1 0 Fe b 20 09 Approximation Algorithms for Bregman Co - clustering and Tensor Clustering ∗

In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9, 17], and tensor clustering [8, 32]. Like k-means, these more general problems also suffer from the NP-hardness of the associated optimization. Researchers have developed approximation algorithms of varying degrees of sophistication for k-means, kmedians, and more recently also for Bregman clustering [2]. However, there seem to be no approximation algorithms for Bregman coand tensor clustering. In this paper we derive the first (to our knowledge) guaranteed methods for these increasingly important clustering settings. Going beyond Bregman divergences, we also prove an approximation factor for tensor clustering with lp norms, which for the l1 norm is a generalization of k-medians to tensors. Through extensive experiments we evaluate the characteristics of our method, and show that our methods also have practical impact.