Papers โบ Families of costs with zero and nonnegative MTW tensor in optimal transport
Families of costs with zero and nonnegative MTW tensor in optimal transport
Du Nguyen
We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on โโฟ with a cost function of form ๐ผ(x, y) = ๐(x^๐ฑy), where ๐ is a scalar function with inverse ๐, $x^{\ft}y$ is a nondegenerate bilinear pairing of vectors x, y belonging to an open subset of โโฟ. 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 ๐โฝยฒโพ - S๐โฝยนโพ + P๐ = 0 with constant coefficients P and S. The resulting inverse functions include {\it Lambert} and {\it generalized inverse hyperbolic\slash trigonometric} functions. The square Euclidean metric and log-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 ๐ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\it power function costs}. We analyze the sinh-type hyperbolic cost, providing examples of ๐ผ-convex functions and divergence.
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