Abstract: Optimal transport between probability distributions is generally an infinite-dimensional optimization problem, yet several important distributional families admit explicit optimal couplings and transport maps. In this talk, I will describe a geometric framework that explains a common source of this tractability: symmetry.

We consider families of probability measures generated by the action of a Lie group on a reference distribution and study optimal transport between measures lying on the same orbit. The group structure reduces the Monge problem to an optimization over the stabilizer subgroup of the reference law. Under suitable conditions, this reduced problem is exact for both the Monge and Kantorovich formulations, and the optimal transport map is itself induced by a group transformation.

For quadratic transport, we give a structural criterion based on a Cartan decomposition that can be verified at the level of the group action and reference distribution. Once established, it applies simultaneously to every pair of measures in the corresponding orbit. For affine actions, the resulting optimal maps and Wasserstein distances can be characterized explicitly through the means and covariance operators of the distributions.

This perspective recovers the classical transport formula for elliptical distributions and gives closed-form optimal transport maps for several non-Euclidean distributional families. Examples include congruence families on the cone of positive-definite matrices, such as Wishart, inverse-Wishart, and matrix beta type II distributions, as well as multivariate normal sinh-arcsinh distributions with a common Gaussian copula. More broadly, the results suggest that symmetry provides a useful principle for identifying probabilistic models in which otherwise difficult optimal transport problems become explicitly solvable.

Bio: Bahar Tașkesen is an Assistant Professor of Operations Management and Liew Family Junior Faculty Fellow at the University of Chicago Booth School of Business. In June 2024, she completed her PhD in the Risk Analytics and Optimization Lab at École polytechnique fédérale de Lausanne (EPFL) in Switzerland. In 2018, she obtained her Bachelor of Science degree in Electrical and Electronics Engineering from the Middle East Technical University.