We study regression problems with distribution-valued responses and mixed distributional and Euclidean predictors. In quadratic cost, the negative gradient of the Kantorovich potential represents, at each source location, the displacement to its matched location under the optimal transport map. By constructing potentials from the Wasserstein barycenter to individual distributions, the proposed Kantorovich regression model approximates the response displacement field as a sum of predictor displacement fields, each adjusted by a functional parameter. Owing to the linear structure, Euclidean predictors can enter as scaling coefficients of $c$-concave parameter potentials. We characterize functional parameter classes ensuring intrinsicness of the model, establish asymptotic theory through uniform convergence of the empirical Wasserstein loss, and derive Gâteaux derivatives leading to first-order optimization algorithms. Real data applications include a mixed-predictor analysis of housing price distributions and an analysis of two-dimensional temperature distributions, demonstrating the flexibility and interpretability of the proposed framework.

翻译：本研究探讨了响应变量为分布值、预测因子混合包含分布型和欧几里得型的回归问题。在二次成本下，Kantorovich势函数的负梯度表示每个源位置在最优传输映射下向其匹配位置的位移。通过构建从Wasserstein重心到个体分布的势函数，所提出的Kantorovich回归模型将响应位移场近似为预测因子位移场的加权和，其中每个位移场通过一个函数参数进行调整。得益于线性结构，欧几里得预测因子可以作为$c$-凹参数势函数的缩放系数引入。我们刻画了确保模型内在性的函数参数类，通过经验Wasserstein损失的一致收敛性建立了渐近理论，并推导了Gâteaux导数以构建一阶优化算法。实际数据应用包括对房价分布的混合预测因子分析以及对二维温度分布的分析，展示了所提出框架的灵活性和可解释性。