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. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

翻译：近期提出的遗传列生成（GenCol）算法在数值实验中已被观察到能够高效且精确地计算一般多边缘问题的高维最优传输方案，但迄今缺乏该算法的理论结果。该算法在由稀疏方案组成的动态更新的低维子流形上求解最优传输线性规划问题。子流形维度仅比最优方案的稀疏支撑集高出一个固定因子$\beta$。本文证明，在$\beta \geq 2$且为两边缘情形时，对于任意代价函数和边缘分布，GenCol算法始终收敛于精确解。该证明依赖于$c$-循环单调性概念。由此衍生，GenCol算法将两边缘最优传输问题的数据复杂度从$O(\ell^2)$严格降至$O(\ell)$（其中$\ell$为单边缘离散化点数），且精度不受任何损失。文末还给出了多边缘情形下收敛行为的一些见解。