Approximation Algorithms for Tensor Clustering

Abstract

We present the first (to our knowledge) approximation algorithm for tensor clustering—a powerful generalization to basic 1D clustering. Tensors are increasingly common in modern applications dealing with complex heterogeneous data and clustering them is a fundamental tool for data analysis and pattern discovery. Akin to their 1D cousins, common tensor clustering formulations are NP-hard to optimize. But, unlike the 1D case, no approximation algorithms seem to be known. We address this imbalance and build on recent co-clustering work to derive a tensor clustering algorithm with approximation guarantees, allowing metrics and divergences (e.g., Bregman) as objective functions. Therewith, we answer two open questions by Anagnostopoulos et al. (2008). Our analysis yields a constant approximation factor independent of data size; a worst-case example shows this factor to be tight for Euclidean co-clustering. However, empirically the approximation factor is observed to be conservative, so our method can also be used in practice.