While statistical modeling of distributional data has gained increased attention, the case of multivariate distributions has been somewhat neglected despite its relevance in various applications. This is because the Wasserstein distance, commonly used in distributional data analysis, poses challenges for multivariate distributions. A promising alternative is the sliced Wasserstein distance, which offers a computationally simpler solution. We propose distributional regression models with multivariate distributions as responses paired with Euclidean vector predictors. The foundation of our methodology is a slicing transform from the multivariate distribution space to the sliced distribution space for which we establish a theoretical framework, with the Radon transform as a prominent example. We introduce and study the asymptotic properties of sample-based estimators for two regression approaches, one based on utilizing the sliced Wasserstein distance directly in the multivariate distribution space, and a second approach based on a new slice-wise distance, employing a univariate distribution regression for each slice. Both global and local Fr\'echet regression methods are deployed for these approaches and illustrated in simulations and through applications. These include joint distributions of excess winter death rates and winter temperature anomalies in European countries as a function of base winter temperature and also data from finance.

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