# Bayesian Multiple Multivariate Density-Density Regression

arXiv: https://arxiv.org/abs/2601.02640 · HTML: https://mlsanity.com/paper/2601.02640/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 6 Jan 2026.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Distributional data analysis of accelerometer data from the NHANES database using nonparametric survey regression models](https://mlsanity.com/paper/2104.01165/) (2021)
- [Sharp bounds for max-sliced Wasserstein distances](https://mlsanity.com/paper/2403.00666/) (2024)
- [Geodesic Optimal Transport Regression](https://mlsanity.com/paper/2312.15376/) (2023)
- [Towards Marginal Fairness Sliced Wasserstein Barycenter](https://mlsanity.com/paper/2405.07482/) (2024)
- [Adaptive Bayesian density regression for high-dimensional data](https://mlsanity.com/paper/1403.2695/) (2014)
- [Density-on-Density Regression](https://mlsanity.com/paper/2307.03642/) (2023)
- [Distribution-in-distribution-out Regression](https://mlsanity.com/paper/2405.11626/) (2024)
- [Bayesian Multivariate Density-Density Regression](https://mlsanity.com/paper/2504.12617/) (2025)
- [Acceleration via silver step-size on Riemannian manifolds with applications to Wasserstein space](https://mlsanity.com/paper/2506.06160/) (2025)

## Built on by (most cited first, 1 of 1)

- [Metric space valued Fr'echet regression](https://mlsanity.com/paper/2602.05225/) (2026)

## Abstract (arXiv)

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.