Smooth marginals induce non-degenerate curvature in the optimal transport functional w.r.t. the cost, enabling local identifiability and stability for inverse optimal transport.
The following proposition comes from [Mena and Niles-Weed, 2019]
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Curvature of optimal transport with respect to the cost and applications to inverse optimal transport
Smooth marginals induce non-degenerate curvature in the optimal transport functional w.r.t. the cost, enabling local identifiability and stability for inverse optimal transport.