We demonstrate the relevance of an algorithm called generalized iterative scaling (GIS) or simultaneous multiplicative algebraic reconstruction technique (SMART) and its rescaled block-iterative version (RBI-SMART) in the field of optimal transport (OT). Many OT problems can be tackled through the use of entropic regularization by solving the Schr\"odinger problem, which is an information projection problem, that is, with respect to the Kullback--Leibler divergence. Here we consider problems that have several affine constraints. It is well-known that cyclic information projections onto the individual affine sets converge to the solution. In practice, however, even these individual projections are not explicitly available in general. In this paper, we exchange them for one GIS iteration. If this is done for every affine set, we obtain RBI-SMART. We provide a convergence proof using an interpretation of these iterations as two-step affine projections in an equivalent problem. This is done in a slightly more general setting than RBI-SMART, since we use a mix of explicitly known information projections and GIS iterations. We proceed to specialize this algorithm to several OT applications. First, we find the measure that minimizes the regularized OT divergence to a given measure under moment constraints. Second and third, the proposed framework yields an algorithm for solving a regularized martingale OT problem, as well as a relaxed version of the barycentric weak OT problem. Finally, we show an approach from the literature for unbalanced OT problems.

翻译：我们证明了一种称为广义迭代缩放（GIS）或同步乘法代数重建技术（SMART）及其重缩放块迭代版本（RBI-SMART）的算法在最优输运（OT）领域中的相关性。许多OT问题可以通过熵正则化求解薛定谔问题来处理，该问题本质上是信息投影问题，即相对于Kullback-Leibler散度的投影。本文考虑具有多个仿射约束的问题。众所周知，循环信息投影到各个仿射集上会收敛到解。然而在实际中，即使这些单独的投影通常也无法显式获得。本文中，我们用一次GIS迭代替换每个投影步骤。若对所有仿射集执行此操作，则得到RBI-SMART。我们通过将这些迭代解释为等效问题中的两步仿射投影，提供了收敛性证明。该证明在比RBI-SMART稍一般的框架下完成，因为混合使用了显式已知的信息投影与GIS迭代。随后我们将该算法具体应用到多个OT场景中。首先，在矩约束条件下，我们找到了最小化正则化OT散度到给定测度的测度。其次和第三，所提出的框架为求解正则化鞅OT问题以及重心弱OT问题的松弛版本提供了算法。最后，我们展示了文献中用于非平衡OT问题的方法。