Gromov-Wasserstein distances between Gaussian distributions. (English) Zbl 1501.60012
Summary: 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.
MSC:
| 60E05 | Probability distributions: general theory |
| 68T09 | Computational aspects of data analysis and big data |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 49Q22 | Optimal transportation |
Keywords:
optimal transport; Wasserstein distance; Gromov-Wasserstein distance; Gaussian distributionsSoftware:
Wasserstein GANReferences:
| [1] | Alvarez-Melis, D., Jegelka, S. and Jaakkola, T. S. (2019). Towards optimal transport with global invariances. Proc. Mach. Learn. Res. 89, 1870-1879. |
| [2] | Anstreicher, K. and Wolkowicz, H. (2000). On Lagrangian relaxation of quadratic matrix constraints. J. Matrix Anal. Appl.22, 41-55. · Zbl 0990.90088 |
| [3] | Arjovsky, M., Chintala, S. and Bottou, L. (2017). Wasserstein generative adversarial networks. Proc. Mach. Learn. Res. 70, 214-223. |
| [4] | Bigot, J.et al. (2017). Geodesic PCA in the Wasserstein space by convex PCA. Ann. Inst. H. Poincaré Prob. Statist.53, 1-26. · Zbl 1362.62065 |
| [5] | Blanchet, J., Kang, Y. and Murthy, K. (2019). Robust Wasserstein profile inference and applications to machine learning. J. Appl. Prob.56, 830-857. · Zbl 1436.62336 |
| [6] | Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math.44, 375-417. · Zbl 0738.46011 |
| [7] | Cai, Y. and Lim, L.-H. (2020). Distances between probability distributions of different dimensions. Preprint, arXiv:2011.00629 [math.ST]. |
| [8] | 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. |
| [9] | Courty, N., Flamary, R., Tuia, D. and Rakotomamonjy, A. (2016). Optimal transport for domain adaptation. IEEE Trans. Pattern Anal. Mach. Intellig.39, 1853-1865. |
| [10] | Dowson, D. C. and Landau, B. V. (1982). The Fréchet distance between multivariate normal distributions. J. Multivar. Anal.12, 450-455. · Zbl 0501.62038 |
| [11] | 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. · Zbl 1285.49012 |
| [12] | Genevay, A., Peyré, G. and Cuturi, M. (2018). Learning generative models with sinkhorn divergences. Proc. Mach. Learn. Res. 84, 1608-1617. |
| [13] | Givens, C. R.et al. (1984). A class of Wasserstein metrics for probability distributions. Mich. Math. J.31, 231-240. · Zbl 0582.60002 |
| [14] | Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika12, 134-139. |
| [15] | James, I. M. (1976). The Topology of Stiefel Manifolds (London Math. Soc. Lecture Note Series24). Cambridge University Press. · Zbl 0337.55017 |
| [16] | Mémoli, F. (2011). Gromov-Wasserstein distances and the metric approach to object matching. Found. Comput. Math.11, 417-487. · Zbl 1244.68078 |
| [17] | 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. · Zbl 1323.68544 |
| [18] | Peyré, G. and Cuturi, M. (2019). Computational optimal transport, with applications to data science. Found. Trends Mach. Learn.11, 355-607. |
| [19] | Peyré, G., Cuturi, M. and Solomon, J. (2016). Gromov-Wasserstein averaging of kernel and distance matrices. J. Mach. Learn. Res.48, 2664-2672. |
| [20] | 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. |
| [21] | 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. |
| [22] | Santambrogio, F. (2015). Optimal Transport for Applied Mathematicians (Progress in Nonlinear Differential Equations and their Applications87). Springer, New York. |
| [23] | 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]. |
| [24] | 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. · Zbl 1206.60016 |
| [25] | Vayer, T. (2020). A contribution to optimal transport on incomparable spaces. Preprint, arXiv:2011.04447 [stat.ML]. |
| [26] | Villani, C. (2003). Topics in Optimal Transportation (Graduate Studies in Math. 58). American Mathematical Society, Providence, RI. · Zbl 1106.90001 |
| [27] | Villani, C. (2008). Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften338). Springer, Berlin. |
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