The Schr\"odinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schr\"odinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schr\"odinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics

翻译：薛定谔桥问题旨在寻找连接两个边缘分布的随机动力系统，该系统需最小化特定传输成本。该问题作为最优传输在随机情形下的推广，因其与扩散模型及流匹配的联系以及在自然科学中的应用而受到关注。然而，现有算法仅适用于两个分布均有样本可用的情形。本文首次提出一种通用方法，用于在其中一个（或两个）分布仅由其未归一化密度给出且无法获取数据样本的情况下建模薛定谔桥。我们的算法受近期扩散采样器训练的离策略强化学习进展启发，将迭代比例拟合（IPF）过程推广至无数据情形。我们在合成问题上验证了所提出的数据到能量IPF的有效性，发现其能成功学习多峰分布间的传输。作为我们强化学习公式的附带结果（该公式假设动力学采用固定时间离散方案），我们发现通过动力学扩散系数的学习，现有数据到数据薛定谔桥算法可获得显著改进。最后，我们将新算法应用于生成模型隐空间中后验分布的采样问题，从而创建了一种无数据的图像到图像转换方法。代码：https://github.com/mmacosha/d2e-stochastic-dynamics