Schrödinger Bridge Over A Compact Connected Lie Group

2026-03-14

AI Summary

This study addresses the optimal control of controlled diffusion processes on compact connected Lie groups, aiming to steer an initial probability density to a prescribed terminal density with minimal control cost. To this end, the authors propose a coordinate-free stochastic optimal control framework that formulates the Schrödinger bridge problem intrinsically via the geometric structure of the Lie group, thereby circumventing the limitations of local parametrizations or Euclidean embeddings. By integrating tools from differential geometry, stochastic control theory, and analysis of Schrödinger systems, they establish existence and uniqueness of solutions and derive an explicit optimal geometric controller. The approach is validated through efficient numerical implementations on SO(2) and SO(3), demonstrating precise and optimal transport of probability densities.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$.

Problem

• Schrödinger bridge
• Lie group
• stochastic optimal control
• probability density steering
• diffusion process

Innovation

• Schrödinger bridge
• Lie group
• coordinate-free formulation
• stochastic optimal control
• geometric controller