ICLR 2025

Published as a conference paper at ICLR 2025
TOPOLOGICAL SCHR ¨ODINGER BRIDGE MATCHING
Maosheng Yang
Delft University of Technology
m.yang-2@tudelft.nl
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 genera-
tive 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 net-
work entities, such as node signals and edge flows. In this work, we propose the
Topological Schr¨odinger Bridge problem (T SBP) for matching signal distribu-
tions on a topological domain, where we set the reference process to follow some
linear tractable topology-aware stochastic dynamics such as topological heat dif-
fusion. For the case of Gaussian boundary distributions, we derive a closed-form
Gaussian topological SB 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 T SB-based models for match-
ing 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 T SB-
based models on both synthetic and real-world networks, emphasizing the role of
topology. Additionally, we discuss the connections of T SB-based models to other
emerging models, and outline future directions for topological signal matching.
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INTRODUCTION
As a fundamental problem in statistics and optimization, matching distributions aims to find a map
that transforms one distribution to anot...