Given a set of $K$ probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

翻译：给定一组 $K$ 个概率密度函数，我们考虑学习一个联合分布的多边缘生成建模问题，该联合分布将这些密度作为边缘分布恢复。该联合分布的结构应识别指定边缘之间的多向对应关系。我们在随机插值框架的推广中形式化了解决该任务的方法，从而构建了基于测度动态传输的高效学习算法。我们的生成模型由速度场和得分场定义，这些场可表征为简单二次目标函数的最小化器，并且定义在一个推广了通常动态传输框架中时间变量的单纯形上。该单纯形上的传输受所有边缘的影响，我们证明了多向对应关系可以被提取。此类对应关系的识别在风格迁移、算法公平性和数据去噪方面具有应用价值。此外，多边缘视角能够实现一种用于降低普通两边缘设置中动态传输成本的算法。我们通过若干数值示例展示了这些能力。