Topological Schr""odinger Bridge Matching

2025-04-07

AI Summary

This paper addresses signal distribution matching on topological structures—such as graphs and simplicial complexes—by proposing the Topological Schrödinger Bridge (TSB) framework, the first to extend Schrödinger bridge theory to topology-aware stochastic dynamics. Methodologically, TSB constructs forward–backward stochastic processes grounded in topological heat diffusion, derives closed-form solutions under Gaussian boundary conditions, and integrates topological neural networks with stochastic differential equations for learnable dynamics modeling and maximum-likelihood training. Key contributions include: (i) establishing the first theoretical framework for Schrödinger bridges on topological domains; (ii) introducing a unified paradigm for topological optimal transport; and (iii) demonstrating empirically that TSB significantly improves matching accuracy for both node signals and edge flows—validated on synthetic and real-world network data, confirming the critical role of topological structure in distribution alignment.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing

Abstract

Given two boundary distributions, the Schr""odinger Bridge (SB) problem seeks the ``most likely`` random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for generative modeling and distribution matching. While these methods perform well in Euclidean domains, they are not directly applicable to topological domains such as graphs and simplicial complexes, which are crucial for data defined over network entities, such as node signals and edge flows. In this work, we propose the Topological Schr""odinger Bridge problem (TSBP) for matching signal distributions on a topological domain. We set the reference process to follow some linear tractable topology-aware stochastic dynamics such as topological heat diffusion. For the case of Gaussian boundary distributions, we derive a closed-form topological SB (TSB) in terms of its time-marginal and stochastic differential. In the general case, leveraging the well-known result, we show that the optimal process follows the forward-backward topological dynamics governed by some unknowns. Building on these results, we develop TSB-based models for matching topological signals by parameterizing the unknowns in the optimal process as (topological) neural networks and learning them through likelihood training. We validate the theoretical results and demonstrate the practical applications of TSB-based models on both synthetic and real-world networks, emphasizing the role of topology. Additionally, we discuss the connections of TSB-based models to other emerging models, and outline future directions for topological signal matching.

Problem

• Extends Schrödinger Bridge to topological domains like graphs
• Derives closed-form solution for Gaussian boundary distributions
• Develops neural models for matching topological signal distributions

Innovation

• Proposes Topological Schrödinger Bridge for graphs
• Uses topology-aware stochastic dynamics reference
• Parameterizes unknowns with neural networks