发表机构
Yau Mathematical Sciences Center, Tsinghua University; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications; Qiuzhen College, Tsinghua University; Department of Mathematical Sciences, Tsinghua University（丘成桐数学科学中心，清华大学; 北京雁栖湖应用数学研究院; 邱士纶研究院，清华大学; 数学科学系，清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
SWAP是一种直接作用于置换空间的可扩展最优传输求解器，通过几何感知提议实现单调下降，确保一对一约束精确满足，并在大规模高维任务中显著降低成本和资源消耗。
AI 中文摘要
大规模和高维离散最优传输问题常常面临可扩展性与精确分配可行性之间的张力：许多可扩展方法优化松弛、分解或稀疏的传输计划，而不是在整个优化过程中保持一对一的分配。我们引入了SWAP，一种直接在置换空间中操作的迭代求解器，适用于等权重点云。从任意可行分配开始，SWAP利用几何感知的提议来识别降低成本的局部重排。因此，每次迭代都是一个有效的置换，一对一约束在整个优化过程中被精确满足，无需舍入或可行性修正。SWAP避免了构建$N\ imes N$成本或耦合矩阵，仅需$O(Nd)$内存。我们建立了单调下降性质，引入了一系列循环稳定性条件，证明了在一般非平行条件下SWAP几乎必然达到成对稳定性，并推导了成对稳定性蕴含全局最优性的充分条件。在包含640,500个点、2,048维的ImageNet分配、合成基准以及MERFISH空间对齐上的实验表明，SWAP在具有挑战性的大规模和高维问题上实现了更低的传输成本，与现有方法相比，运行时间和内存显著减少。
英文摘要
Large-scale and high-dimensional discrete optimal transport often involves a tension between scalability and exact assignment feasibility: many scalable approaches optimize relaxed, factorized, or sparse transport plans rather than maintaining a one-to-one assignment throughout optimization. We introduce SWAP, an iterative solver that operates directly in permutation space for equally weighted point clouds. Starting from any feasible assignment, SWAP uses geometry-aware proposals to identify cost-decreasing local rearrangements. Consequently, every iterate is a valid permutation, so the one-to-one constraints are satisfied exactly throughout the optimization without rounding or feasibility correction. SWAP avoids constructing an $N\times N$ cost or coupling matrix and requires only $O(Nd)$ memory. We establish monotone descent, introduce a family of cycle-stability conditions, prove that SWAP reaches pairwise stability almost surely under generic nonparallel conditions, and derive sufficient conditions under which pairwise stability implies global optimality. Experiments on an ImageNet assignment with 640,500 points in 2,048 dimensions, synthetic benchmarks, and MERFISH spatial alignment demonstrate that SWAP achieves lower transport costs on challenging large-scale and high-dimensional problems, with substantially reduced runtime and memory compared with
