Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-20T13:18:13.568Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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