The Sinkhorn algorithm is the most popular method for solving the entropy minimization problem called the Schr\"odinger problem: in the non-degenerate cases, the latter admits a unique solution towards which the algorithm converges linearly. Here, motivated by recent applications of the Schr\"odinger problem with respect to structured stochastic processes (such as increasing ones), we study the Sinkhorn algorithm in degenerate cases where it might happen that no solution exist at all. We show that in this case, the algorithm ultimately alternates between two limit points. Moreover, these limit points can be used to compute the solution of a relaxed version of the Schr\"odinger problem, which appears as the $\Gamma$-limit of a problem where the marginal constraints are replaced by asymptotically large marginal penalizations, exactly in the spirit of the so-called unbalanced optimal transport. Finally, our work focuses on the support of the solution of the relaxed problem, giving its typical shape and designing a procedure to compute it quickly. We showcase promising numerical applications related to a model used in cell biology.

翻译：Sinkhorn算法是求解称为薛定谔问题的熵最小化问题的最流行方法：在非退化情形下，该问题存在唯一解，算法以线性速率收敛至此解。本文受薛定谔问题在结构化随机过程（如递增过程）中最新应用的启发，研究当问题可能完全无解时的退化情形下Sinkhorn算法的行为。我们证明在此情况下，算法最终会在两个极限点之间交替震荡。此外，这两个极限点可用于计算薛定谔问题松弛版本的解，该松弛问题表现为边际约束被渐近放大的边际惩罚项所替代问题的$\Gamma$-极限——这正是不平衡最优传输的核心思想。最后，我们的工作聚焦于松弛问题解的支撑集，刻画其典型形态并设计快速计算流程。我们展示了与细胞生物学模型相关的数值应用前景。