Abstract
AbstractInspired by the recent theory of Entropy-Transport problems and by the$${\mathbf {D}}$$D-distance of Sturm onnormalisedmetric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the “pure transport”$${\mathbf {D}}$$D-distance and introducing a new class of “pure entropic” distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.
Funder
H2020 European Research Council
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Analysis
Reference49 articles.
1. Alvarez-Melis, D., Jaakkola, T.S.: “Gromov-Wasserstein Alignment of Word Embedding Spaces”, Proc. 2018 Conference on Empirical Methods in Natural Language Processing, pp. 1881–1890 (2018)
2. Ambrosio, L.: “Calculus, heat flow and curvature-dimension bounds in metric measure spaces”, In: Proceedings of the ICM 2018, Rio de Janeiro, Vol. 1, pp. 301–340
3. Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure. Trans. Amer. Math. Soc. 367(7), 4661–4701 (2015)
4. Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014)
5. Ambrosio, L., Mondino, A., Savaré, G.: “Nonlinear diffusion equations and curvature conditions in metric measure spaces”, Mem. Amer. Math. Soc., 262(1270) (2019)
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献