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On the matrix Monge–Kantorovich problem

Published online by Cambridge University Press:  05 August 2019

YONGXIN CHEN
Affiliation:
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA email: [email protected]
WILFRID GANGBO
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA, USA email: [email protected]
TRYPHON T. GEORGIOU
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA email: [email protected]
ALLEN TANNENBAUM
Affiliation:
Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY, USA email: [email protected]

Abstract

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

Type
Papers
Copyright
© Cambridge University Press 2019

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Footnotes

This project was supported by AFOSR grants (FA9550-17-1-0435 and FA9550-18-1-0502), grants from the National Center for Research Resources (P41-RR-013218) and the National Institute of Biomedical Imaging and Bioengineering (P41-EB-015902), National Science Foundation grants (DMS-1160939, 1665031, 1807664, 1839441 and 1901599), grants from the National Institutes of Health (1U24CA18092401A1 and R01-AG048769) and a postdoctoral fellowship through Memorial Sloan Kettering Cancer Center.

References

Ambrosio, L. & Gigli, N. (2012) A user’s guide to optimal mass transport. In: Modelling and Optimisation of Flows on Networks, Lecture Notes in Mathematics, Vol. 2062, Springer, New York, pp. 1155.Google Scholar
Carlen, E. & Maas, J. (2014) An analog of the 2-Wasserstein metric in non-Commutative probability under which the fermionic Fokker–Planck equation is gradient flow for the entropy. Commun. Math. Phys. 331(3), 887926.CrossRefGoogle Scholar
Carlen, E. & Maas, J. (2017) Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. J. Funct. Anal. 273, 18101869. Original version https://arxiv.org/abs/1609.01254, 2016.CrossRefGoogle Scholar
Chen, Y., Georgiou, T. & Tannenbaum, A. (2018) Matrix optimal mass transport: a quantum mechanical approach. IEEE Trans. Autom. Control 63, 26122619. Original version https://arxiv.org/abs/1610.03041, 2016.CrossRefGoogle Scholar
Chen, Y., Gangbo, W., Georgiou, T. & Tannenbaum, A. (2017) On the matrix Monge-Kantorovich problem. https://arxiv.org/abs/1701.02826.Google Scholar
Chen, Y., Haber, E., Yamamoto, K., Georgiou, T. & Tannenbaum, A. (2018) An efficient algorithm for matrix-valued and vector-valued optimal mass transport. J. Sci. Comput. 77, 79100. https://arxiv.org/abs/1706.08841.CrossRefGoogle Scholar
Benamou, J.-D. & Brenier, Y. (2000) A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numerische Math. 84, 375393.CrossRefGoogle Scholar
Dittmann, J. (1993) On the Riemannian geometry of finite dimensional mixed states. Semin. Sophus Lie 3, 7387.Google Scholar
Ekeland, I. & Teman, R. (1976) Convex Analysis and Variational Problems, SIAM, Philadelphia.Google Scholar
Gangbo, W. & McCann, R. J. (1996) The geometry of optimal transportation. Acta Math. 177(2), 113161.CrossRefGoogle Scholar
Gozlan, N., Roberto, C. & Samson, P.-M. (2012) Hamilton-Jacobi equations on metric spaces and transport-entropy inequalities. https://arxiv.org/abs/1203.2783https://arxiv.org/abs/1203.2783.Google Scholar
Gustafson, S. & Sigal, I. M. (2011) Mathematical Concepts of Quantum Mechanics, Springer, New York.CrossRefGoogle Scholar
Jordan, R., Kinderlehrer, D. & Otto, F. (1998) The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 117.CrossRefGoogle Scholar
Kantorovich, L. V. (1948) On a problem of Monge. Uspekhi Mat. Nauk. 3, 225226.Google Scholar
McCann, R. (1997) A convexity principle for interacting gases. Adv. Math. 128, 153179.CrossRefGoogle Scholar
Mertens, J.-F., Sorin, S. & Zamir, S. (1994) Repeated Games. Part A: Background Material. CORE Discussion Paper No 9420, Université Catholique de Louvain.Google Scholar
Mittnenzweig, M. & Mielke, A. (2016) An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models. https://arxiv.org/abs/1609.05765.Google Scholar
Ning, L. & Georgiou, T. T. (2014) Metrics for matrix-valued measures via test functions. In: 53rd IEEE Conference on Decision and Control, pp. 26422647.CrossRefGoogle Scholar
Ning, L., Georgiou, T. T. & Tannenbaum, A. (2015) On matrix–valued Monge-Kantorovich optimal mass transport. IEEE Trans. Autom. Control 60(2), 373382.CrossRefGoogle ScholarPubMed
Otto, F. (2001) The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101174.CrossRefGoogle Scholar
Rachev, S. & Rüschendorf, L. (1998) Mass Transportation Problems, Probability and Its Applications, Vols. I and II, Springer, New York.Google Scholar
Tannenbaum, E., Georgiou, T. & Tannenbaum, A. (2010) Signals and control aspects of optimal mass transport and the Boltzmann entropy. In: 49th IEEE Conference on Decision and Control (CDC).CrossRefGoogle Scholar
Villani, C. (2003) Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, AMS, Providence, RI.Google Scholar