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.
Finally, we note that all methods, including ours, find only a locally optimal solution to the NTF problem. Finding the global solution is generally much harder; for instance, Vavasis @cite_34 proves it is NP-hard for an NMF model that fits the data exactly.
@cite_34
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The Nonnegative Matrix Factorization (NMF) problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where the factorization is computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. We initiate a study of when this problem is solvable in polynomial time. Consider a nonnegative m x n matrix @math and a target inner-dimension r. Our results are the following: - We give a polynomial-time algorithm for exact and approximate NMF for every constant r. Indeed NMF is most interesting in applications precisely when r is small. We complement this with a hardness result, that if exact NMF can be solved in time (nm)o(r), 3-SAT has a sub-exponential time algorithm. Hence, substantial improvements to the above algorithm are unlikely. - We give an algorithm that runs in time polynomial in n, m and r under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first polynomial-time algorithm that provably works under a non-trivial condition on the input matrix and we believe that this will be an interesting and important direction for future work. Non-negative matrix factorization (NMF), i.e. V ap WH where both V, W and H are non-negative has become a widely used blind source separation technique due to its part based representation. The NMF decomposition is not in general unique and a part based representation not guaranteed. However, imposing sparseness both improves the uniqueness of the decomposition and favors part based representation. Sparseness in the form of attaining as many zero elements in the solution as possible is appealing from a conceptional point of view and corresponds to minimizing reconstruction error with an L0 norm constraint. In general, solving for a given L0 norm is an NP hard problem thus convex relaxation to regularization by the L1 norm is often considered, i.e., minimizing (1 2||V - WH||F 2 + lambda||H||1).An open problem is to control the degree of sparsity lambda imposed. We here demonstrate that a full regularization path for the L1 norm regularized least squares NMF for fixed W can be calculated at the cost of an ordinary least squares solution based on a modification of the least angle regression and selection (LARS) algorithm forming a non-negativity constrained LARS (NLARS). With the full regularization path, the L1 regularization strength lambda that best approximates a given L0 can be directly accessed and in effect used to control the sparsity of H. The MATLAB code for the NLARS algorithm is available for download. Nonnegative matrix factorization (NMF) and its extensions such as Nonnegative Tensor Factorization (NTF) have become prominent techniques for blind sources separation (BSS), analysis of image databases, data mining and other information retrieval and clustering applications. In this paper we propose a family of efficient algorithms for NMF NTF, as well as sparse nonnegative coding and representation, that has many potential applications in computational neuroscience, multi-sensory processing, compressed sensing and multidimensional data analysis. We have developed a class of optimized local algorithms which are referred to as Hierarchical Alternating Least Squares (HALS) algorithms. For these purposes, we have performed sequential constrained minimization on a set of squared Euclidean distances. We then extend this approach to robust cost functions using the alpha and beta divergences and derive flexible update rules. Our algorithms are locally stable and work well for NMF-based 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, these algorithms can be tuned to different noise statistics by adjusting a single parameter. Extensive experimental results confirm the accuracy and computational performance of the developed algorithms, especially, with usage of multi-layer hierarchical NMF approach [3]. We introduce an algorithm for a non-negative 3D tensor factorization for the purpose of establishing a local parts feature decomposition from an object class of images. In the past, such a decomposition was obtained using non-negative matrix factorization (NMF) where images were vectorized before being factored by NMF. A tensor factorization (NTF) on the other hand preserves the 2D representations of images and provides a unique factorization (unlike NMF which is not unique). The resulting "factors" from the NTF factorization are both sparse (like with NMF) but also separable allowing efficient convolution with the test image. Results show a superior decomposition to what an NMF can provide on all fronts - degree of sparsity, lack of ghost residue due to invariant parts and efficiency of coding of around an order of magnitude better. Experiments on using the local parts decomposition for face detection using SVM and Adaboost classifiers demonstrate that the recovered features are discriminatory and highly effective for classification.