On a general matrix valued unbalanced optimal transport and its fully discretization: dynamic formulation and convergence framework
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In this work, we present a rather general class of transport distances over the space of positive semidefinite matrix valued Radon measures, called the weighted Wasserstein Bures distance, and consider the convergence property of their fully discretized counterparts. These distances are defined via a generalization of Benamou Brenier formulation of the quadratic optimal transport, based on a new weighted action functional and an abstract matricial continuity equation. It gives rise to a convex optimization problem. We shall give a complete characterization of its minimizer (i.e., the geodesic) and discuss some topological and geometrical properties of these distances. Some recently proposed models: the interpolation distance by Chen et al. [18] and the Kantorovich Bures distance by Brenier et al. [11], as well as the well studied Wasserstein Fisher Rao distance [43, 19, 40], fit in our model. The second part of this work is devoted to the numerical analysis of the fully discretization of the new transport model. We reinterpret the convergence framework proposed very recently by Lavenant [41] for the quadratic optimal transport from the perspective of Lax equivalence theorem and extend it to our general problem. In view of this abstract framework, we suggest a concrete fully discretized scheme inspired by the finite element theory, and show the unconditional convergence under mild assumptions. In particular, these assumptions are removed in the case of Wasserstein Fisher Rao distance due to the existence of a static formulation.
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