A parameter-less algorithm for tensor co-clustering

The majority of the data produced by human activities and modern cyber-physical systems involve complex relations among their features. Such relations can be often represented by means of tensors, which can be viewed as generalization of matrices and, as such, can be analyzed by using higher-order extensions of existing machine learning methods, such as clustering and co-clustering. Tensor co-clustering, in particular, has been proven useful in many applications, due to its ability of coping with n-modal data and sparsity. However, setting up a co-clustering algorithm properly requires the specification of the desired number of clusters for each mode as input parameters. This choice is already difficult in relatively easy settings, like flat clustering on data matrices, but on tensors it could be even more frustrating. To face this issue, we propose a new tensor co-clustering algorithm that does not require the number of desired co-clusters as input, as it optimizes an objective function based on a measure of association across discrete random variables (called Goodman and Kruskal’s τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document}) that is not affected by their cardinality. We introduce different optimization schemes and show their theoretical and empirical convergence properties. Additionally, we show the effectiveness of our algorithm on both synthetic and real-world datasets, also in comparison with state-of-the-art co-clustering methods based on tensor factorization and latent block models.