论文标题
当schrödinger问题没有解决方案时，sindhorn算法的收敛性
Convergence of the Sinkhorn algorithm when the Schrödinger problem has no solution
论文作者
论文摘要
sindhorn算法是解决熵最小化问题的最流行方法，称为schrödinger问题：在非分类情况下，后者接受了一种独特的解决方案，该解决方案是算法收集线性的独特解决方案。在这里，由于Schrödinger问题在结构化随机过程（例如增加）方面的最新应用，我们研究了脱杂种算法在退化的情况下，可能发生根本不存在解决方案的情况。我们表明，在这种情况下，算法最终在两个限制点之间交替。此外，这些限制点可用于计算schrödinger问题的放松版本的解决方案，该解决方案似乎是$γ$ - 限制问题的限制，在这种问题中，边缘约束被渐近较大的边缘惩罚所取代，正是本着所谓的不平衡最佳运输的精神。最后，我们的工作着重于对放松问题的解决方案的支持，给出了典型的形状并设计了一个快速计算的程序。我们展示了与细胞生物学中使用的模型相关的有希望的数值应用。
The Sinkhorn algorithm is the most popular method for solving the entropy minimization problem called the Schrödinger 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 Schrödinger 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 Schrödinger problem, which appears as the $Γ$-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.
