高维数据聚类的概率$\ell_1$方法
A Probabilistic $\ell_1$ Method for Clustering High Dimensional Data
摘要：
一般来说，聚类问题是NP难的，并且对于非平凡的实例不能建立全局最优性。对于高维数据，基于距离的聚类或分类方法面临着额外的困难，即在非常高维的空间中距离的不可靠性。我们提出了一个基于距离的迭代法聚类数据在非常高维的空间，使用$\ell_1$度量，是不太敏感的高维比欧几里德距离。对于$\mathbb{R}^n$中的$K$簇，问题分解为通过概率耦合的$K$问题，并且迭代简化为寻找线上的点的$Kn$加权中值。该算法的复杂度与数据空间的维数成线性关系，并且随着维数的增加，其性能有明显的提高。
英文摘要：
In general, the clustering problem is NP-hard, and global optimality cannot be established for non-trivial instances. For high-dimensional data, distance-based methods for clustering or classification face an additional difficulty, the unreliability of distances in very high-dimensional spaces. We propose a distance-based iterative method for clustering data in very high-dimensional space, using the $\ell_1$-metric that is less sensitive to high dimensionality than the Euclidean distance. For $K$ clusters in $\mathbb{R}^n$, the problem decomposes to $K$ problems coupled by probabilities, and an iteration reduces to finding $Kn$ weighted medians of points on a line. The complexity of the algorithm is linear in the dimension of the data space, and its performance was observed to improve significantly as the dimension increases.
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