The Schr\"odinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds. We provide the code at https://github.com/Daniil-Selikhanovych/ASBM.

翻译：薛定谔桥（SB）问题为结合最优传输与扩散模型提供了一个强大的框架。近期一种解决SB问题的有前景方法是迭代马尔可夫拟合（IMF）过程，该过程在连续时间随机过程的马尔可夫投影与互易投影之间交替进行。然而，由于使用了随机微分方程数值求解器的多步计算，IMF过程构建的模型具有较长的推理时间。为克服这一限制，我们提出了一种新颖的离散时间IMF（D-IMF）过程，其中随机过程的学习被替换为仅学习离散时间下的少数转移概率。其显著优势在于，在实践中可以自然地利用去噪扩散生成对抗网络（DD-GAN）这一已成熟的对抗性生成建模技术来实现。我们证明，D-IMF过程能够提供与IMF相当的非配对域转换质量，而仅需数步生成步骤而非数百步。代码发布于https://github.com/Daniil-Selikhanovych/ASBM。