Flexible HALS algorithms for sparse non-negative matrix/tensor factorization

In this paper we propose a family of new algorithms for non-negative matrix/tensor factorization (NMF/NTF) and sparse nonnegative coding and representation that has many potential applications in computational neuroscience, multi-sensory, multidimensional data analysis and text mining. We have developed a class of local algorithms which are extensions of hierarchical alternating least squares (HALS) algorithms proposed by us in . For these purposes, we have performed simultaneous constrained minimization of a set of robust cost functions called alpha and beta divergences. Our algorithms are locally stable and work well for the NMF blind source separation (BSS) not only for the over-determined case but also for an under-determined (over-complete) case (i.e., for a system which has less sensors than sources) if data are sufficiently sparse. The NMF learning rules are extended and generalized for N-th order nonnegative tensor factorization (NTF). Moreover, new algorithms can be potentially accommodated to different noise statistics by just adjusting a single parameter. Extensive experimental results confirm the validity and high performance of the developed algorithms, especially, with usage of the multi-layer hierarchical approach .