发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种带对数对偶更新的熵Bregman ADMM算法求解离散最优传输问题，证明了其R-线性收敛性，并在二维网格上利用核结构加速，数值实验显示比HOT和HALO等求解器有显著加速。
AI 中文摘要
本文研究了一种带有对数对偶更新的熵Bregman交替方向乘子法（BADMM），用于求解离散最优传输（OT）问题。该算法使用简单的行和列归一化，无需内部迭代，从而支持高效的CPU和GPU实现。然而，该方法对于一般OT问题的收敛性尚未被建立。从一个严格正可行的原始点和一个零乘子出发，我们建立了对于一般OT代价、任意固定乘子步长在(0,2)内且足够大的固定惩罚项的原始-对偶序列的$R$-线性收敛性，即使原始极限位于边界上也是如此。我们的分析进一步将原始极限刻画为初始点在最优解集上的熵Bregman投影，并建立了极限OT原始-对偶解的严格互补性。对于二维网格上的可分离代价，我们利用核结构来加速矩阵-向量乘积并减少存储需求。数值实验显示，对于一般代价矩阵，该方法具有竞争力的性能，并且在二维网格边缘的平方欧氏代价上，相较于最先进的求解器（包括HOT和HALO）实现了显著的加速。最大的测试问题，每个边缘有$2048\ imes2048$个网格点，在单个GPU上约100秒内求解完成。
英文摘要
This paper studies an entropy Bregman alternating direction method of multipliers (BADMM) with logarithmic dual update for solving discrete optimal transport (OT) problems. The algorithm uses simple row and column normalizations without inner iterations, allowing efficient CPU and GPU implementations. However, the convergence of this method for general OT problems has not been established. Starting from a strictly positive feasible primal point and a zero multiplier, we establish $R$-linear convergence of the primal--dual sequence for general OT costs with any fixed multiplier stepsize in $(0,2)$ and a sufficiently large fixed penalty, even when the primal limit lies on the boundary. Our analysis further characterizes the primal limit as the entropy Bregman projection of the initial point onto the optimal solution set and establishes strict complementarity of the limiting OT primal--dual solution. For separable costs on two-dimensional grids, we exploit the kernel structure to accelerate matrix--vector products and reduce storage requirements. Numerical experiments show competitive performance for general cost matrices and substantial speedups over state-of-the-art solvers, including HOT and HALO, for squared Euclidean costs on two-dimensional-grid marginals. The largest tested problem, with $2048\times2048$ grid points in each marginal, is solved in approximately 100 seconds on a single GPU.
Comments49 pages, 2 figures
