脚手架约束子集动态规划用于精确SSE聚类

中文摘要 AI 辅助

精确的欧几里得K均值将n个观测值划分为K个无标签簇，但无约束搜索通常是指数级的。我们利用数据驱动的几何图对精确子集动态规划进行预处理：因此，仅允许连通顶点子集作为簇，而平方误差和（SSE）损失保持不变。一个剩余集递推最小化固定K或惩罚的SSE，并在每个剩余集的连通分量上进行精确分解。我们研究的核心问题是，在保持无约束最优的同时，可以移除多少计算支持。图的包含关系给出了单调覆盖和支持关系，瓶颈阈值识别了嵌套层次中的第一个覆盖图。对于固定的K和维度，在紧球支撑和密度界限下，保留每个观测值的q=O(log n)个最近邻，以趋于1的概率保持经验SSE最优，使用完整图边的O(log n/n)比例。具有不等权重和协方差的截断高斯混合满足这些条件。我们提供的速率是充分上界，而非暗示多项式优化复杂性的结果。目标匹配的合成和全数据比较评估了覆盖、压缩和参考标签一致性。作为次要应用，我们展示了所提出的脚手架预处理如何用于提高保持无约束混合后验的分裂-合并提议的效率。

英文摘要

Exact Euclidean \(K\)-means partitions \(n\) observations into \(K\) unlabelled clusters, but the unrestricted search is generally exponential. We use data-derived geometric graphs to precondition an exact subset dynamic program: as a result only connected vertex subsets are admitted as clusters, while sum-of-squared-errors (SSE) loss is unchanged. A remaining-set recurrence minimises fixed-\(K\) or penalised SSE, with exact factorisation over the connected components of each remaining set. The central question we study is how much computational support can be removed while preserving an unrestricted optimum. Graph inclusion gives monotone coverage and support relations, and a bottleneck threshold identifies the first covering graph in a nested hierarchy. For fixed \(K\) and dimension, under compact ball support and density bounds, retaining \(q=O(\log n)\) nearest neighbours per observation preserves an empirical SSE optimum with probability tending to one, using an \(O(\log n/n)\) fraction of complete-graph edges. Truncated Gaussian mixtures with unequal weights and covariances satisfy these conditions. The rate we provide is a sufficient upper bound rather than a result implying polynomial optimisation complexity. Objective-matched synthetic and full-data comparisons assess coverage, compression, and reference-label agreement. As a secondary application, we illustrate how the proposed scaffold preconditioning can be utilized to improve the efficiency of split-merge proposals that preserve unrestricted mixture posteriors.

发表机构

• University of Nottingham（诺丁汉大学）
• University of Birmingham（伯明翰大学）

机构由 AI 辅助整理，请以论文原文为准。