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**1 - 4**of**4**### The why and how of nonnegative matrix factorization

- REGULARIZATION, OPTIMIZATION, KERNELS, AND SUPPORT VECTOR MACHINES. CHAPMAN & HALL/CRC
, 2014

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### A Fast Hierarchical Alternating Least Squares Algorithm for Orthogonal Nonnegative Matrix Factorization

"... Abstract Nonnegative Matrix Factorization (NMF) is a popular technique in a variety of fields due to its component-based representation with physical interpretablity. NMF finds a nonnegative hidden structures as oblique bases and coefficients. Recently, Orthogonal NMF (ONMF), imposing an orthogonal ..."

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Abstract Nonnegative Matrix Factorization (NMF) is a popular technique in a variety of fields due to its component-based representation with physical interpretablity. NMF finds a nonnegative hidden structures as oblique bases and coefficients. Recently, Orthogonal NMF (ONMF), imposing an orthogonal constraint into NMF, has been gathering a great deal of attention. ONMF is more appropriate for the clustering task because the resultant constrained matrix consisting of the coefficients can be considered as an indicator matrix. All traditional ONMF algorithms are based on multiplicative update rules or project gradient descent method. However, these algorithms are slow in convergence compared with the state-ofthe-art algorithms used for regular NMF. This is because they update a matrix in each iteration step. In this paper, therefore, we propose to update the current matrix columnwisely using Hierarchical Alternating Least Squares (HALS) algorithm that is typically used for NMF. The orthogonality and nonnegativity constraints are both utilized efficiently in the column-wise update procedure. Through experiments on six real-life datasets, it was shown that the proposed algorithm converges faster than the other conventional ONMF algorithms due to a smaller number of iterations, although the theoretical complexity is the same. It was also shown that the orthogonality is also attained in an earlier stage.

### Chapter 7 Nonnegative Matrix Factorization for Interactive Topic Modeling and Document Clustering

"... Abstract Nonnegative matrix factorization (NMF) approximates a nonnegative matrix by the product of two low-rank nonnegative matrices. Since it gives semantically meaningful result that is easily interpretable in clustering applications, NMF has been widely used as a clustering method especially for ..."

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Abstract Nonnegative matrix factorization (NMF) approximates a nonnegative matrix by the product of two low-rank nonnegative matrices. Since it gives semantically meaningful result that is easily interpretable in clustering applications, NMF has been widely used as a clustering method especially for document data, and as a topic modeling method. We describe several fundamental facts of NMF and introduce its optimization framework called block coordinate descent. In the context of clustering, our framework provides a flexible way to extend NMF such as the sparse NMF and the weakly-supervised NMF. The former provides succinct representations for better interpretations while the latter flexibly incorporate extra information and user feedback in NMF, which effectively works as the basis for the visual analytic topic modeling system that we present. Using real-world text data sets, we present quantitative experimental results showing the superiority of our framework from the following aspects: fast conver-gence, high clustering accuracy, sparse representation, consistent output, and user interactivity. In addition, we present a visual analytic system called UTOPIAN (User-driven Topic modeling based on Interactive NMF) and show several usage scenarios. Overall, our book chapter cover the broad spectrum of NMF in the context of clustering and topic modeling, from fundamental algorithmic behaviors to practical visual analytics systems.