We propose the first approach for multiple multivariate density-density regression (MDDR), making it possible to consider the regression of a multivariate density-valued response on multiple multivariate density-valued predictors. The core idea is to define a fitted distribution using a sliced Wasserstein barycenter (SWB) of push-forwards of the predictors and to quantify deviations from the observed response using the sliced Wasserstein (SW) distance. Regression functions, which map predictors' supports to the response support, and barycenter weights are inferred within a generalized Bayes framework, enabling principled uncertainty quantification without requiring a fully specified likelihood. The inference process can be seen as an instance of an inverse SWB problem. We establish theoretical guarantees, including the stability of the SWB under perturbations of marginals and barycenter weights, sample complexity of the generalized likelihood, and posterior consistency. For practical inference, we introduce a differentiable approximation of the SWB and a smooth reparameterization to handle the simplex constraint on barycenter weights, allowing efficient gradient-based MCMC sampling. We demonstrate MDDR in an application to inference for population-scale single-cell data. Posterior analysis under the MDDR model in this example includes inference on communication between multiple source/sender cell types and a target/receiver cell type. The proposed approach provides accurate fits, reliable predictions, and interpretable posterior estimates of barycenter weights, which can be used to construct sparse cell-cell communication networks.

翻译：我们提出了首个多元密度-密度多重回归（MDDR）方法，使得考虑多元密度值响应变量对多个多元密度值预测变量的回归成为可能。其核心思想是：通过预测变量前推映射的切片瓦瑟斯坦重心（SWB）定义拟合分布，并利用切片瓦瑟斯坦（SW）距离量化与观测响应变量的偏差。回归函数（将预测变量支撑集映射至响应变量支撑集）与重心权重在广义贝叶斯框架下进行推断，无需完全指定似然函数即可实现严格的不确定性量化。该推断过程可视为逆SWB问题的一个实例。我们建立了理论保证，包括边缘分布与重心权重扰动下SWB的稳定性、广义似然的样本复杂性以及后验一致性。为实现实际推断，我们引入了SWB的可微近似及处理重心权重单纯形约束的光滑重参数化方法，从而支持基于梯度的高效MCMC采样。我们在群体规模单细胞数据推断中展示了MDDR的应用。该案例中MDDR模型的后验分析包括对多源/发送细胞类型与目标/接收细胞类型间通信的推断。所提方法能够提供精确的拟合、可靠的预测以及可解释的重心权重复验估计，这些估计可用于构建稀疏的细胞间通信网络。