Tensor factorizations with nonnegativity constraints have found application in analysing data from cyber traffic, social networks, and other areas. We consider application data best described as being generated by a Poisson process e.g. count data, which leads to sparse tensors that can be modelled by sparse factor matrices. In this paper, we investigate efficient techniques for computing an appropriate canonical polyadic tensor factorization based on the Kullback–Leibler divergence function. We propose novel subproblem solvers within the standard alternating block variable approach. Our new methods exploit structure and reformulate the optimization problem as small independent subproblems. We employ bound-constrained Newton and quasi-Newton methods. We compare our algorithms against other codes, demonstrating superior speed for high accuracy results and the ability to quickly find sparse solutions.
A new variant ‘PMF’ of factor analysis is described. It is assumed that X is a matrix of observed data and σ is the known matrix of standard deviations of elements of X. Both X and σ are of dimensions n × m. The method solves the bilinear matrix problem X = GF + E where G is the unknown left hand factor matrix (scores) of dimensions n × p, F is the unknown right hand factor matrix (loadings) of dimensions p × m, and E is the matrix of residuals. The problem is solved in the weighted least squares sense: G and F are determined so that the Frobenius norm of E divided (element-by-element) by σ is minimized. Furthermore, the solution is constrained so that all the elements of G and F are required to be non-negative. It is shown that the solutions by PMF are usually different from any solutions produced by the customary factor analysis (FA, i.e. principal component analysis (PCA) followed by rotations). Usually PMF produces a better fit to the data than FA. Also, the result of PF is guaranteed to be non-negative, while the result of FA often cannot be rotated so that all negative entries would be eliminated. Different possible application areas of the new method are briefly discussed. In environmental data, the error estimates of data can be widely varying and non-negativity is often an essential feature of the underlying models. Thus it is concluded that PMF is better suited than FA or PCA in many environmental applications. Examples of successful applications of PMF are shown in companion papers.   Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although bound-constrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this letter, we propose two projected gradient methods for NMF, both of which exhibit strong optimization properties. We discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach. A simple Matlab code is also provided.        SUMMARY In this paper a modification of the standard algorithm for non-negativity-constrained linear least squares regression is proposed. The algorithm is specifically designed for use in multiway decomposition methods such as PARAFAC and N-mode principal component analysis. In those methods the typical situation is that there is a high ratio between the numbers of objects and variables in the regression problems solved. Furthermore, very similar regression problems are solved many times during the iterative procedures used. The algorithm proposed is based on the de facto standard algorithm NNLS by Lawson and Hanson, but modified to take advantage of the special characteristics of iterative algorithms involving repeated use of non-negativity constraints. The principle behind the NNLS algorithm is described in detail and a comparison is made between this standard algorithm and the new algorithm called FNNLS (fast NNLS). © 1997 John Wiley & Sons, Ltd. Is perception of the whole based on perception of its parts? There is psychological1 and physiological2,3 evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations4,5. But little is known about how brains or computers might learn the parts of objects. Here we demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text. This is in contrast to other methods, such as principal components analysis and vector quantization, that learn holistic, not parts-based, representations. Non-negative matrix factorization is distinguished from the other methods by its use of non-negativity constraints. These constraints lead to a parts-based representation because they allow only additive, not subtractive, combinations. When non-negative matrix factorization is implemented as a neural network, parts-based representations emerge by virtue of two properties: the firing rates of neurons are never negative and synaptic strengths do not change sign. Abstract A novel fixed point algorithm for positive tensor factorization (PTF) is introduced. The update rules efficiently minimize the reconstruction error of a positive tensor over positive factors. Tensors of arbitrary order can be factorized, which extends earlier results in the literature. Experiments show that the factors of PTF are easier to interpret than those produced by methods based on the singular value decomposition, which might contain negative values. We also illustrate the tendency of PTF to generate sparsely distributed codes.