AI 中文总结
本文提出二点局部最优性概念，通过边界点筛选算法BPS-2PLS，在降低计算成本的同时实现更强的局部最优保证，并在多数基准上取得更优聚类结果。
AI 中文摘要
Lloyd算法和用于$k$-均值的离散局部（D-局部）优化方法（Li等人，2025）仅提供弱局部最优性保证，且其解的质量对初始化仍然敏感。在本文中，我们引入$r$-点局部最优性，在该定义下，对至多$r$个样本的重新分配不会降低目标函数，并重点关注$r=2$的情况。主要计算障碍是对$d$维空间中$n$个样本和$k$个簇进行穷举二点认证的$\nmathcal{O}(n^2(k^2+d))$成本。为解决这一挑战，我们证明：（i）D-局部最优的每个改进二点移动必须涉及两个重新分配共享的簇，且（ii）只有证书定义的边界点才能参与改进对。利用这一结构，我们提出边界点筛选二点局部搜索（BPS-2PLS），该算法终止于二点局部最优。对于固定的$k,d$和非消失的簇占用，在从有界支撑且有界密度的分布或高斯混合中进行独立同分布采样时，保留候选数$m$满足$m=\mathcal{O}_{\mathbb{P}}(\log n)$。在十二个基准测试中，BPS-2PLS在十个上达到最低的平均WCSS。在子采样研究中，在最大测试规模下，筛选平均保留0.10%至2.81%的样本。代码可在以下网址获取：此https URL。
英文摘要
Lloyd's algorithm and the discrete local (D-local) optimization method (Li et al., 2025) for $k$-means provide only weak local-optimality guarantees, and their solution quality remains sensitive to initialization. In this paper, we introduce $r$-point local optimality, under which no reassignment of at most $r$ samples decreases the objective function, and focus on $r=2$. The main computational obstacle is the $\mathcal{O}(n^2(k^2+d))$ cost of exhaustive two-point certification for $n$ samples in $d$ dimensions and $k$ clusters. To address this challenge, we prove that (i) every improving two-point move of a D-local optimum must involve a cluster shared by both reassignments, and (ii) only certificate-defined boundary points can participate in an improving pair. Exploiting this structure, we propose Boundary-Point-Screened Two-Point Local Search (BPS-2PLS), which terminates at a two-point local optimum. For fixed $k,d$ and nonvanishing cluster occupancy, the number $m$ of retained candidates satisfies $m=\mathcal{O}_{\mathbb{P}}(\log n)$ under i.i.d. sampling from a bounded-support distribution with bounded density or from a Gaussian mixture. Across twelve benchmarks, BPS-2PLS attains the lowest available mean WCSS on ten. In a subsampling study, screening retains 0.10% to 2.81% of samples on average at the largest tested sizes. The code is available at https://github.com/lwl-learning/BPS-2PLS.
Comments30 pages, including appendices. Code available at https://github.com/lwl-learning/BPS-2PLS
