Smoothed Analysis of the k-Means Method

Abstract

The k -means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k -means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this article, we settle the smoothed running time of the k -means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/ σ , where σ is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k -means method will run in expected polynomial time on that input set.

Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics

“…For the k-means problem, Kanungo et al [37] showed that local search achieves a 9`ε-approximation in general metrics and this remains the best known approximation guarantee so far even for fixed d. There are also a variety of results for k-means and k-median when the input has some stability conditions (see for example [10,8,14,13,18,40,45]) or in the context of smoothed analysis (see for example [6,5]). …”

Identification of nonlinear behavior with clustering techniques in car crash simulations for better model reduction

“…The computational complexity of Algorithm 1 is O(n·K · d · ω) with ω the number of iterations until satisfactory convergence is achieved in line 6. Even though ω can grow exponentially in n [18], it is in average (via smoothed analysis) polynomial in n [19]. For real data, it often can be observed that ω does not grow that fast and is considered proportional to n.…”

A traffic-based evolutionary algorithm for network clustering

“…The overall cost is therefore O(kn 2 ) for each iteration, which, in our case, can be simplified to O(n 2 ). Though it has been shown that, in the very worst case, the algorithm requires an exponential number of iterations [36], it has been recognized since long that the performance of the K-means algorithm exhibits a stark contrast between practical observations and theoretical bounds [37]. In our case, we have observed an approximately constant number of iterations, so that we can safely assume that the overall cost of the K-means algorithm is O(n 2 ).…”