Multimarginal generative modeling with stochastic interpolants
Michael S. Albergo Nicholas M. Boffi Michael Lindsey Eric Vanden-Eijnden
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.