@inproceedings{d894f88d409245c08b1e5d6c9ddb8c89,

title = "Worst-case and smoothed analysis of $k$-means clustering with Bregman divergences",

abstract = "The $k$-means algorithm is the method of choice for clustering large-scale data sets and it performs exceedingly well in practice. Most of the theoretical work is restricted to the case that squared Euclidean distances are used as similarity measure. In many applications, however, data is to be clustered with respect to other measures like, e.g., relative entropy, which is commonly used to cluster web pages. In this paper, we analyze the running-time of the $k$-means method for Bregman divergences, a very general class of similarity measures including squared Euclidean distances and relative entropy. We show that the exponential lower bound known for the Euclidean case carries over to almost every Bregman divergence. To narrow the gap between theory and practice, we also study $k$-means in the semi-random input model of smoothed analysis. For the case that $n$ data points in $\mathbb{R}^d$ are perturbed by noise with standard deviation $\sigma,$ we show that for almost arbitrary Bregman divergences the expected running-time is bounded by poly($n^{\sqrt k}, 1/\sigma$) and $k^{kd}$ poly($n, 1/\sigma$).",

keywords = "$k$-Means, Bregman divergence, IR-68761, METIS-264223, Relative entropy, Kullback-Leibler divergence, Itakura-Saito Divergence, Generalized I-Divergence, Clustering, Mahalanobis Distance, Machine Learning, EWI-16970, Smoothed Analysis",

author = "Bodo Manthey and Heiko R{\"o}glin",

note = "10.1007/978-3-642-10631-6_103 ; null ; Conference date: 16-12-2009 Through 18-12-2009",

year = "2009",

doi = "10.1007/978-3-642-10631-6_103",

language = "Undefined",

isbn = "978-3-642-10630-9",

series = "Lecture Notes in Computer Science",

publisher = "Springer",

pages = "1024--1033",

editor = "Yingfei Dong and Dingzhu Du and Oscar Ibarra",

booktitle = "Proceedings of the 20th International Symposium on Algorithms and Computation, ISAAC 2009",

address = "Netherlands",

}