caa a22 4500 445331607 CHVBK 20180317142739.0 cr unu---uuuuu 170323e20110801xx s 000 0 eng 10.1007/s10208-011-9093-5 doi (NATIONALLICENCE)springer-10.1007/s10208-011-9093-5 Mémoli Facundo Department of Mathematics, Stanford University, California, USA aut Gromov-Wasserstein Distances and the Metric Approach to Object Matching [Elektronische Daten] [Facundo Mémoli] This paper discusses certain modifications of the ideas concerning the Gromov-Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov-Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. Asecond goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented. SFoCM, 2011 Gromov-Hausdorff distances nationallicence Gromov-Wasserstein distances nationallicence Data analysis nationallicence Shape matching nationallicence Mass transport nationallicence Metric measure spaces nationallicence Foundations of Computational Mathematics Springer-Verlag 11/4(2011-08-01), 417-487 1615-3375 11:4<417 2011 11 10208 https://doi.org/10.1007/s10208-011-9093-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10208-011-9093-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Mémoli Facundo Department of Mathematics, Stanford University, California, USA aut NATIONALLICENCE 773 0- Foundations of Computational Mathematics Springer-Verlag 11/4(2011-08-01), 417-487 1615-3375 11:4<417 2011 11 10208 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer