The Sinkhorn algorithm is the most popular method for solving the Schr\"odinger problem: it is known to converge as soon as the latter has a solution, and with a linear rate when the solution has the same support as the reference coupling. Motivated by recent applications of the Schr\^odinger problem where structured stochastic processes lead to degenerate situations with possibly no solution, we show that the Sinkhorn algorithm still gives rise in this case to exactly two limit points, that 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 marginal penalizations. These results also allow to develop a theoretical procedure for characterizing the support of the solution - both in the original and in the relaxed problem - for any reference coupling and marginal constraints. We showcase promising numerical applications related to a model used in cell biology.

翻译：Sinkhorn算法是解决Schr\'odinger问题最受欢迎的方法:当Schr\'odinger问题有解决办法时,人们就知道它会立即汇合,当解决办法具有与参照组合相同的支持时,它就会以线性速度汇合。由于Schr ⁇ odinger问题最近的应用,结构性的随机分析过程导致情况恶化,而可能没有解决办法,因此我们表明Sinkhorn算法仍然在此案中产生精确的两个限制点,可以用来计算Schr\'odinger问题的宽松版本的解决方案,这个版本似乎是一个边际限制被边际惩罚取代的问题的$\Gamma$-范围。这些结果还有助于形成理论程序,将任何参考组合和边际限制的解决方案的支持定性为原始的和宽松问题。我们展示了与细胞生物学中使用的模型有关的有希望的数字应用。