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On the matrix Monge–Kantorovich problem

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.

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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.

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On the matrix Monge–Kantorovich problem

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