Bi-stochastic Matrix Approximation Framework for Data Co-clustering

Abstract

The matrix approximation approaches like Singular Value Decomposition SVD and Non-negative Matrix Tri-Factorization (NMTF) have recently been shown to be useful and effective to tackle the co-clustering problem. In this work, we embed the co-clustering in a Bistochastic Matrix Approximation (BMA) framework and we derive from the double kmeans objective function a new formulation of the criterion to optimize. First, we show that the double k-means is equivalent to algebraic problem of BMA under some suitable constraints. Secondly, we propose an iterative process seeking for the optimal simultaneous partitions of rows and columns data, the solution is given as the steady state of a markov chain process. We develop two iterative algorithms; the first consists in learning rows and columns similarities matrices and the second consists in obtaining the simultaneous rows and columns partitions. Numerical experiments on simulated and real datasets demonstrate the interest of our approach which does not require the knowledge of the number of co-clusters.