Abstract
We present a new Linear Programming model that formulates the problem of computing the Kantorovich-Wasserstein distance associated with a truncated ground distance. The key idea of our model is to consider only the quantity of mass that is transported to nearby points and to ignore the quantity of mass that should be transported between faraway pairs of locations. The proposed model has a number of variables that depends on the threshold value used in the definition of the set of nearby points. Using a small threshold value, we can obtain a significant speedup. We use our model to numerically evaluate the percentage gap between the true Wasserstein distance and the truncated Wasserstein distance, using a set of standard grey scale images.
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Altschuler, J., Weed, J., Rigollet, P.: Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration. Adv. Neural Inf. Proces. Syst. 1961–1971 (2017)
Amaldi, E., Coniglio, S., Gualandi, S.: Coordinated cutting plane generation via multi-objective separation. Math. Program. 143(1), 87–110 (2014)
Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein GAN. arXiv preprint. arXiv:1701.07875 (2017)
Auricchio, G., Bassetti, F., Gualandi, S., Veneroni, M.: Computing Kantorovich-Wasserstein distances on d-dimensional histograms using (d + 1)-partite graphs. Adv. Neural Inf. Proces. Syst. 5793–5803 (2018)
Bassetti, F., Regazzini, E.: Asymptotic properties and robustness of minimum dissimilarity estimators of location-scale parameters. Theory Probab. Appl. 50(2), 171–186 (2006)
Bassetti, F., Bodini, A., Regazzini, E.: On minimum Kantorovich distance estimators. Stat. Probab. Lett. 76(12), 1298–1302 (2006)
Bassetti, F., Gualandi, S., Veneroni, M.: On the computation of Kantorovich-Wasserstein distances between 2D-histograms by uncapacitated minimum cost flows. arXiv preprint, arXiv:1804.00445 (2018)
Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)
Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport. Adv. Neural Inf. Proces. Syst. 2292–2300 (2013)
Cuturi, M., Doucet, A.: Fast computation of Wasserstein barycenters. In: International Conference on Machine Learning, pp. 685–693 (2014)
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, New York (2012)
Flood, M.M.: On the Hitchcock distribution problem. Pac. J. Math. 3(2), 369–386 (1953)
Frogner, C., Zhang, C., Mobahi, H., Araya, M., Poggio, T.A.: Learning with a Wasserstein loss. Adv. Neural Inf. Proces. Syst. 2053–2061 (2015)
Goldberg, A.V., Tardos, É., Tarjan, R.: Network flow algorithm. Technical report, Cornell University Operations Research and Industrial Engineering (1989)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics. Springer, New York (2009)
Levina, E., Bickel, P.: The Earth mover’s distance is the Mallows distance: some insights from statistics. In: Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001. ICCV 2001, vol. 2, pp. 251–256. IEEE (2001)
Ling, H., Okada, K.: An efficient earth mover’s distance algorithm for robust histogram comparison. IEEE Trans. Pattern Anal. Mach. Intell. 29(5), 840–853 (2007)
Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Oper. Res. 41(2), 338–350 (1993)
Pele, O., Werman, M.: Fast and robust earth mover’s distances. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 460–467. IEEE (2009)
Rubner, Y., Tomasi, C., Guibas, L.J.: A metric for distributions with applications to image databases. In: Sixth International Conference on Computer Vision, 1998, pp. 59–66. IEEE (1998)
Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40(2), 99–121 (2000)
Schrieber, J., Schuhmacher, D., Gottschlich, C.: Dotmark–a benchmark for discrete optimal transport. IEEE Access 5, 271–282 (2017)
Schrijver, A.: On the history of the transportation and maximum flow problems. Math. Program. 91(3), 437–445 (2002)
Solomon, J., Rustamov, R., Guibas, L., Butscher, A.: Wasserstein propagation for semi-supervised learning. In: International Conference on Machine Learning, pp. 306–314 (2014)
Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A., Du, T., Guibas, L.: Convolutional Wasserstein distances: efficient optimal transportation on geometric domains. ACM Trans. Graph. (TOG) 34(4), 66 (2015)
Acknowledgements
This research was partially supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati”, University of Pavia.
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Auricchio, G., Gualandi, S., Veneroni, M. (2019). The Maximum Nearby Flow Problem. In: Paolucci, M., Sciomachen, A., Uberti, P. (eds) Advances in Optimization and Decision Science for Society, Services and Enterprises. AIRO Springer Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-34960-8_3
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DOI: https://doi.org/10.1007/978-3-030-34960-8_3
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