Abstract
This paper presents a family of generative Linear Programming models that permit to compute the exact Wasserstein Barycenter of a large set of two-dimensional images. Wasserstein Barycenters were recently introduced to mathematically generalize the concept of averaging a set of points, to the concept of averaging a set of clouds of points, such as, for instance, two-dimensional images. In Machine Learning terms, the Wasserstein Barycenter problem is a generative constrained optimization problem, since the values of the decision variables of the optimal solution give a new image that represents the “average” of the input images. Unfortunately, in the recent literature, Linear Programming is repeatedly described as an inefficient method to compute Wasserstein Barycenters. In this paper, we aim at disproving such claim. Our family of Linear Programming models rely on different types of Kantorovich-Wasserstein distances used to compute a barycenter, and they are efficiently solved with a modern commercial Linear Programming solver. We numerically show the strength of the proposed models by computing and plotting the barycenters of all digits included in the classical MNIST dataset.
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MNIST dataset. http://yann.lecun.com/exdb/mnist/. Accessed 29 Nov 2018
Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43(2), 904–924 (2011)
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Alfred P. Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts (1988)
Anderes, E., Borgwardt, S., Miller, J.: Discrete Wasserstein Barycenters: optimal transport for discrete data. Math. Methods Oper. Res. 84(2), 389–409 (2016)
Auricchio, G., Bassetti, F., Gualandi, S., Veneroni, M.: Computing Kantorovich-Wasserstein distances on d-dimensional histograms using (d+1)-partite graphs. In: Advances in Neural Information Processing Systems (2018)
Bassetti, F., Gualandi, S., Veneroni, M.: On the computation of Kantorovich-Wasserstein distances between 2D-histograms by uncapacitated minimum cost flows (2018). arXiv preprint: arXiv:1804.00445
Claici, S., Chien, E., Solomon, J.: Stochastic Wasserstein Barycenters (2018). arXiv preprint: arXiv:1802.05757
Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Advances in Neural Information Processing Systems, pp. 2292–2300 (2013)
Cuturi, M., Doucet, A.: Fast computation of Wasserstein Barycenters. In: International Conference on Machine Learning, pp. 685–693 (2014)
Flood, M.M.: On the Hitchcock distribution problem. Pac. J. Math. 3(2), 369–386 (1953)
Santambrogio, F.: Optimal Transport for Applied Mathematicians, pp. 99–102. Birkäuser, New York (2015)
Schrijver, A.: On the history of the transportation and maximum flow problems. Math. Program. 91(3), 437–445 (2002)
Sinkhorn, R., Knopp, P.: Concerning nonnegative matrices and doubly stochastic matrices. Pac. J. Math. 21(2), 343–348 (1967)
Solomon, J., et al.: Convolutional Wasserstein distances: efficient optimal transportation on geometric domains. ACM Trans. Graph. (TOG) 34(4), 66 (2015)
Staib, M., Claici, S., Solomon, J.M., Jegelka, S.: Parallel streaming Wasserstein Barycenters. In: Advances in Neural Information Processing Systems, pp. 2647–2658 (2017)
Vershik, A.M.: Long history of the Monge-Kantorovich transportation problem. Math. Intell. 35(4), 1–9 (2013)
Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71050-9
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Auricchio, G., Bassetti, F., Gualandi, S., Veneroni, M. (2019). Computing Wasserstein Barycenters via Linear Programming. In: Rousseau, LM., Stergiou, K. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019. Lecture Notes in Computer Science(), vol 11494. Springer, Cham. https://doi.org/10.1007/978-3-030-19212-9_23
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DOI: https://doi.org/10.1007/978-3-030-19212-9_23
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