On Convergence of Approximate Schr\"{o}dinger Bridge with Bounded Cost

The Schr\"odinger bridge has demonstrated promising applications in generative models. It is an entropy-regularized optimal-transport (EOT) approach that employs the iterative proportional fitting (IPF) algorithm to solve an alternating projection problem. However, due to the complexity of finding precise solutions for the projections, approximations are often required. In our study, we study the convergence of the IPF algorithm using approximated projections and a bounded cost function. Our results demonstrate an approximate linear convergence with bounded perturbations. While the outcome is not unexpected, the rapid linear convergence towards smooth trajectories suggests the potential to examine the efficiency of the Schrödinger bridge compared to diffusion models.