Curvature of optimal transport with respect to the cost and applications to inverse optimal transport

Abstract

We study the inverse optimal transport problem of recovering the ground cost from an optimal transport plan. In discrete settings, this problem reduces to inverse linear programming and is intrinsically ill-posed, exhibiting non-identifiability and flat directions. We show that in the continuous setting, the regularity of the marginals fundamentally alters the structure of the inverse problem. Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in H\"older spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs (i.e. Mahalanobis parametrizations), the ground cost can be uniquely recovered up to the intrinsic invariances from a single optimal coupling under a natural spanning condition. We further show that this identifiability property is generic under arbitrarily small perturbations of the marginals, while settings where the optimal transport map is affine (for instance Gaussian or elliptical marginals) remain degenerate. Finally, we establish precise bounds on the bias and statistical variance of inverse optimal transport under entropic regularization. These results reveal a structural parallel between forward and inverse optimal transport: regularity of the marginals ensures smooth optimal maps in the forward problem, while non-degeneracy of the induced transport plan yields curvature and local invertibility in the inverse problem.