Families of costs with zero and nonnegative MTW tensor in optimal transport

Code

• github.com/dnguyend/regularmtw
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Abstract

We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on R^n with a cost function of form c(x, y) = u(x^ty), where u is a scalar function with inverse s, x^y is a nondegenerate bilinear pairing of vectors x, y belonging to an open subset of R^n. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form s^(2) - Ss^(1) + Ps = 0 with constant coefficients P and S. The resulting inverse functions include Lambert and generalized inverse hyperbolic trigonometric functions. The square Euclidean metric and -type costs are equivalent to instances of these solutions. The optimal map for the family is also explicit. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of s using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of power function costs. We analyze the -type hyperbolic cost, providing examples of c-convex functions and divergence.

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