On the Contraction Coefficient of the Schrödinger Bridge for
  Stochastic Linear Systems
Alexis M. H. Teter Yongxin Chen Abhishek Halder
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.