Optimal Transport on Lie Group Orbits

In its most general form, the optimal transport problem is an infinite-dimensional optimization problem, yet certain notable instances admit closed-form solutions. We identify the common source of this tractability as \textit{symmetry} and formalize it using Lie group theory. Fixing a Lie group action on the outcome space and a reference distribution, we study optimal transport between measures lying on the same Lie group orbit of the reference distribution. In this setting, the Monge problem admits an explicit upper bound given by an optimization problem over the stabilizer subgroup of the reference distribution. The reduced problem's dimension scales with that of the stabilizing subgroup and, in the tractable cases we study, is either zero or finite. Under mild regularity conditions, a feasible point of this reduced problem whose induced transport map satisfies a $c$-convex first-order certificate makes the upper bound tight for both the Monge and Kantorovich formulations, with the optimal map realized by a group element. For the quadratic cost on a finite-dimensional Hilbert space and affine-induced actions, the $c$-convex certificate reduces to an algebraic condition: the candidate map must have self-adjoint positive semidefinite linear part. We give a structural criterion, based on Cartan theory, that guarantees this condition. When the linear image of the acting group admits a global Cartan decomposition and its fixed-point subgroup is contained in the linear image of the stabilizer of the reference law, the compact component can be absorbed by the stabilizer, yielding a transport map with a self-adjoint positive definite linear part. This orbit-based viewpoint unifies known closed-form solutions, such as elliptical distributions, and yields new closed-form solutions for Wishart, inverse-Wishart, and matrix beta type~II distributions under the squared Frobenius cost.