将流模型方法扩展到树结构代价，高效求解熵正则化沃瑟斯坦中位数。
Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres
- 基于流模型的迭代马尔可夫拟合，逐步匹配路径分布
- 在树结构代价下实现比传统方法更快的收敛与更优稳定性
- 适用于生成模型中的多分布融合任务，如图像中位数生成
基于流的生成建模近期实现了对分布间薛定谔桥（Schrödinger Bridge, SB）的高效计算，这是一种针对二次代价的熵正则化最优传输（OT）动态形式。成功的迭代马尔可夫拟合（IMF）方法通过一系列桥接匹配步骤求解SB问题，相较于传统的迭代比例拟合（IPF），具有更优的性质和实用性。在标准设置之外，最优传输可推广至多边际情形，即最小化多个边缘分布之间的联合代价。其中，树结构代价尤为重要，其特殊情形可恢复沃瑟斯坦中位数。本文将IMF方法扩展至树结构的薛定谔桥问题。所提出的算法在树结构设定下继承了IMF相比IPF的诸多优势。在沃瑟斯坦中位数情况下，本方法可视为将广泛使用的固定点法扩展为使用基于流的熵正则化OT求解器，且每轮仅需简单的桥接匹配步骤。
原文摘要 · Abstract (English)
Recent advances in flow-based generative modelling have provided scalable methods for computing the Schrödinger Bridge (SB) between distributions, a dynamic form of entropy-regularised Optimal Transport (OT) for the quadratic cost. The successful Iterative Markovian Fitting (IMF) procedure solves the SB problem via sequential bridge-matching steps, presenting an elegant and practical approach with many favourable properties over the more traditional Iterative Proportional Fitting (IPF) procedure. Beyond the standard setting, optimal transport can be generalised to the multi-marginal case in which the objective is to minimise a cost defined over several marginal distributions. Of particular importance are costs defined over a tree structure, from which Wasserstein barycentres can be recovered as a special case. In this work, we extend the IMF procedure to solve for the tree-structured SB problem. Our resulting algorithm inherits the many advantages of IMF over IPF approaches in the tree-based setting. In the case of Wasserstein barycentres, our approach can be viewed as extending the widely used fixed-point approach to use flow-based entropic OT solvers, while requiring only simple bridge-matching steps at each iteration.
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