高维空间中的色聚类
Chromatic Clustering in High Dimensional Space
摘要：
本文研究了高维空间中的一种新的聚类问题--{ \em色聚类}。彩色聚类试图将一组彩色点划分为组（或聚类），以便没有组包含具有相同颜色的点，并且优化某个目标函数。在本文中，我们考虑两个变种的问题，色$k$ -均值聚类（表示为$k$ -CMeans）和色$k$ -中位数聚类（表示为$k$ -CMedians），并研究其硬度和近似解决方案。对于$k$ -CMeans，我们表明，额外的着色约束破坏了几个关键属性（如局部属性）在现有的$k$ -手段技术（普通点），并显着复杂化的问题。对于色聚类问题，没有FPTAS，即使$k=2$。为了克服额外的困难，我们开发了一个独立的结果，称为{ \em单纯形引理}，它使我们能够有效地近似的平均点的未知点集通过一个固定的维单纯形。单纯形的一个很好的特点是它与原始空间的维数无关，因此可以用于非常高维空间中的问题。与单纯形引理，再加上几个随机抽样技术，我们表明，$(1+ε)$ -近似的$k$ -CMeans可以实现在近线性时间通过球剥离算法。对于$k$ -CMedians，我们表明，一个类似的球体剥离算法存在实现常数近似解。
英文摘要：
In this paper, we study a new type of clustering problem, called {\em Chromatic Clustering}, in high dimensional space. Chromatic clustering seeks to partition a set of colored points into groups (or clusters) so that no group contains points with the same color and a certain objective function is optimized. In this paper, we consider two variants of the problem, chromatic $k$-means clustering (denoted as $k$-CMeans) and chromatic $k$-medians clustering (denoted as $k$-CMedians), and investigate their hardness and approximation solutions. For $k$-CMeans, we show that the additional coloring constraint destroys several key properties (such as the locality property) used in existing $k$-means techniques (for ordinary points), and significantly complicates the problem. There is no FPTAS for the chromatic clustering problem, even if $k=2$. To overcome the additional difficulty, we develop a standalone result, called {\em Simplex Lemma}, which enables us to efficiently approximate the mean point of an unknown point set through a fixed dimensional simplex. A nice feature of the simplex is its independence with the dimensionality of the original space, and thus can be used for problems in very high dimensional space. With the simplex lemma, together with several random sampling techniques, we show that a $(1+ε)$-approximation of $k$-CMeans can be achieved in near linear time through a sphere peeling algorithm. For $k$-CMedians, we show that a similar sphere peeling algorithm exists for achieving constant approximation solutions.
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