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This research establishes a rigorous mathematical equivalence between stochastic optimal control and path space formulations for Schrödinger bridges on compact Lie groups, bridging geometric probability with machine learning theory. By demonstrating that the problem reduces to relative entropy minimization solvable via static Sinkhorn recursions, the work offers a novel computational pathway for high-dimensional transport problems relevant to generative modeling and diffusion processes. The provided open-source code facilitates practical experimentation with these advanced theoretical frameworks on specific manifolds like the torus.
On the Equivalence of Stochastic Control and Path Space Formulations for Schr\"odinger Bridges over Compact Connected Lie Groups
arXiv:2609.13758v1 Announce Type: cross Abstract: We establish the equivalence between the stochastic optimal control and path space formulations of the Schr\"odinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP
Abstract:We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint this http URL result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: this https URL
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Systems and Control (eess.SY); Probability (math.PR); Machine Learning (stat.ML)
arXiv:2609.13758 [math.OC]
(or arXiv:2609.13758v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.13758 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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