The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor $\beta$. Here we prove that for $\beta \geq 2$ and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from $O(\ell^2)$ to $O(\ell)$ without any loss in accuracy, where $\ell$ is the number of discretization points for a single marginal.

翻译：近期提出的遗传列生成（Genetic Column Generation, GenCol）算法在数值实验中已被观察到能够高效且精确地计算通用多边缘问题的高维最优运输方案，但该算法的理论结果迄今尚属空白。该算法在一个由稀疏方案构成的动态更新的低维子流形上求解OT线性规划问题。该子流形的维度仅比最优方案的稀疏支撑集多出固定倍数$\beta$。本文证明，对于$\beta \geq 2$且在两边缘情形下，GenCol算法始终收敛于精确解，且适用于任意代价函数和边缘分布。该证明依赖于c-循环单调性概念。作为衍生结果，GenCol算法严格地将求解两边缘OT问题的数据复杂度从$O(\ell^2)$降低至$O(\ell)$，且不损失任何精度，其中$\ell$为单个边缘的离散化点数。