Multimarginal generative modeling with stochastic interpolants
This addresses the problem of identifying correspondences across multiple datasets for researchers and practitioners in generative modeling, though it appears incremental as it extends an existing framework.
The paper tackles the multimarginal generative modeling problem of learning a joint distribution that recovers multiple probability densities as marginals, formalizing an approach within a generalization of the stochastic interpolant framework to identify multi-way correspondences. The result is efficient learning algorithms built upon dynamical transport of measure, with applications to style transfer, algorithmic fairness, and data decorruption.
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.