改进的k-Means平滑分析方法
Improved Smoothed Analysis of the k-Means Method
摘要：
k-means方法是一种广泛使用的聚类算法。其显著特点之一是实践速度快。然而，其最坏情况下的运行时间是指数级的，在实际和理论性能之间留下了差距。Arthur和Vassilvitskii（FOCS 2006）旨在缩小这一差距，他们证明了$\poly(n^k, σ^{-1})$对k均值方法平滑运行时间的限制，其中n是数据点的数量，$σ$是高斯扰动的标准差。这个界限，虽然比最坏情况下的界限，仍然是远远大于在实践中观察到的运行时间。我们改进的平滑分析的k-means方法，显示两个上界的预期运行时间的k-means。首先，我们证明了期望的运行时间是有界的多项式在$n^{\sqrt k}$和$σ^{-1}$。其次，我们证明了一个上界的$k^{kd} \cdot \poly(n, σ^{-1})$，其中d是数据空间的维数。多项式是独立的k和d，我们得到一个多项式界的预期运行时间为$k, d \in O(\sqrt{\log n/\log \log n})$。最后，我们表明，k-均值运行在平滑的多项式时间的一维实例。
英文摘要：
The k-means method is a widely used clustering algorithm. One of its distinguished features is its speed in practice. Its worst-case running-time, however, is exponential, leaving a gap between practical and theoretical performance. Arthur and Vassilvitskii (FOCS 2006) aimed at closing this gap, and they proved a bound of $\poly(n^k, σ^{-1})$ on the smoothed running-time of the k-means method, where n is the number of data points and $σ$ is the standard deviation of the Gaussian perturbation. This bound, though better than the worst-case bound, is still much larger than the running-time observed in practice. We improve the smoothed analysis of the k-means method by showing two upper bounds on the expected running-time of k-means. First, we prove that the expected running-time is bounded by a polynomial in $n^{\sqrt k}$ and $σ^{-1}$. Second, we prove an upper bound of $k^{kd} \cdot \poly(n, σ^{-1})$, where d is the dimension of the data space. The polynomial is independent of k and d, and we obtain a polynomial bound for the expected running-time for $k, d \in O(\sqrt{\log n/\log \log n})$. Finally, we show that k-means runs in smoothed polynomial time for one-dimensional instances.
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