Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SBs between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SBs, and various other examples. The implementation is publicly available at https://github.com/georgeRapa/GMMflow.

翻译：薛定谔桥是一种扩散过程，它能在有限时间内将给定的初始分布引导至目标终态分布，同时最小化适当的代价泛函。尽管近期文献中已提出多种计算薛定谔桥的方法，但大多数方法需要计算成本高昂的训练方案，即使对于求解低维问题也是如此。本研究提出了一种可行策略集合的解析参数化方法，用于将动态系统的分布从一个高斯混合模型引导至另一个。该方法不依赖标准的非凸优化技术，而是将集合内的最优策略近似为低维线性规划问题的解，其维度随各混合模型中分量数量线性增长。所提方法可自然推广至更广泛的动态系统类别，如可控线性时变系统，从而为高斯混合模型间的多边际动量薛定谔桥——一个具有挑战性的分布插值问题——提供高效解决方案。我们在中低维度问题中展示了该方法的潜力，包括自编码器隐空间中的图像到图像转换、利用多边际动量薛定谔桥学习细胞动力学，以及其他多种应用案例。实现代码已公开于 https://github.com/georgeRapa/GMMflow。