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：逐步施加高斯噪声可将复杂的数据分布变换为近似高斯分布。逆转这一动态过程便定义了一个生成模型。当正向加噪过程由随机微分方程（SDE）给出时，Song等人（2021）展示了如何通过分数匹配估计关联逆向时间SDE的非齐次漂移项。该方法的一个局限性在于，正向时间SDE必须运行足够长的时间，以使最终分布近似为高斯分布。相比之下，求解薛定谔桥问题（SB），即路径空间上的熵正则化最优传输问题，可生成在有限时间内从数据分布采样的扩散过程。我们提出扩散薛定谔桥（DSB），即迭代比例拟合（IPF）流程的一种原创近似方法，用于求解SB问题，并提供了理论分析及生成建模实验。DSB的第一次迭代恢复了Song等人（2021）提出的方法，同时允许使用更短的时间间隔，因为后续DSB迭代减少了正向（或逆向）SDE在最终时间边缘分布与先验（或数据）分布之间的偏差。除了生成建模，DSB还提供了一种广泛适用的计算最优传输工具，作为连续状态空间版本中流行的Sinkhorn算法（Cuturi, 2013）的对应物。