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 the intrinsic structure 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：我们研究响应变量为分布值、预测变量包含分布与欧几里得变量的回归问题。在二次代价下，Kantorovich势的负梯度表示每个源位置在最优传输映射下到其匹配位置的位移。本文通过从Wasserstein重心到个体分布构建势函数，提出Kantorovich回归模型将响应位移场近似为各预测器位移场之和，并通过函数参数调整。借助线性结构，欧几里得预测器可作为c-凹参数势的缩放系数引入模型。我们刻画了确保模型内在结构的函数参数类，通过经验Wasserstein损失的一致收敛性建立渐近理论，推导出用于一阶优化算法的Gateaux导数。实际数据应用包括住房价格分布混合预测分析及二维温度分布分析，验证了所提出框架的灵活性与可解释性。