On the Contraction Coefficient of the Schr\"odinger Bridge for Stochastic Linear Systems

• URL: http://arxiv.org/abs/2309.06622v1
• Date: Tue, 12 Sep 2023 22:24:05 GMT
• Title: On the Contraction Coefficient of the Schr\"odinger Bridge for Stochastic Linear Systems
• Authors: Alexis M.H. Teter, Yongxin Chen, Abhishek Halder
• Abstract summary: A popular method to numerically solve the Schr"odinger bridge problems is via contractive fixed point recursions. We study a priori estimates for the contraction coefficients associated with the convergence of respective Schr"odinger systems.
• Score: 15.022863946000495
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Schr\"{o}dinger bridge is a stochastic optimal control problem to steer a given initial state density to another, subject to controlled diffusion and deadline constraints. A popular method to numerically solve the Schr\"{o}dinger bridge problems, in both classical and in the linear system settings, is via contractive fixed point recursions. These recursions can be seen as dynamic versions of the well-known Sinkhorn iterations, and under mild assumptions, they solve the so-called Schr\"{o}dinger systems with guaranteed linear convergence. In this work, we study a priori estimates for the contraction coefficients associated with the convergence of respective Schr\"{o}dinger systems. We provide new geometric and control-theoretic interpretations for the same. Building on these newfound interpretations, we point out the possibility of improved computation for the worst-case contraction coefficients of linear SBPs by preconditioning the endpoint support sets.

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