The Schrödinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence anal…