  We present a formulation of flow matching as variational inference, which we
refer to as variational flow matching (VFM). Based on this formulation we
develop CatFlow, a flow matching method for categorical data. CatFlow is easy
to implement, computationally efficient, and achieves strong results on graph
generation tasks. In VFM, the objective is to approximate the posterior
probability path, which is a distribution over possible end points of a
trajectory. We show that VFM admits both the CatFlow objective and the original
flow matching objective as special cases. We also relate VFM to score-based
models, in which the dynamics are stochastic rather than deterministic, and
derive a bound on the model likelihood based on a reweighted VFM objective. We
evaluate CatFlow on one abstract graph generation task and two molecular
generation tasks. In all cases, CatFlow exceeds or matches performance of the
current state-of-the-art models.
