We develop necessary and sufficient conditions and a novel provably consistent and efficient algorithm for discovering topics (latent factors) from observations (documents) that are realized from a probabilistic mixture of shared latent factors that have certain properties. Our focus is on the class of topic models in which each shared latent factor contains a novel word that is unique to that factor, a property that has come to be known as separability. Our algorithm is based on the key insight that the novel words correspond to the extreme points of the convex hull formed by the row-vectors of a suitably normalized word co-occurrence matrix. We leverage this geometric insight to establish polynomial computation and sample complexity bounds based on a few isotropic random projections of the rows of the normalized word co-occurrence matrix. Our proposed random-projections-based algorithm is naturally amenable to an efficient distributed implementation and is attractive for modern web-scale distributed data mining applications.
Nonnegative Matrix Factorization (NMF) is an alternative approach for topic estimation. NMF-based methods exploit the fact that both the topic matrix @math and the mixing weights are nonnegative and attempt to decompose the empirical observation matrix @math into a product of a nonnegative topic matrix @math and the matrix of mixing weights by minimizing a cost function of the form @cite_36 @cite_9 @cite_30 @cite_15 where @math is some measure of closeness and @math is a regularization term which enforces desirable properties, e.g., sparsity, on @math and the mixing weights. The NMF problem, however, is also known to be non-convex and @math -hard @cite_6 in general. Sub-optimal strategies such as alternating minimization, greedy gradient descent, and heuristics are used in practice @cite_9 .
@cite_30 @cite_36 @cite_9 @cite_6 @cite_15
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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. Nonnegative matrix factorization (NMF) determines a lower rank approximation of a matrix @math where an integer @math is given and nonnegativity is imposed on all components of the factors @math and @math . NMF has attracted much attention for over a decade and has been successfully applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative, such as chemical concentrations in experimental results or pixels in digital images, NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for NMF based on alternating nonnegativity constrained least squares (NMF ANLS) and the active set-based fast algorithm for nonnegativity constrained least squares with multiple right-hand side vectors, and we discuss its convergence properties and a rigorous convergence criterion based on the Karush-Kuhn-Tucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by @math -norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior. Nonnegative matrix factorization (NMF) has beenwidely applied in many domains. In document analysis, it hasbeen increasingly used in topic modeling applications, where aset of underlying topics are revealed by a low-rank factor matrixfrom NMF. However, it is often the case that the resulting topicsgive only general topic information in the data, which tends notto convey much information. To tackle this problem, we proposea novel ensemble model of nonnegative matrix factorizationfor discovering high-quality local topics. Our method leveragesthe idea of an ensemble model, which has been successfulin supervised learning, into an unsupervised topic modelingcontext. That is, our model successively performs NMF givena residual matrix obtained from previous stages and generatesa sequence of topic sets. Our algorithm for updating the inputmatrix has novelty in two aspects. The first lies in utilizing theresidual matrix inspired by a state-of-the-art gradient boostingmodel, and the second stems from applying a sophisticatedlocal weighting scheme on the given matrix to enhance thelocality of topics, which in turn delivers high-quality, focusedtopics of interest to users. We evaluate our proposed method bycomparing it against other topic modeling methods, such as afew variants of NMF and latent Dirichlet allocation, in termsof various evaluation measures representing topic coherence, diversity, coverage, computing time, and so on. We also presentqualitative evaluation on the topics discovered by our methodusing several real-world data sets.