Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification

Analysis of high dimensional data in modern applications, such as neuroscience, text mining, spectral analysis or chemometrices naturally requires tensor decomposition methods. The Tucker decompositions allow us to extract hidden factors (component matrices) with a different dimension in each mode and investigate interactions among various modes. The Alternating Least Squares (ALS) algorithms have been confirmed effective and efficient in most of tensor decompositions, especially, Tucker with orthogonality constraints. However, for nonnegative Tucker decomposition (NTD), standard ALS algorithms suffer from unstable convergence properties, demand high computational cost for large scale problems due to matrix inversion and often return suboptimal solutions. Moreover, they are quite sensitive with respect to noise, and can be relatively slow in the special case when the data are nearly collinear. In this paper, we propose a new algorithm for nonnegative Tucker decomposition based on constrained minimization of a set of local cost functions and Hierarchical Alternating Least Squares (HALS). The developed HALS NTD algorithm sequentially updates components, hence avoids matrix inversion, and is suitable for large-scale problems. The proposed algorithm is also regularized with additional constraint terms such as sparseness, orthogonality, smoothness, and especially discriminant constraints for classification problems. Extensive experiments confirm the validity and higher performance of the developed algorithm in comparison with other existing algorithms.