Multiscale co-clustering for tensor data based on canonical polyadic decomposition and slice-wise factorization

Abstract

The analysis of tensor data is necessary in many applications. Similar to bi-clustering of matrix data, multiscale co-clustering can simultaneously extract coherent patterns along all or partial modes of a tensor. However, numerical methods for co-clustering have not reached the level of maturity of bi-clustering. In this paper, we present a theoretical framework to perform co-clustering for multidimensional data based on tensor and matrix decomposition. According to the proposed principle, we first develop an alternative algorithm for tensor canonical decomposition with full-rank constraint on slice-wise factorization (FRSF). Owing to the least squares principle and the constraint inherent in the resolution of multidimensional data, FRSF provides a natural way to avoid two-factor degeneracy and renders the resolved profiles stable with respect to high dimensionality. FRSF maintains a high convergence rate and greatly reduces the computational complexity with the compression technique based on matrix singular value decomposition. Furthermore, the algebraic expression of co-clusters in tensor data can be mapped to some linear structures in the factor spaces of FRSF. We employ a linear grouping algorithm to identify these geometrical patterns in the factor spaces. Finally, the combination of the linear grouping points along every mode successfully supports the detection of co-clusters in tensor data. On the basis of the proposed framework, a flexible and fast FRSF-based co-clustering algorithm is developed. Extensive simulations and experiment data analysis demonstrate the validity and efficiency of FRSF and the proposed co-clustering algorithm.