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

TL;DR

This paper investigates the contraction properties of Schr"odinger bridge systems for stochastic linear dynamics, offering new geometric insights and methods to enhance computational efficiency through preconditioning techniques.

Contribution

It provides novel geometric and control-theoretic interpretations of contraction coefficients and suggests improved computation methods via preconditioning for Schr"odinger bridge problems.

Findings

New geometric interpretations of contraction coefficients.

Control-theoretic insights into Schr"odinger systems.

Potential for improved computation through preconditioning.

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.