Families of costs with zero and nonnegative MTW tensor in optimal transport and the c-divergences

Abstract

We study the information geometry of $\bcc$-divergences from families of costs of the form $\mathsf{c}(x, \barx) =\mathsf{u}(x^{\mathfrak{t}}\barx)$ through the optimal transport point of view. Here, $\mathsf{u}$ is a scalar function with inverse $\mathsf{s}$, $x^{\ft}\barx$ is a nondegenerate bilinear pairing of vectors $x, \barx$ belonging to an open subset of $\mathbb{R}^n$. We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on $\mathbb{R}^n$ with this cost. 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 $\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$ 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 may be written explicitly in terms of the potential function. 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 $\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\it power function costs}. We express the divergence geometry of the $\mathsf{c}$-divergence in terms of the Kim-McCann metric, including a $\mathsf{c}$-Crouzeix identity and a formula for the primal connection. We analyze the $\sinh$-type hyperbolic cost, providing examples of $\mathsf{c}$-convex functions, which are used to construct a new \emph{local form} of the $\alpha$-divergences on probability simplices. We apply the optimal maps to sample the multivariate $t$-distribution.