On the Contraction Coefficient of the Schrödinger Bridge for Stochastic Linear Systems
This work offers incremental improvements in computational efficiency for solving Schrödinger bridge problems, primarily benefiting researchers in stochastic control and optimal transport.
The paper tackles the problem of estimating contraction coefficients for the convergence of Schrödinger bridge systems in stochastic linear settings, providing new geometric and control-theoretic interpretations and suggesting improved computation methods through preconditioning of endpoint support sets.
Schrödinger 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 Schrödinger 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 Schrödinger systems with guaranteed linear convergence. In this work, we study a priori estimates for the contraction coefficients associated with the convergence of respective Schrödinger 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.