] in Remark 2.3) and then prove that this convergence coincides with the convergence induced by the distance \(\textbf{PL}_p\) for very \(p\geq 1\).
In the last section, the authors discover an upper bound of the conic Gromov-Wasserstein distance in terms of the SET-distance.
Reviewer: Jialong Deng (Beijing)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 60B10 | Convergence of probability measures |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
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