Schrödinger Bridge Problem Equivalence Demonstrated in Lie Group Dynamics

This paper establishes a formal equivalence between two distinct mathematical frameworks for solving the Schrödinger bridge problem (SBP) when applied to compact connected Lie groups. The research demonstrates that the stochastic optimal control formulation is mathematically identical to the path space formulation. This equivalence offers analytical benefits, aiding in proving the existence and uniqueness of solutions, and provides probabilistic insights into the nature of the Schrödinger bridge as the most probable deviation from uncontrolled dynamics. Computationally, the findings enable the use of static Sinkhorn recursions for direct problem-solving and optimal path measure computation, with a numerical illustration provided on the torus. AI

IMPACT This research contributes to the theoretical underpinnings of generative models and diffusion processes, potentially influencing future algorithmic development.

RANK_REASON Academic paper detailing a theoretical equivalence in a mathematical problem. [lever_c_demoted from research: ic=1 ai=0.7]

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• Girsanov theorem
• Lie Groups
• Schrödinger Bridges
• Sinkhorn recursions
• torus
• Wiener measure

• paper
• other

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