A1030
Title: Geodesic optimal transport regression
Authors: Changbo Zhu - University of Notre Dame (United States) [presenting]
Hans-Georg Mueller - University of California Davis (United States)
Abstract: Classical regression models do not cover non-Euclidean data that reside in a general metric space, while the current literature on non-Euclidean regression, by and large, has focused on scenarios where either predictors or responses are random objects, i.e., non-Euclidean, but not both. Geodesic optimal transport regression models are proposed for the case where both predictors and responses lie in a common geodesic metric space, and predictors may include not only one but also several random objects. This provides an extension of classical multiple regression to the case where both predictors and responses reside in non-Euclidean metric spaces, a scenario that has not been considered before. It is based on the concept of optimal geodesic transports, which is defined as an extension of the notion of optimal transports in distribution spaces to more general geodesic metric spaces, where optimal transports are characterized as transports along geodesics. The proposed regression models cover many spaces of practical statistical interest, including one-dimensional distributions viewed as elements of the 2-Wasserstein space and multidimensional distributions with the Fisher-Rao metric represented as data on the Hilbert sphere. Also included are data on finite-dimensional Riemannian manifolds, with an emphasis on spheres, covering directional and compositional data, as well as data that consist of symmetric positive definite matrices.
