We study distribution-on-distribution regression problems in which a response distribution depends on multiple distributional predictors. Such settings arise naturally in applications where the outcome distribution is driven by several heterogeneous distributional sources, yet remain challenging due to the nonlinear geometry of the Wasserstein space. We propose an intrinsic regression framework that aggregates predictor-specific transported distributions through a weighted Fréchet mean in the Wasserstein space. The resulting model admits multiple distributional predictors, assigns interpretable weights quantifying their relative contributions, and defines a flexible regression operator that is invariant to auxiliary construction choices, such as the selection of a reference distribution. From a theoretical perspective, we establish identifiability of the induced regression operator and derive asymptotic guarantees for its estimation under a predictive Wasserstein semi-norm, which directly characterizes convergence of the composite prediction map. Extensive simulation studies and a real data application demonstrate the improved predictive performance and interpretability of the proposed approach compared with existing Wasserstein regression methods.

翻译：我们研究了响应分布依赖于多个分布预测因子的分布对分布回归问题。这类问题自然出现在结果分布由多个异质分布源驱动的应用场景中，但由于Wasserstein空间的非线性几何结构，此类问题仍然具有挑战性。我们提出了一个内蕴回归框架，通过Wasserstein空间中的加权Fréchet均值来聚合预测因子特定的传输分布。所得模型能够容纳多个分布预测因子，分配可解释的权重以量化其相对贡献，并定义了一个灵活的回归算子，该算子对辅助构造选择（如参考分布的选择）具有不变性。从理论角度，我们证明了所诱导回归算子的可识别性，并在预测性Wasserstein半范数下推导了其估计的渐近保证，该半范数直接刻画了复合预测映射的收敛性。大量的模拟研究和实际数据应用表明，与现有的Wasserstein回归方法相比，所提方法具有更优的预测性能和可解释性。