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Abstract

The Euclidean K-means problem is fundamental to clustering and over the years it has been
intensely investigated. More recently, generalizations such as Bregman k-means [8], co-clustering [10],
and tensor (multi-way) clustering [40] have also gained prominence. A well-known computational difficulty
encountered by these clustering problems is the NP-Hardness of the associated optimization task,
and commonly used methods guarantee at most local optimality. Consequently, approximation algorithms
of varying degrees of sophistication have been developed, though largely for the basic Euclidean
K-means (or `1-norm K-median) problem. In this paper we present approximation algorithms for several
Bregman clustering problems by building upon the recent paper of Arthur and Vassilvitskii [5]. Our algorithms
obtain objective values within a factor O(logK) for Bregman k-means, Bregman co-clustering,
Bregman tensor clustering, and weighted kernel k-means. To our knowledge, except for some special
cases, approximation algorithms have not been considered for these general clustering problems. There
are several important implications of our work: (i) under the same assumptions as Ackermann et al. [1]
it yields a much faster algorithm (non-exponential in K, unlike [1]) for information-theoretic clustering,
(ii) it answers several open problems posed by [4], including generalizations to Bregman co-clustering,
and tensor clustering, (iii) it provides practical and easy to implement methods—in contrast to several
other common approximation approaches.