We give what appears to be the first explicit, easily computable bound on the transportation cost distance with respect to the weighted Hamming metric. The bound follows from Kantorovich duality and a novel inequality, which amounts to bounding the maximal value of certain linear programs and may be of independent interest. We give two application to concentration of measure for dependent processes and pose some open problems and directions for future work.
References
Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In SODA, pages 1079–1087, 2004. MR2291000 Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In SODA, pages 1079–1087, 2004. MR2291000
Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Math. Journal, 19:357–367, 1967. MR221571 10.2748/tmj/1178243286 euclid.tmj/1178243286
Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Math. Journal, 19:357–367, 1967. MR221571 10.2748/tmj/1178243286 euclid.tmj/1178243286
Moses S. Charikar. Similarity estimation techniques from rounding algorithms. In STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 380–388, New York, NY, USA, 2002. ACM. MR2121163 1192.68226 Moses S. Charikar. Similarity estimation techniques from rounding algorithms. In STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 380–388, New York, NY, USA, 2002. ACM. MR2121163 1192.68226
Jean-René Chazottes, Pierre Collet, Christof Külske, and Frank Redig. Concentration inequalities for random fields via coupling. Probability Theory and Related Fields, 137(1-2):201–225, 2007. MR2278456 1111.60070 10.1007/s00440-006-0026-1 Jean-René Chazottes, Pierre Collet, Christof Külske, and Frank Redig. Concentration inequalities for random fields via coupling. Probability Theory and Related Fields, 137(1-2):201–225, 2007. MR2278456 1111.60070 10.1007/s00440-006-0026-1
Jean-René Chazottes and Frank Redig. Concentration inequalities for Markov processes via coupling. Electron. J. Probab., 14:no. 40, 1162–1180, 2009. MR2511280 1191.60023 Jean-René Chazottes and Frank Redig. Concentration inequalities for Markov processes via coupling. Electron. J. Probab., 14:no. 40, 1162–1180, 2009. MR2511280 1191.60023
Chandra Chekuri, Sanjeev Khanna, Joseph Naor, and Leonid Zosin. Approximation algorithms for the metric labeling problem via a new linear programming formulation. In SODA, pages 109–118, 2001. MR1958397 0989.90104 Chandra Chekuri, Sanjeev Khanna, Joseph Naor, and Leonid Zosin. Approximation algorithms for the metric labeling problem via a new linear programming formulation. In SODA, pages 109–118, 2001. MR1958397 0989.90104
Leonid (Aryeh) Kontorovich and Kavita Ramanan. Concentration Inequalities for Dependent Random Variables via the Martingale Method. Ann. Probab., 36(6):2126–2158, 2008. MR2478678 1154.60310 10.1214/07-AOP384 euclid.aop/1229696598
Leonid (Aryeh) Kontorovich and Kavita Ramanan. Concentration Inequalities for Dependent Random Variables via the Martingale Method. Ann. Probab., 36(6):2126–2158, 2008. MR2478678 1154.60310 10.1214/07-AOP384 euclid.aop/1229696598
Gábor Lugosi. Concentration-of-measure inequalities, http://www.econ.upf.es/~lugosi/anu.ps. 2003. http://www.econ.upf.es/~lugosi/anu.ps Gábor Lugosi. Concentration-of-measure inequalities, http://www.econ.upf.es/~lugosi/anu.ps. 2003. http://www.econ.upf.es/~lugosi/anu.ps
Katalin Marton. Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration. Ann. Probab., 24(2):857–866, 1996. MR1404531 0865.60017 10.1214/aop/1039639365 euclid.aop/1039639365
Katalin Marton. Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration. Ann. Probab., 24(2):857–866, 1996. MR1404531 0865.60017 10.1214/aop/1039639365 euclid.aop/1039639365
Katalin Marton. Measure concentration and strong mixing. Studia Scientiarum Mathematicarum Hungarica, 19(1-2):95–113, 2003. MR2002993 10.1556/SScMath.40.2003.1-2.8 Katalin Marton. Measure concentration and strong mixing. Studia Scientiarum Mathematicarum Hungarica, 19(1-2):95–113, 2003. MR2002993 10.1556/SScMath.40.2003.1-2.8
Colin McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, volume 141 of LMS Lecture Notes Series, pages 148–188. Morgan Kaufmann Publishers, San Mateo, CA, 1989. MR1036755 0712.05012 Colin McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, volume 141 of LMS Lecture Notes Series, pages 148–188. Morgan Kaufmann Publishers, San Mateo, CA, 1989. MR1036755 0712.05012
Shmuel Peleg, Michael Werman, and Hillel Rom. A unified approach to the change of resolution: Space and gray-level. IEEE Trans. Pattern Anal. Mach. Intell., 11(7):739–742, 1989. Shmuel Peleg, Michael Werman, and Hillel Rom. A unified approach to the change of resolution: Space and gray-level. IEEE Trans. Pattern Anal. Mach. Intell., 11(7):739–742, 1989.
Emmanuel Rio. Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Sér. I Math., 330(10):905–908, 2000. MR1771956 10.1016/S0764-4442(00)00290-1 Emmanuel Rio. Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Sér. I Math., 330(10):905–908, 2000. MR1771956 10.1016/S0764-4442(00)00290-1
Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover's distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99–121, 2000. Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover's distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99–121, 2000.
Paul-Marie Samson. Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab., 28(1):416–461, 2000. MR1756011 1044.60061 10.1214/aop/1019160125 euclid.aop/1019160125
Paul-Marie Samson. Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab., 28(1):416–461, 2000. MR1756011 1044.60061 10.1214/aop/1019160125 euclid.aop/1019160125
L. N. Vasershtein. Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii, 5(3):64–72, 1969. MR314115 L. N. Vasershtein. Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii, 5(3):64–72, 1969. MR314115
Cédric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. MR1964483 1106.90001 Cédric Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. MR1964483 1106.90001
Cédric Villani. Optimal Transport: Old and New, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. MR2459454 Cédric Villani. Optimal Transport: Old and New, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. MR2459454