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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: yongchen@iastate.edu
TRYPHON T. GEORGIOU
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA email: tryphon@uci.edu
ALLEN TANNENBAUM
Affiliation:
Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY, USA email: allen.tannenbaum@stonybrook.edu
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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).

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