This note details the Wasserstein-2 Barycenter for distributions which are spherically equivalent. Separating random variables which have spherically equivalent distributions are the radial component distributions. We show the Barycenter corresponds to a quantile averaging of the distributions. An illustration involving Student-t distributions is presented.
@article{GHAFFARI2023109788,title={W2 barycenters for radially related distributions},journal={Statistics & Probability Letters},publisher={Elsevier},volume={195},pages={109788},year={2023},issn={0167-7152},doi={https://doi.org/10.1016/j.spl.2023.109788},url={https://www.sciencedirect.com/science/article/pii/S0167715223000123},author={Ghaffari, N. and Walker, S.G.},keywords={Optimal map, Radial families, Spherical distributions, Wasserstein barycenters},}
Recent findings for optimal transport maps between distribution functions sharing the same copula show that componentwise the solution is the optimal map between the marginal distributions. This is an important discovery since in the multivariate setting optimal maps are difficult to find and only known in a few special cases. In this paper, we extend the result on common copulas by showing that orthonormal transformations of variables sharing a common copula also have a known optimal map. We illustrate this by establishing optimal maps between members of a class of scale mixture of normal distributions.
@article{GHAFFARI2021108989,title={Parseval’s identity and optimal transport maps},journal={Statistics & Probability Letters},publisher={Elsevier},volume={170},pages={108989},year={2021},issn={0167-7152},doi={https://doi.org/10.1016/j.spl.2020.108989},url={https://www.sciencedirect.com/science/article/pii/S0167715220302923},author={Ghaffari, N. and Walker, S.G.},keywords={Positive definite matrix, Orthonormal matrix, Copula, Optimal transportation, Convex function, Barycenter},}