寻找最优多边缘量子桥，本质是度量值顶点的最小生成树问题。
Optimal Multimarginal Schrödinger Bridge: Minimum Spanning Tree over Measure-valued Vertices
- 将多边缘量子桥优化转化为度量值顶点的最小生成树问题。
- 通过构建边权为双边缘最优值与端点熵之和的完全图求解。
- 适用于需建模多变量相关结构的统计推断与生成建模场景。
多边缘量子桥（MSB）用于寻找具有已知统计特性及关联结构的随机向量组间的最优耦合。在传统MSB中，关联结构以度量值顶点的无向连通图为先验给定。本文提出并求解了在所有可能图结构中寻找最优MSB的问题，发现该问题等价于度量值顶点上的最小生成树问题。求解分为两步：第一步构建完全图，边权为对应双边缘量子桥最优值与端点熵之和；第二步在此加权完全图上求解标准最小生成树。数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
The Multimarginal Schrödinger Bridge (MSB) finds the optimal coupling among a collection of random vectors with known statistics and a known correlation structure. In the MSB formulation, this correlation structure is specified \emph{a priori} as an undirected connected graph with measure-valued vertices. In this work, we formulate and solve the problem of finding the optimal MSB in the sense we seek the optimal coupling over all possible graph structures. We find that computing the optimal MSB amounts to solving the minimum spanning tree problem over measure-valued vertices. We show that the resulting problem can be solved in two steps. The first step constructs a complete graph with edge weight equal to a sum of the optimal value of the corresponding bimarginal SB and the entropies of the endpoints. The second step solves a standard minimum spanning tree problem over that complete weighted graph. Numerical experiments illustrate the proposed solution.
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