For simultaneous optimal transport, the paper proves existence, characterizes discrete optimal mixing points, and claims Monge and Kantorovich solutions coincide when density ratios are simple functions, but that claim fails for singular measures.
On semi-discrete sub-partitions of vector-valued measures
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abstract
We introduce a concept of optimal transport for vector-valued measures and its dual formulation. In this note we concentrate on the semi-discrete case and show some fundamental differences between the scalar and vector cases. A manifestation of this difference is the possibility of non-existence of optimal solution for the dual problem for feasible primer problems.
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2024 1verdicts
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On a problem of optimal mixing
For simultaneous optimal transport, the paper proves existence, characterizes discrete optimal mixing points, and claims Monge and Kantorovich solutions coincide when density ratios are simple functions, but that claim fails for singular measures.