Improved Smoothed Analysis of the k-Means Method

Abstract: 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 $\poly(n^k, \sigma<sup>{-1})$ on the smoothed running-time of the k-means method, where n is the number of data points and $\sigma$ 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 $\sigma<sup>{-1}$. Second, we prove an upper bound of $k^{kd} \cdot \poly(n, \sigma<sup>{-1})$, 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.