On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups
Authors: Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar
Organizations: Dept. of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Applied Mathematics at the University of California at Santa Cruz, CA, 95064, USA · Dept. of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA
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 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 T2. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP
Figures & tables
Fig. 1: Evolution of the Schrödinger bridge probability density between endpoint distributions ρ0 and ρ1 on the torus T2 . In blue is shown the continuous density function ρσopt(⋅,t) – the solution of ( 7 ) computed by solving the boundary-coupled system of heat PDEs on the Lie group as in [ 10 ] . In orange are the scattered data points transported along Popt – the minimizer of ( 32 ) computed by solving the static Sinkhorn recursions.
Department of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA
Department of Mechanical Engineering, MIT, Cambridge, MA 02139 · SEAS, Harvard University, Cambridge, MA 02138 · SEAS, Physics and OEB, Harvard University, Cambridge, MA 02138