This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely-used iterative Bregman projections algorithm (or Sinkhorn--Knopp algorithm). We first propose to rely on regularized nonlinear acceleration schemes. In practice, such approaches lead to fast algorithms, but their global convergence is not ensured. Hence, we next propose a new algorithm with convergence guarantees. The idea is to overrelax the Bregman projection operators, allowing for faster convergence. We propose a simple method for establishing global convergence by ensuring the decrease of a Lyapunov function at each step. An adaptive choice of overrelaxation parameter based on the Lyapunov function is constructed. We also suggest a heuristic to choose a suitable asymptotic overrelaxation parameter, based on a local convergence analysis. Our numerical experiments show a gain in convergence speed by an order of magnitude in certain regimes.

翻译：文章描述了一套快速计算正规化最佳运输问题解决方案的方法。 它概括并改进了广泛使用的迭接Bregman预测算法( 或 Sinkhorn- Knopp 算法 ) 。 我们首先建议依赖非线性正常加速计划。 实际上, 此类方法会导致快速算法, 但无法确保它们的全球趋同。 因此, 我们接下来提出一种新的算法, 并提供趋同保证。 其想法是过度放松布雷格曼投影操作员, 以便更快地趋同 。 我们提出一种简单的方法, 通过确保每步减少一个 Lyapunov 函数来建立全球趋同。 构建基于 Lyapunov 函数的过度放松参数的适应性选择 。 我们还建议, 根据本地趋同分析, 选择一个适合的无弹性过度减速参数。 我们的数字实验显示, 在某些制度中, 以一个数量级的顺序在趋同速度上有所增益 。