# The Gromov–Wasserstein Distance: A Brief Overview

## Abstract

**:**

## 1. Introduction

#### 1.1. Notation and Background Concepts

## 2. The Gromov–Hausdorff Distance

**Example 1.**

**Theorem 1.**

**Example 2.**

**Theorem 2**

## 3. A Metric on ${\mathcal{M}}^{w}$

**Example 3.**

**Example 4.**

**Remark 1.**

**Theorem 3**

**Example 5.**

**Question 1.**

#### 3.1. Pre-Compactness

**Theorem 4**

**Remark 2.**

#### 3.2. Completeness

**Claim 1.**

**Claim 2.**

#### 3.3. Other Properties: Geodesics and Alexandrov Curvature

**Proposition 1**

**Theorem 5**

#### 3.4. The Metric ${d}_{\mathcal{G}\mathcal{W},p}$ in Applications

**Remark 3.**

**Proposition 3**

**Remark 4.**

**Remark 5.**

## 4. Discussion and Outlook

## Conflicts of Interest

## References

- Burago, D.; Burago, Y.; Ivanov, S. A Course in Metric Geometry. In AMS Graduate Studies in Math; American Mathematical Society: Providence, RI, USA, 2001; Volume 33. [Google Scholar]
- Petersen, P. Gromov-Hausdorff convergence of metric spaces. In Differential Geometry: Riemannian Geometry, Proceedings of the Symposium in Pure Mathematics, Los Angeles, CA, USA, 8–18 July 1990; 1990. [Google Scholar]
- Mémoli, F. Gromov-Wasserstein distances and the metric approach to object matching. Found. Comput. Math.
**2011**, 11, 417–487. [Google Scholar] [CrossRef] - Sakai, T. Riemannian geometry. In Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1996; Volume 149. [Google Scholar]
- Sturm, K.-T. The space of spaces: Curvature bounds and gradient flows on the space of metric measure spaces. Mathematics
**2012**. [Google Scholar] - Mémoli, F. On the use of Gromov-Hausdorff distances for shape comparison. In Proceedings of the Point Based Graphics 2007, Prague, Czech Republic, 2–3 September 2007.
- Osada, R.; Funkhouser, T.; Chazelle, B.; Dobkin, D. Shape distributions. ACM Trans. Graph.
**2002**, 21, 807–832. [Google Scholar] [CrossRef] - Boutin, M.; Kemper, G. On reconstructing n-point configurations from the distribution of distances or areas. Adv. in Appl. Math.
**2004**, 32, 709–735. [Google Scholar] [CrossRef] - Brinkman, D.; Olver, P.J. Invariant histograms. Am. Math. Mon.
**2012**, 119, 4–24. [Google Scholar] - Sturm, K.-T. On the geometry of metric measure spaces. I. Acta Math.
**2006**, 196, 65–131. [Google Scholar] [CrossRef] - Mémoli, F. Gromov-Hausdorff distances in Euclidean spaces. In Proceedings of the 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, CVPR Workshops 2008, Anchorage, AK, USA, 23–28 June 2008.
- Mémoli, F. A spectral notion of Gromov-Wasserstein distances and related methods. Appl. Comput. Harmon. Anal.
**2011**, 30, 363–401. [Google Scholar] [CrossRef]

© 2014 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Mémoli, F.
The Gromov–Wasserstein Distance: A Brief Overview. *Axioms* **2014**, *3*, 335-341.
https://doi.org/10.3390/axioms3030335

**AMA Style**

Mémoli F.
The Gromov–Wasserstein Distance: A Brief Overview. *Axioms*. 2014; 3(3):335-341.
https://doi.org/10.3390/axioms3030335

**Chicago/Turabian Style**

Mémoli, Facundo.
2014. "The Gromov–Wasserstein Distance: A Brief Overview" *Axioms* 3, no. 3: 335-341.
https://doi.org/10.3390/axioms3030335