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.
NMF in the LS case was first proposed by Paatero and Tapper @cite_36 and also studied by Bro [p. 169] BroPhD . Lee and Seung later consider the problem for both LS and K-L formulations and introduce multiplicative updates based on the convex subproblems @cite_12 @cite_0 . Their work is extended to tensors by Welling and Weber @cite_11 . Many other works have been published on the LS versions of NMF @cite_32 @cite_49 @cite_42 @cite_19 @cite_44 and NTF @cite_15 @cite_1 @cite_45 @cite_2 @cite_6 .
@cite_36 @cite_42 @cite_1 @cite_32 @cite_6 @cite_0 @cite_19 @cite_44 @cite_45 @cite_49 @cite_2 @cite_15 @cite_12 @cite_11
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In this paper we study Nonnegative Tensor Factorization (NTF) based on the Kullback---Leibler (KL) divergence as an alternative Csiszar---Tusnady procedure. We propose new update rules for the aforementioned divergence that are based on multiplicative update rules. The proposed algorithms are built on solid theoretical foundations that guarantee that the limit point of the iterative algorithm corresponds to a stationary solution of the optimization procedure. Moreover, we study the convergence properties of the optimization procedure and we present generalized pythagorean rules. Furthermore, we provide clear probabilistic interpretations of these algorithms. Finally, we discuss the connections between generalized Probabilistic Tensor Latent Variable Models (PTLVM) and NTF, proposing in that way algorithms for PTLVM for arbitrary multivariate probabilistic mass functions. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788--791), which introduced an algorithm based on Multiplicative Updates (MU). More recently, another class of methods called Hierarchical Alternating Least Squares (HALS) was introduced that seems to be much more efficient in practice. In this paper, we consider the problem of approximating a not necessarily nonnegative matrix with the product of two nonnegative matrices, which we refer to as Nonnegative Factorization (NF) ; this is the subproblem that HALS methods implicitly try to solve at each iteration. We prove that NF is NP-hard for any fixed factorization rank, using a reduction to the maximum edge biclique problem. We also generalize the multiplicative updates to NF, which allows us to shed some light on the differences between the MU and HALS algorithms for NMF and give an explanation for the better performance of HALS. Finally, we link stationary points of NF with feasible solutions of the biclique problem to obtain a new type of biclique finding algorithm (based on MU) whose iterations have an algorithmic complexity proportional to the number of edges in the graph, and show that it performs better than comparable existing methods. Nonnegative Matrix Factorization (NMF) is an effective dimension reduction method for non-negative dyadic data, and has proven to be useful in many areas, such as text mining, bioinformatics and image processing. NMF is usually formulated as a constrained non-convex optimization problem, and many algorithms have been developed for solving it. Recently, a coordinate descent method, called FastHals, has been proposed to solve least squares NMF and is regarded as one of the state-of-the-art techniques for the problem. In this paper, we first show that FastHals has an inefficiency in that it uses a cyclic coordinate descent scheme and thus, performs unneeded descent steps on unimportant variables. We then present a variable selection scheme that uses the gradient of the objective function to arrive at a new coordinate descent method. Our new method is considerably faster in practice and we show that it has theoretical convergence guarantees. Moreover when the solution is sparse, as is often the case in real applications, our new method benefits by selecting important variables to update more often, thus resulting in higher speed. As an example, on a text dataset RCV1, our method is 7 times faster than FastHals, and more than 15 times faster when the sparsity is increased by adding an L1 penalty. We also develop new coordinate descent methods when error in NMF is measured by KL-divergence by applying the Newton method to solve the one-variable sub-problems. Experiments indicate that our algorithm for minimizing the KL-divergence is faster than the Lee & Seung multiplicative rule by a factor of 10 on the CBCL image dataset.