A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy 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 an adjustable 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 any of our generative models built upon stochastic dynamics. By contrast, we establish that generative models based upon a deterministic dynamics must, in addition, control the Fisher divergence between the target and the model. We also construct estimators for the likelihood and the cross-entropy of interpolant-based generative models, discuss connections with other stochastic bridges, and demonstrate that such models recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant.

翻译：本文提出了一类统合基于流和基于扩散的生成模型的生成模型框架。这些模型扩展了Albergo & Vanden-Eijnden (2023)提出的框架，允许使用一类称为"随机插值"的广泛连续时间随机过程，在有限时间内精确衔接任意两个概率密度函数。这些插值通过结合两个指定密度的数据与一个额外潜变量构建而成，该潜变量以灵活方式塑造桥接过程。研究表明，随机插值的时间依赖概率密度函数满足一阶输运方程以及具有可调扩散系数的正向和反向福克-普朗克方程族。在考虑单个样本的时间演化时，这一视角直接导出基于概率流方程或具有可调噪声水平的随机微分方程的确定性和随机生成模型。这些模型中的漂移系数是时间依赖的速度场，其特性表现为简单二次目标函数的唯一极小化器，其中一项是针对插值密度分数的新目标函数。值得注意的是，我们证明最小化这些二次目标函数能够控制基于随机动力学构建的任何生成模型的似然性。相比之下，我们确立基于确定性动力学的生成模型必须额外控制目标分布与模型之间的费舍尔散度。我们还构建了基于插值的生成模型的似然函数和交叉熵估计器，讨论与其他随机桥接方法的联系，并证明当显式对插值进行优化时，此类模型可恢复两个目标密度之间的薛定谔桥。