Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).
翻译:逐步应用高斯噪音可以将复杂的数据分布转换到约高斯。 扭转这一动态将定义一个基因模型。 当通过一个随机分化(SDE), Song 等人(2021年) 给出了前点点点点进程时, 演示如何使用分比法来估计相关反向时间SDE的不均匀漂移时间。 这种方法的一个局限性是, 远期 SDE 必须运行足够长的时间, 才能使最终分布达到约高斯。 相反, 解决 Schr\" odinger Bridge (SB) 的问题, 也就是说, 路径空间上的一个螺旋式固定最佳运输问题(SB), 即, 产生在有限时间内从数据分布中生成样本的分数。 我们展示了 difulation SB (DSB), 这是用于解决 SB 问题的原始比例调整程序, 并且提供理论模型分析, 以便最终的模型实验可以大致化为高斯。 首期SB 恢复 Song 等人(2021) 提议的方法,, 即, 以恒点固定固定的正态最佳的正位最佳最佳最佳最佳最佳运输方法, 将SB (SB) 的后期比值 降低前值的后基值 的后值的后值的后值的后值的后值,, 的后期的SB (SB), 的后值 的后值 的后值,,, 后值 后值 后值 后值 后值 后期的 后期的 后值 的 的 的 的, 。