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Optimal Transportation with an Oscillation-Type Cost: The One-Dimensional Case

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Abstract

The main result of this paper is the existence of an optimal transport map T between two given measures μ and ν, for a cost which considers the maximal oscillation of T at scale δ, given by ω δ (T) : =  sup|x − y| < δ |T(x) − T(y)|. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.

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References

  1. 1.

    Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On area preserving mappings of minimal distortion. In: System Theory: Modeling, Analysis and Control (Cambridge, MA, 1999), pp. 275–286. Kluwer Internat. Ser. Engrg. Comput. Sci., 518, Kluwer Acad. Publ., Boston, MA (2000)

  2. 2.

    C.U. Center for International Earth Science Information Network (CIESIN) and C.I. de Agricultura Tropical (CIAT): Gridded population of the world, version 3 (gpwv3) (2005)

  3. 3.

    Granieri, L., Maddalena, F.: A metric approach to elastic reformations. Available on http://cvgmt.sns.it/ (2012, preprint)

  4. 4.

    Kantorovich, L.: On the transfer of masses. Dokl. Acad. Nauk. USSR 37, 7–8 (1942)

  5. 5.

    Louet, J.: Problèmes variationnels entre le transport optimal et l’élasticité incompressible. PhD thesis in preparation

  6. 6.

    Louet, J., Santambrogio, F.: A sharp inequality for transport maps in W 1,p(ℝ) via approximation. Appl. Math. Lett. 25(3), 648–653 (2012)

  7. 7.

    Mascetti, S., Freni, D., Bettini, C., Wang, X.-S., Jajodia, S.: Privacy in geo-social networks: proximity notification with untrusted service providers and curious buddies. VLDB J. 20(4), 541–566 (2011)

  8. 8.

    Mascetti, S., Bettini, C., Freni, D.: Longitude: centralized privacy-preserving computation of users’ proximity. In: Proc. of the 6th VLDB Workshop on Secure Data Management (SDM ’09), LNCS, vol. 5776. Springer (2009)

  9. 9.

    Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, pp. 666–704 (1781)

  10. 10.

    Peyré, G., Rabin, J.: Wasserstein regularization of imaging problems. In: Proc. ICIP’11, pp. 1541–1544 (2011)

  11. 11.

    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, AMS (2003)

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Correspondence to Filippo Santambrogio.

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Lesesvre, D., Pegon, P. & Santambrogio, F. Optimal Transportation with an Oscillation-Type Cost: The One-Dimensional Case. Set-Valued Var. Anal 21, 541–556 (2013). https://doi.org/10.1007/s11228-013-0229-4

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Keywords

  • Monge-Kantorovich
  • Optimal transportation
  • Modulus of continuity
  • Monotone transports
  • Privacy respect

Mathematics Subject Classifications (2010)

  • Primary 49J45; Secondary 49J05
  • 46N10