发表机构
College of Connected Computing; Vanderbilt University（互联计算学院; 范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对圆上的部分最优传输问题，提出时间复杂度为O(N log N)的PAWC算法，可高效返回所有传输基数的最优成本，在实验中展现出比现有方法更优的鲁棒性与计算效率。
AI 中文摘要
部分最优传输用于比较两个测度，同时使部分质量不匹配，这使其对异常值、遮挡和杂乱具有鲁棒性。研究关注的量通常是整个轮廓——即每个传输基数下的最优成本，因为实际中很少预先知道合适的传输量；在实直线上，PAWL算法可在O(N log N)时间内返回该轮廓。许多数据是周期性的而非线性的，例如角度、相位、方向、一天中的时间、色调，以及投影到大圆上得到的所有方向。在圆上，同一问题会产生全局环流，或等价的优化割，朴素的精确方法通过在每个支撑间隙运行一次直线算法来处理，时间复杂度为O(N² log N)。我们证明该因子N是不必要的，直线结构以无割形式存在，且自由间隙不变量在每一步提供一个割，使得所有先前的局部更新仍为有效的直线更新。这得到PAWC算法：一种精确的O(N log N)时间、O(N)内存算法，可在一次运行中返回所有K+1个成本、嵌套活动集和计划，以及一个对每个基数同时最优的单一间隙。通过对大圆进行切片，该算法可扩展到ℝ^(d-1)。实验表明，当N=4096时，整个轮廓的计算成本为0.56毫秒，而通用求解器单次传输分数的成本为1.5秒；在被遮挡、杂乱的mpeg-7形状上，固定描述符仅改变成本时，该算法保留了66%的干净数据检索分数，而平衡圆形OT的该值为16%；在ℝ²上，它将球形切片Wasserstein对受污染目标（合成和真实目标）的拟合误差降低了一半。代码可在提供的URL获取。
英文摘要
Partial optimal transport compares two measures while leaving part of the mass unmatched, which is what makes it robust to outliers, occlusion, and clutter. The quantity of interest is usually the whole profile - the optimal cost at every transported cardinality - because the right amount to transport is rarely known in advance, and on the real line the PAWL algorithm returns that profile in $O(N\log N)$. Much data is periodic rather than linear: angles, phases, orientations, time of day, hue, and every direction obtained by projecting onto a great circle. On the circle the same problem acquires a global circulation, or equivalently an optimized cut, which the naive exact method handles by running the line algorithm once per support gap, at $O(N^{2}\log N)$. We show that this factor $N$ is unnecessary. The line structure survives in cut-free form, and a free-gap invariant supplies, at every step, a cut at which all previous local updates remain valid line updates. This yields PAWC: an exact $O(N\log N)$ time, $O(N)$ memory algorithm returning all $K+1$ costs, nested active sets and plans in one run, together with a single gap that is simultaneously optimal for every cardinality. Slicing over great circles extends it to $\mathbb{S}^{d-1}$. Empirically the whole profile costs $0.56$ms at $N=4096$ against $1.5$s for a single transported fraction from a general solver; on occluded, cluttered mpeg-7 shapes, holding the descriptor fixed and varying only the cost, it retains $66\%$ of the clean-data retrieval score against $16\%$ for balanced circular OT, and on $\mathbb{S}^{2}$ it halves the fitting error of spherical sliced Wasserstein against contaminated targets, synthetic and real. Code is available at https://github.com/mint-vu/Partial_Wasserstein_on_Circles.
