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

arXiv: https://arxiv.org/abs/2401.00953 · HTML: https://mlsanity.com/paper/2401.00953/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 49 references that are in the map.
- First posted on arXiv 1 Jan 2024.

## Builds on (most cited first, 9 of 9)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces](https://mlsanity.com/paper/1911.02536/) (2019)
- [When Optimal Transport Meets Information Geometry](https://mlsanity.com/paper/2206.14791/) (2022)
- [Aligning Hyperbolic Representations: an Optimal Transport-based approach](https://mlsanity.com/paper/2012.01089/) (2020)
- [Geometry in global coordinates in mechanics and optimal transport](https://mlsanity.com/paper/2307.10017/) (2023)

## Abstract (arXiv)

We study the information geometry of bcc-divergences from families of costs of the form mathsfc(x, barx) =mathsfu(x^mathfraktbarx) through the optimal transport point of view. Here, mathsfu is a scalar function with inverse mathsfs, xftbarx is a nondegenerate bilinear pairing of vectors x, barx belonging to an open subset of ℝn. We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on ℝn 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 mathsfs(2) - Smathsfs(1) + Pmathsfs = 0 with constant coefficients P and S. The resulting inverse functions include it Lambert and it generalized inverse hyperbolicslash 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 mathsfs 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 mathsfc-divergence in terms of the Kim-McCann metric, including a mathsfc-Crouzeix identity and a formula for the primal connection. We analyze the sinh-type hyperbolic cost, providing examples of mathsfc-convex functions, which are used to construct a new local form of the α-divergences on probability simplices. We apply the optimal maps to sample the multivariate t-distribution.