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. In this paper we propose geodesic optimal transport regression models 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 we define as an extension of the notion of optimal transports in distribution spaces to more general geodesic metric spaces, where we characterize optimal transports as transports along geodesics. The proposed regression models cover the relation between non-Euclidean responses and vectors of non-Euclidean predictors in many spaces of practical statistical interest. These include one-dimensional distributions viewed as elements of the 2-Wasserstein space and multidimensional distributions with the Fisher-Rao metric that are 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. We illustrate the utility of geodesic optimal transport regression with data on summer temperature distributions and human mortality.

翻译：经典回归模型无法处理存在于一般度量空间中的非欧几里得数据，而当前关于非欧几里得回归的文献主要关注预测变量或响应变量为随机对象（即非欧几里得）的情形，但并未同时考虑两者。本文针对预测变量和响应变量均位于同一测地度量空间，且预测变量可包含一个或多个随机对象的情况，提出了测地最优传输回归模型。这将经典多元回归扩展至预测变量和响应变量均位于非欧几里得度量空间的情形，而这一场景此前尚未被研究。该模型基于测地最优传输的概念，我们将其定义为分布空间中最优传输概念向更一般测地度量空间的推广，其中最优传输被表征为沿测地线的传输。所提出的回归模型涵盖了众多具有实际统计意义的空间中非欧几里得响应变量与非欧几里得预测变量向量之间的关系，包括：将一维分布视为2-瓦瑟斯坦空间中的元素，以及采用费希尔-饶度量并表示为希尔伯特球面上数据的多维分布；还包括有限维黎曼流形上的数据（重点讨论球面），涵盖方向数据和成分数据，以及由对称正定矩阵构成的数据。我们通过夏季温度分布和人类死亡率数据展示了测地最优传输回归的实用性。