Foundations of Computational Mathematics

, Volume 11, Issue 4, pp 417–487

Gromov–Wasserstein Distances and the Metric Approach to Object Matching


DOI: 10.1007/s10208-011-9093-5

Cite this article as:
Mémoli, F. Found Comput Math (2011) 11: 417. doi:10.1007/s10208-011-9093-5


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. A second 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.


Gromov–Hausdorff distances Gromov–Wasserstein distances Data analysis Shape matching Mass transport Metric measure spaces 

Mathematics Subject Classification (2000)

68T10 53C23 54E35 60D05 68U05 

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityCaliforniaUSA

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