Fast nonnegative tensor factorization for very large-scale problems using two-stage procedure

Parallel factor analysis (PARAFAC) is a multi-way decomposition method which allows to find hidden factors from the raw tensor data. Recently, the nonnegative tensor factorization (NTF), a variant of the model with nonnegativity constraints imposed on hidden factors has attracted interesting due to meaningful representation with many potential applications in neuroscience, bioinformatics, chemometrics etc. NTF algorithms can be easily extended from algorithms for nonnegative matrix factorization (NMF) by forming learning rules on the unfolding tensor. However, they often compute Khatri-Rao products of factors which lead to large matrices, and require large memory for temporal variables. Hence decomposition of large-scale tensor is still a challenging problem for NTF. PARAFAC by alternating least squares (ALS) can explain the raw tensor by a small number of rank-one tensor with a high fitness. Based on this advantage, we propose a new fast NTF algorithm which factorizes the approximate tensor obtained from the PARAFAC. Our new algorithm computes Hadamard products, therefore it is extremely fast in comparison with all the existing NTF algorithms. Extensive experiments confirm the validity, high performance and high speed of the developed algorithm.