Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

We introduce a class of generative models based on the stochastic interpolant framework proposed in Albergo & Vanden-Eijnden (2023) that unifies flow-based and diffusion-based methods. We first show how to construct a broad class of continuous-time stochastic processes whose time-dependent probability density function bridges two arbitrary densities exactly in finite time. These `stochastic interpolants' are built by combining data from the two densities with an additional latent variable, and the specific details of the construction can be leveraged to shape the resulting time-dependent density in a flexible way. We then show that the time-dependent density of the stochastic interpolant satisfies a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion; upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with a tunable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. Remarkably, we show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics; by contrast, we show that generative models based upon a deterministic dynamics must, in addition, control the Fisher divergence between the target and the model. Finally, we construct estimators for the likelihood and the cross-entropy of interpolant-based generative models, and demonstrate that such models recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant.