Block Decomposition for Very Large-Scale Nonnegative Tensor Factorization

Nonnegative parallel factor analysis (PARAFAC) (also called nonnegative tensor factorization - NTF) allows to find nonnegative factors hidden under the raw tensor data which have many potential applications in neuroscience, bioinformatics, chemometrics etc [1], [2]. NTF algorithms can be easily established based on the unfolding tensor and Khatri-Rao products of factors [1], [3]. This kind of algorithms leads to large matrices, and requires large memory for temporal variables. Hence decomposition of large-scale tensor is still a challenging problem for NTF. To deal with this problem, a new tensor factorization scheme is proposed, in which the data tensor will be divided into a grid of multiple of small-sized subtensors, then processed in two stages: PARAFAC for the subtensors, and construction of full factors for the whole data. The two new algorithms compute Hadamard products, and perform on relatively small matrices. Therefore they are extremely fast in comparison with all the existing NTF algorithms. Extensive experiments confirm the validity, high performance and high speed of the developed algorithms.