Co-clustering from Tensor Data

Abstract

With the exponential growth of collected data in different fields like recommender system (user, items), text mining (document, term), bioinformatics (individual, gene), co-clustering which is a simultaneous clustering of both dimensions of a data matrix, has become a popular technique. Co-clustering aims to obtain homogeneous blocks leading to an easy simultaneous interpretation of row clusters and column clusters. Many approaches exist, in this paper we rely on the latent block model (LBM) which is flexible allowing to model different types of data matrices. We extend its use to the case of a tensor (3D matrix) data in proposing a Tensor LBM (TLBM) allowing different relations between entities. To show the interest of TLBM, we consider continuous and binary datasets. To estimate the parameters, a variational EM algorithm is developed. Its performances are evaluated on synthetic and real datasets to highlight different possible applications.

A Appendix: Update and \(\forall i,k,j,\ell \)

To obtain the expression of , we maximize the above soft criterion $F_C(\tilde{\mathbf {z}},\tilde{\mathbf {w}};\varOmega )$ with respect to , subject to the constraint . The corresponding Lagrangian, up to terms which are not function of , is given by:

Taking derivatives with respect to , we obtain:

Setting this derivative to zero yields: Summing both sides over all $k'$ yields $\exp (\beta + 1)= \sum _{k'} \pi _{k'} \exp (\sum _{j,\ell }w_{j\ell }\log (\varPhi (\mathbf {x}_{ij},\varvec{\lambda }_{k'\ell })).$ Plugging $\exp (\beta )$ in leads to: In the same way, we can estimate maximizing $F_C(\tilde{\mathbf {z}},\tilde{\mathbf {w}};\varOmega )$ with respect to , subject to the constraint ; we obtain