Abstract
This paper is about similarity between objects that can be represented as points in metric measure spaces. A metric measure space is a metric space that is also equipped with a measure. For example, a network with distances between its nodes and weights assigned to its nodes is a metric measure space. Given points x and y in different metric measure spaces or in the same space, how similar are they? A well known approach is to consider x and y similar if their neighborhoods are similar. For metric measure spaces, similarity between neighborhoods is well captured by the Gromov-Hausdorff-Prokhorov distance, but it is \(\mathsf {NP}\)-hard to compute this distance even in quite simple cases. We propose a tractable alternative: the radial distribution distance between the neighborhoods of x and y. The similarity measure based on the radial distribution distance is coarser than the similarity based on the Gromov-Hausdorff-Prokhorov distance but much easier to compute.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, R., Delmas, J.-F., Hoscheit, P.: A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18(14), 1–21 (2013)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows. In: Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics. Birkhäuser (2005)
Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: Proceedings of the 34th International Conference on Machine Learning, ICML 2017, pp. 214–223 (2017)
Athreya, S., Löhr, W., Winter, A.: Invariance principle for variable speed random walks on trees. Ann. Probab. 45(2), 625–667 (2017)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society (2001)
Edwards, D.A.: The structure of superspace. In: Studies in Topology, pp. 121–133. Academic Press (1975)
Gromov, M.: Groups of polynomial growth and expanding maps (with an appendix by Jacques tits). Publications Mathématiques de l’IHÉS 53, 53–78 (1981)
Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics. Birkhäuser (1999). Based on the 1981 French original
Lei, T.: Scaling limit of random forests with prescribed degree sequences. Bernoulli 25(4A), 2409–2438 (2019)
Mémoli, F.: On the use of Gromov-Hausdorff distances for shape comparison. In: Proceedings of the Symposium on Point Based Graphics, Prague 2007, pp. 81–90 (2007)
Mémoli, F.: Gromov-Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11(4), 417–487 (2011)
Mémoli, F., Smith, Z., Wan, Z.: Gromov-Hausdorff distances on \(p\)-metric spaces and ultrametric spaces. ArXiv e-prints (2019)
Miermont, G.: Tessellations of random maps of arbitrary genus. Annales Scientifiques de L’École Normale Supérieure 42(5), 725–781 (2009)
Ning, X., Desrosiers, C., Karypis, G.: A comprehensive survey of neighborhood-based recommendation methods. In: Recommender Systems Handbook, pp. 37–76. Springer, Boston (2015)
Rabadán, R., Blumberg, A.: Topological Data Analysis for Genomics and Evolution: Topology in Biology. Cambridge University Press (2019)
Schmiedl, F.: Computational aspects of the Gromov-Hausdorff distance and its application in non-rigid shape matching. Discrete Comput. Geom. 57(4), 854–880 (2017)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Dantsin, E., Wolpert, A. (2020). Similarity Between Points in Metric Measure Spaces. In: Satoh, S., et al. Similarity Search and Applications. SISAP 2020. Lecture Notes in Computer Science(), vol 12440. Springer, Cham. https://doi.org/10.1007/978-3-030-60936-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-60936-8_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-60935-1
Online ISBN: 978-3-030-60936-8
eBook Packages: Computer ScienceComputer Science (R0)