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Interpolation of matrices and matrix-valued densities: The unbalanced case

Published online by Cambridge University Press:  08 May 2018

YONGXIN CHEN ,
TRYPHON T. GEORGIOU  and
ALLEN TANNENBAUM
YONGXIN CHEN
Affiliation:
Department of Electrical and Computer Engineering, Iowa State University, IA, 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]
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Abstract

We propose unbalanced versions of the quantum mechanical version of optimal mass transport that is based on the Lindblad equation describing open quantum systems. One of them is a natural interpolation framework between matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance, and the second is an interpolation between Wasserstein distance and Frobenius norm. We also give analogous results for the matrix-valued density measures, i.e., we add a spatial dependency on the density matrices. This might extend the applications of the framework to interpolating matrix-valued densities/images with unequal masses.

Keywords

Optimal mass transportquantum mechanicsmatrix-valued densitiesFisher-Rao informationWasserstein metric
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 3 , June 2019 , pp. 458 - 480
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

This project was supported by AFOSR grants (FA9550-15-1-0045 and FA9550-17-1-0435), ARO grant (W911NF-17-1-049), 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 (NSF ECCS-1509387), NCI grant (1U24CA18092401A1), NIA grant (R01 AG053991), Breast Cancer Research Foundation, and a grant from the National Institutes of Health (P30-CA-008748).

References

[1] Benamou, J.-D. (2010) Numerical resolution of an unbalanced mass transport problem. ESAIM: Math. Model. Numer. Anal. 37 (2), 851–868.Google Scholar
[2] Benamou, J.-D. & Brenier, Y. (2000) A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393.Google Scholar
[3] Boyd, S. & Vandenberghe, L. (2004) Convex Optimization, Cambridge University Press, Cambridge, England.Google Scholar
[4] Carlen, E. & Maas, J. (2017) Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. J. Funct. Anal. 273, 1810–1869. https://arxiv.org/abs/1609.01254.Google Scholar
[5] Chen, Y., Gangbo, W., Georgiou, T. T. & Tannenbaum, A. (2017) On the matrix Monge–Kantorovich problem. Preprint arXiv:1701.02826.Google Scholar
[6] Chen, Y., Georgiou, T. T. & Tannenbaum, A. (2017) Matrix optimal mass transport: A quantum mechanical approach. IEEE Trans. Autom. Control (99), DOI: 10.1109/TAC.2017.2767707.Google Scholar
[7] Chen, Y., Georgiou, T. T. & Pavon, M. (2016) On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint. J. Optim. Theory Appl. 169 (2), 671–691.Google Scholar
[8] Chen, Y., Haber, E., Yamamoto, K., Georgiou, T. T. & Tannenbaum, A. (2018) An efficient algorithm for matrix-valued and vector-valued optimal mass transport. J. Sci. Comput., https://doi.org/10.1007/s10915-018-0696-8.Google Scholar
[9] Chizat, L., Schmitzer, B., Peyré, G. & Vialard, F.-X. (2018) An interpolating distance between optimal transport and Fisher-Rao. Found. Comp. Math. 18, 1–44. https://arxiv.org/pdf/1506.06430.pdf.Google Scholar
[10] Gustafson, S. & Sigal, I. M. (2011) Mathematical Concepts of Quantum Mechanics, Springer, New York.Google Scholar
[11] Jordan, R., Kinderlehrer, D. & Otto, F. (1998) The variational formulation of the Fokker–Planck equation SIAM J. Math. Anal. 29, 1–17.Google Scholar
[12] Kantorovich, L. V. (1948) On a problem of Monge. Uspekhi Mat. Nauk. 3, 225–226.Google Scholar
[13] Kondratyev, S., Monsaingeon, L. & Vorotnikov, D. (2016) A new optimal trasnport distance on the space of finite Radon measures. Adv. Differ. Equ. 21, 1117–1164.Google Scholar
[14] Liero, M., Mielke, A. & Savaré, G. (2016) Optimal transport in competition with reaction: The Hellinger–Kantorovich distance and geodesic curves. SIAM J. Math. Anal. 48 (4), 2869–2911.Google Scholar
[15] Liero, M., Mielke, A. & Savaré, G. (2018) Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Invent. Math. 211, 969–1117. https://arxiv.org/abs/1508.07941.Google Scholar
[16] Mittnenzweig, M. & Mielke, A. (2017) An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models. J. Stat. Phys. 167, 205–233. https://arxiv.org/abs/1609.05765.Google Scholar
[17] Mueller, M., Karasev, P., Kolesov, I. & Tannenbaum, A. (2013) Optical flow estimation for flame detection in videos. IEEE Trans. Image Process. 22 (2), 2786–2797.Google Scholar
[18] Ning, L., Georgiou, T. & Tannenbaum, A. (2015) On matrix–valued Monge–Kantorovich optimal mass transport. IEEE Trans. Autom. Control 60 (2), 373–382.Google Scholar
[19] Rachev, S. & Rüschendorf, L. (1998) Mass Transportation Problems, Volumes I and II, Probability and Its Applications, Springer, New York.Google Scholar
[20] Tannenbaum, E., Georgiou, T. & Tannenbaum, A. (2010) Signals and control aspects of optimal mass transport and the Boltzmann entropy. In: Proceedings of the 49th IEEE Conf. Decision Control (CDC).Google Scholar
[21] Uhlmann, A. (1992) The metric of Bures and the geometric phase. In: Quantum Groups and Related Topics, edited by Gielerak, R., Kluwer Academic Publishers, Dordrecht, pp. 267–274.Google Scholar
[22] Villani, C. (2003) Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, AMS, Providence, RI.Google Scholar