An All-at-Once Approach to Nonnegative Tensor Factorizations

Tensors can be viewed as multilinear arrays or generalizations of the notion of matrices. Tensor decompositions have applications in various fields such as psychometrics, signal processing, numerical linear algebra and data mining. When the data are nonnegative, the nonnegative tensor factorization (NTF) better reflects the underlying structure. With NTF it is possible to extract information from a given dataset and construct lower-dimensional bases that capture the main features of the set and concisely describe the original data. Nonnegative tensor factorizations are commonly computed as the solution of a nonlinear bound-constrained optimization problem. Some inherent difficulties must be taken into consideration in order to achieve good solutions. Many existing methods for computing NTF optimize over each of the factors separately; the resulting algorithms are often slow to converge and difficult to control. We propose an all-at-once approach to computing NTF. Our method optimizes over all factors simultaneously and combines two regularization strategies to ensure that the factors in the decomposition remain bounded and equilibrated in norm. We present numerical experiments that illustrate the effectiveness of our approach. We also give an example of digital-inpainting, where our algorithm is employed to construct a basis that can be used to restore digital images.