  Generative models inspired by dynamical transport of measure -- such as flows
and diffusions -- construct a continuous-time map between two probability
densities. Conventionally, one of these is the target density, only accessible
through samples, while the other is taken as a simple base density that is
data-agnostic. In this work, using the framework of stochastic interpolants, we
formalize how to \textit{couple} the base and the target densities, whereby
samples from the base are computed conditionally given samples from the target
in a way that is different from (but does preclude) incorporating information
about class labels or continuous embeddings. This enables us to construct
dynamical transport maps that serve as conditional generative models. We show
that these transport maps can be learned by solving a simple square loss
regression problem analogous to the standard independent setting. We
demonstrate the usefulness of constructing dependent couplings in practice
through experiments in super-resolution and in-painting.
