用于线性排列和层次聚类的多层剥离法
Multi Layer Peeling for Linear Arrangement and Hierarchical Clustering
AI总结：
该研究提出多层剥离技术，为度量空间下的最大线性排列和最大层次聚类问题设计了1-ε近似方案，显著优于以往的0.5和0.74近似比。
摘要：
我们提出了一种新的多层剥离技术，用于对度量空间中的点进行聚类。一个众所周知的非参数目标是将度量空间嵌入到更简单的结构化度量空间中，例如直线（即线性排列）或二叉树（即层次聚类）。度量空间中相近的点应被映射到直线/树中相近的点/叶节点；同理，度量空间中相距较远的点在直线或树上也应相距较远。我们特别研究了应用于度量的最大线性排列问题（引用自《最大线性排列的近似算法》）和最大层次聚类问题（引用自《层次聚类：目标函数与算法》）。\n 我们为这些目标设计了近似方案（对于任意常数ε>0，可实现1-ε的近似比）。这尤其表明，通过考虑度量特性，可以显著改进以往的近似结果（最大线性排列为0.5，最大层次聚类为0.74）。我们的核心技术称为多层剥离，包括递归剥离那些远离度量空间“核心”的点。当核心变成一个足够稠密的加权度量空间（即平均距离至少为直径的常数倍），或者核心对目标函数的内部贡献可忽略不计时，递归终止。有趣的是，线性排列场景下的算法比层次聚类场景下的算法要复杂得多，并且使用了精细得多的剥离策略。
英文摘要：
We present a new multi-layer peeling technique to cluster points in a metric space. A well-known non-parametric objective is to embed the metric space into a simpler structured metric space such as a line (i.e., Linear Arrangement) or a binary tree (i.e., Hierarchical Clustering). Points which are close in the metric space should be mapped to close points/leaves in the line/tree; similarly, points which are far in the metric space should be far in the line or on the tree. In particular we consider the Maximum Linear Arrangement problem \cite{Approximation_algorithms_for_maximum_linear_arrangement} and the Maximum Hierarchical Clustering problem \cite{Hierarchical_Clustering:_Objective_Functions_and_Algorithms} applied to metrics. We design approximation schemes ($1 - ε$ approximation for any constant $ε> 0$) for these objectives. In particular this shows that by considering metrics one may significantly improve former approximations ($0.5$ for Max Linear Arrangement and $0.74$ for Max Hierarchical Clustering). Our main technique, which is called multi-layer peeling, consists of recursively peeling off points which are far from the "core" of the metric space. The recursion ends once the core becomes a sufficiently densely weighted metric space (i.e. the average distance is at least a constant times the diameter) or once it becomes negligible with respect to its inner contribution to the objective. Interestingly, the algorithm in the Linear Arrangement case is much more involved than that in the Hierarchical Clustering case, and uses a significantly more delicate peeling.
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