The dynamic Schr\"odinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a reference process. We propose a novel sampling-based iterative algorithm, the iterated diffusion bridge mixture transport (IDBM), aimed at solving the dynamic Schr\"odinger bridge problem. The IDBM procedure exhibits the attractive property of realizing a valid coupling between the target measures at each step. We perform an initial theoretical investigation of the IDBM procedure, establishing its convergence properties. The theoretical findings are complemented by numerous numerical experiments illustrating the competitive performance of the IDBM procedure across various applications. Recent advancements in generative modeling employ the time-reversal of a diffusion process to define a generative process that approximately transports a simple distribution to the data distribution. As an alternative, we propose using the first iteration of the IDBM procedure as an approximation-free method for realizing this transport. This approach offers greater flexibility in selecting the generative process dynamics and exhibits faster training and superior sample quality over longer discretization intervals. In terms of implementation, the necessary modifications are minimally intrusive, being limited to the training loss computation, with no changes necessary for generative sampling.

翻译：动态薛定谔桥问题旨在寻找一个随机过程，该过程定义了两个目标概率测度之间的输运，同时以最优方式满足相对于参考过程在KL散度意义上最接近的准则。我们提出了一种新颖的基于采样的迭代算法——迭代扩散桥混合输运（IDBM），旨在求解动态薛定谔桥问题。IDBM过程具有在每个步骤实现目标测度间有效耦合的优良性质。我们对IDBM过程进行了初步的理论研究，建立了其收敛性质。理论发现辅以大量数值实验，展示了IDBM过程在各种应用中的竞争性能。近期生成式建模的进展利用扩散过程的时间反转来定义一个生成过程，该过程近似地将简单分布输运到数据分布。作为替代方案，我们提出使用IDBM过程的第一次迭代作为一种无需近似的方法来实现这种输运。该方法在选择生成过程动力学方面提供了更大的灵活性，并在更长的离散化区间上展现出更快的训练速度和更优的样本质量。在实现层面，所需修改侵入性极小，仅局限于训练损失计算，生成采样无需任何改动。