Optimal Multimarginal Schrödinger Bridge: Minimum Spanning Tree over Measure-valued Vertices

2025-09-12

AI Summary

This paper addresses the problem of automatically learning the optimal graph structure for optimal probabilistic coupling among multiple random vectors in multi-marginal Schrödinger bridges (MSBs). The method formulates graph structure optimization as a minimum spanning tree (MST) problem over the space of probability measures: edge weights of a complete graph are defined via pairwise marginal Schrödinger bridge transport costs and entropy regularization terms, and the optimal topology is obtained via classical MST algorithms; this is then integrated with joint optimization of graph structure and coupling, yielding a two-stage framework. Theoretically, the problem is proven to be decomposable. Numerical experiments demonstrate accurate recovery of ground-truth connectivity across diverse measure configurations, preserving coupling fidelity while substantially improving interpretability and computational efficiency. This work establishes, for the first time, a systematic connection between MSB graph learning and combinatorial optimization.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Finding optimal coupling over all graph structures
• Solving minimum spanning tree with measure-valued vertices
• Determining optimal Multimarginal Schrödinger Bridge configuration

Innovation

• Minimum spanning tree over measure-valued vertices
• Optimal coupling over all graph structures
• Bimarginal Schrödinger Bridge edge weights