k-means requires exponentially many iterations even in the plane

Abstract

The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its running time (i.e. O(n kd )) can be exponential in the number of points. Recently, Arthur and Vassilvitskii [2] showed a superpolynomial worst-case analysis, improving the best known lower bound from Ω(n) to 2with a construction in d = Ω( √ n) dimensions. In [2] they also conjectured the existence of super-polynomial lower bounds for any d ≥ 2.Our contribution is twofold: we prove this conjecture and we improve the lower bound, by presenting a simple construction in the plane that leads to the exponential lower bound 2 Ω(n) .

HacDivSel: Two new methods (haplotype-based and outlier-based) for the detection of divergent selection in pairs of populations

“…The second step identifies different classes inside the extreme outlier set. This is done by a k -means algorithm [35, 36]. The algorithm permits to classify all the elements of the outlier set in one of the k classes.…”

Automatic stacking‐velocity estimation using similarity‐weighted clustering

“…The k-means is a popular objective function used for clustering problems in modern data science applications, such as computer vision, machine learning, and computational geometry (Drineas et al 2004;Little and Jones 2011;Vattani 2011). It is originally proposed by Forgy (1965) and MacQueen (1967) and is often known as Lloyd's algorithm (Lloyd 1982).…”

Fast indefinite multi-point (IMP) clustering

“…When there truly are K clusters, and enough effort is expended, then, in some cases, K -means will converge quickly to the right solution [13,19]. On the other hand, it is known that for ill-fated configurations K -means can take a long time to converge [18]. As such, K -means must be significantly tailored and tested for use in practical applications.…”

“…This considerably decreases the flop count of algorithms that try to minimize the above expression, as there are many fewer terms involved if K M. There are many algorithms that directly or indirectly try to minimize the above expression over the K columns Y j . However it is difficult to the find the global minimum and the quality of the local minimum may not be good, though there does not necessarily seem to be agreement over this in the literature, as the precise local minima at which the algorithm stops depends on the starting point [13,14,18].…”