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Gromov–Wasserstein distances between Gaussian distributions

Part of: Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  18 August 2022

Julie Delon ,
Agnes Desolneux  and
Antoine Salmona
Julie Delon*
Affiliation:
Université de Paris
Agnes Desolneux*
Affiliation:
CNRS and ENS Paris-Saclay
Antoine Salmona*
Affiliation:
ENS Paris-Saclay
*
*Postal address: Université de Paris, CNRS, MAP5 UMR 8145 and Institut Universitaire de France, 45 rue des Saints-Pères, 75006 Paris, France. Email: [email protected]
**Postal address: ENS Paris-Saclay, CNRS, Centre Borelli, UMR 9010, 4 avenue des sciences, 91190 Gif-sur-Yvette, France. Email: [email protected]
***Postal address: ENS Paris-Saclay, CNRS, Centre Borelli, UMR 9010, 4 avenue des sciences, 91190 Gif-sur-Yvette, France. Email: [email protected]
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Abstract

Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.

Keywords

Optimal transportWasserstein distanceGromov–Wasserstein distanceGaussian distributions

MSC classification

Primary: 60E99: None of the above, but in this section
Secondary: 62H25: Factor analysis and principal components; correspondence analysis
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1178 - 1198
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Alvarez-Melis, D., Jegelka, S. and Jaakkola, T. S. (2019). Towards optimal transport with global invariances. Proc. Mach. Learn. Res. 89, 1870–1879.Google Scholar
Anstreicher, K. and Wolkowicz, H. (2000). On Lagrangian relaxation of quadratic matrix constraints. J. Matrix Anal. Appl. 22, 41–55.CrossRefGoogle Scholar
Arjovsky, M., Chintala, S. and Bottou, L. (2017). Wasserstein generative adversarial networks. Proc. Mach. Learn. Res. 70, 214–223.Google Scholar
Bigot, J. et al. (2017). Geodesic PCA in the Wasserstein space by convex PCA. Ann. Inst. H. Poincaré Prob. Statist. 53, 1–26.CrossRefGoogle Scholar
Blanchet, J., Kang, Y. and Murthy, K. (2019). Robust Wasserstein profile inference and applications to machine learning. J. Appl. Prob. 56, 830–857.CrossRefGoogle Scholar
Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417.CrossRefGoogle Scholar
Cai, Y. and Lim, L.-H. (2020). Distances between probability distributions of different dimensions. Preprint, arXiv:2011.00629 [math.ST].Google Scholar
Chowdhury, S. and Needham, T. (2020). Gromov–Wasserstein averaging in a Riemannian framework. In Proc. IEEE/CVF Conf. Computer Vision and Pattern Recognition Workshops, Curran Associates, Red Hook, NY, pp. 842–843.Google Scholar
Courty, N., Flamary, R., Tuia, D. and Rakotomamonjy, A. (2016). Optimal transport for domain adaptation. IEEE Trans. Pattern Anal. Mach. Intellig. 39, 1853–1865.CrossRefGoogle ScholarPubMed
Dowson, D. C. and Landau, B. V. (1982). The Fréchet distance between multivariate normal distributions. J. Multivar. Anal. 12, 450–455.CrossRefGoogle Scholar
Galichon, A. et al. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Prob. 24, 312–336.Google Scholar
Genevay, A., Peyré, G. and Cuturi, M. (2018). Learning generative models with sinkhorn divergences. Proc. Mach. Learn. Res. 84, 1608–1617.Google Scholar
Givens, C. R. et al. (1984). A class of Wasserstein metrics for probability distributions. Mich. Math. J. 31, 231–240.Google Scholar
Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134–139.CrossRefGoogle Scholar
James, I. M. (1976). The Topology of Stiefel Manifolds (London Math. Soc. Lecture Note Series 24). Cambridge University Press.Google Scholar
Mémoli, F. (2011). Gromov–Wasserstein distances and the metric approach to object matching. Found. Comput. Math. 11, 417–487.Google Scholar
Pele, O. and Taskar, B. (2013). The tangent earth mover’s distance. In Proc. First Int. Conf. Geometric Science of Information, eds F. Nielsen and F. Barbaresco. Springer, New York, pp. 397–404.CrossRefGoogle Scholar
Peyré, G. and Cuturi, M. (2019). Computational optimal transport, with applications to data science. Found. Trends Mach. Learn. 11, 355–607.CrossRefGoogle Scholar
Peyré, G., Cuturi, M. and Solomon, J. (2016). Gromov–Wasserstein averaging of kernel and distance matrices. J. Mach. Learn. Res. 48, 2664–2672.Google Scholar
Rabin, J., Ferradans, S. and Papadakis, N. (2014). Adaptive color transfer with relaxed optimal transport. In Proc. IEEE Int. Conf. Image Processing, pp. 4852–4856.CrossRefGoogle Scholar
Rabin, J., Peyré, G., Delon, J. and Bernot, M. (2011). Wasserstein barycenter and its application to texture mixing. In Proc. Third Int. Conf. Scale Space and Variational Methods in Computer Vision, eds A. M. Bruckstein, B. M. Haar Romeny, A. M. Bronstein, and M. M. Bronstein. Springer, New York, pp. 435–446.Google Scholar
Santambrogio, F. (2015). Optimal Transport for Applied Mathematicians (Progress in Nonlinear Differential Equations and their Applications 87). Springer, New York.Google Scholar
Sturm, K.-T. (2012). The space of spaces: Curvature bounds and gradient flows on the space of metric measure spaces. Preprint, arXiv:1208.0434 [math.MG].Google Scholar
Takatsu, A. On Wasserstein geometry of Gaussian measures. In Probabilistic Approach to Geometry, eds M. Kotani, M. Hino, and T. Kumagai. Mathematical Society of Japan, Tokyo, pp. 463–472.Google Scholar
Vayer, T. (2020). A contribution to optimal transport on incomparable spaces. Preprint, arXiv:2011.04447 [stat.ML].Google Scholar
Villani, C. (2003). Topics in Optimal Transportation (Graduate Studies in Math. 58). American Mathematical Society, Providence, RI.Google Scholar
Villani, C. (2008). Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften 338). Springer, Berlin.Google Scholar