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Schrödinger Bridge Over A Compact Connected Lie Group

Hamza Mahmood, Abhishek Halder, Adeel Akhtar

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 $\mathsf{SO}(3)$.

Schrödinger Bridge Over A Compact Connected Lie Group

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 and .
Paper Structure (10 sections, 7 theorems, 34 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. A Geometric Coordinate-free Solution
  5. Preparatory results
  6. Conditions for optimality and the Schrödinger system
  7. Numerical Results
  8. SBP on SO(2)
  9. SBP on SO(3)
  10. Concluding Remarks

Key Result

Proposition 1

Lee2025 Let $\mathsf{G}$ be a unimodular Lie group with its Lie algebra $\mathfrak{g}$, and $n:=\dim\mathfrak{g}$. Consider an inner product $\langle \cdot, \cdot \rangle_{\mathfrak{g}} : \mathfrak{g} \times \mathfrak{g} \to \mathbb{R}$ and an orthonormal basis $\{e_1^{\mathfrak{g}}, e_2^{\mathfrak{

Figures (2)

  • Figure 1: PDFs $\rho^{\mathrm{opt}}(\cdot,t)$, $t\in[0,1]$, for numerical example in Sec. \ref{['subsec:SO2numericalexample']}.
  • Figure 2: Main plot: PDFs $\rho^{\mathrm{opt}}(\cdot,t)$, $t\in[0,1]$ for numerical example in Sec. \ref{['subsec:SO3numericalexample']}. Inset: Sinkhorn recursion convergence for pair $(\varphi_1,\widehat{\varphi}_0)$ w.r.t. $d_{H}$.

Theorems & Definitions (16)

  • Definition 1: Stratonovich SDE for the Kinematics on a Compact Lie Group
  • Remark 1: Unimodularity and compact Lie groups
  • Proposition 1: Brownian Motion on a Unimodular Lie Group
  • Definition 2: Closed solid cone in a Banach space Bus1973
  • Definition 3: Hilbert's projective metric Bus1973
  • Remark 2
  • Lemma 1: Interior of $C_{\geq 0}(\mathsf{G})$
  • Lemma 2
  • Lemma 3
  • Remark 3
  • ...and 6 more