With smooth positive marginals, the second variation of continuous OT in the cost is non-degenerate modulo invariances, giving local identifiability of inverse OT and unique recovery for bilinear costs under a spanning condition.
Giovanni Conforti and Luca Tamanini
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Curvature of optimal transport with respect to the cost and applications to inverse optimal transport
With smooth positive marginals, the second variation of continuous OT in the cost is non-degenerate modulo invariances, giving local identifiability of inverse OT and unique recovery for bilinear costs under a spanning condition.