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Nonnegative Matrix Factorizations for Clustering: A Survey
Published in Charu C. Aggarwal, Chandan K. Reddy, Data Clustering, 2018
Matrix-based methodologies are rapidly becoming a significant part of data mining as they are amenable to vigorous analysis and can benefit from the well-established knowledge in linear algebra accumulated through centuries. In particular, NMF factorizes an input nonnegative matrix into two nonnegative matrices of lower rank and has the capability to solve challenging data mining problems. Thanks to the data mining capabilities, NMF has attracted a lot of recent attention and has been used in a variety of fields. This chapter provides a comprehensive review of nonnegative matrix factorization methods for clustering by outlining the theoretical foundations on NMF for clustering and providing an overview of different variants on NMF formulations. We also examine the practical issues in NMF algorithms and summarize recent advances on using NMF-based methods for solving many other clustering problems.
Recommendation Systems
Published in Yulei Wu, Fei Hu, Geyong Min, Albert Y. Zomaya, Big Data and Computational Intelligence in Networking, 2017
Nonnegative matrix factorization (NMF) is a matrix factorization technique approximating M≈UVΤ under the constraint that all entries in U and V should be nonnegative. This requirement can be important in some applications where the representation of each element is inherently nonnegative, or it seeks low-rank matrices which are enforced to have only nonnegative values. For example, a text document is represented as a vector of nonnegative numbers with the term-frequency encoding. Each element in this representation is the number of appearances of each term in the document, so it is nonnegative. Another example is image processing. Digital images are represented by a matrix of pixel intensities, which are inherently nonnegative. In natural sciences such as chemistry or biology, chemical concentrations or gene expressions are also nonnegative [6]. In recommendation systems, a rating matrix is also usually nonnegative. Although other matrix factorization methods may allow negative entries in factorized low-rank matrices, it still makes sense to enforce nonnegativity as the original data is nonnegative.
Conic optimization-based algorithms for nonnegative matrix factorization
Published in Optimization Methods and Software, 2023
Valentin Leplat, Yurii Nesterov, Nicolas Gillis, François Glineur
Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix, , as the product of two smaller nonnegative matrices, and , where K is a given parameter known as the factorization rank. One aims at finding the best approximation, that is, the one that minimizes the discrepancy between V and the product WH, often measured by the Frobenius norm of their difference, . Despite the fact that NMF is NP-hard in general [30] (see also [27]), it has been used successfully in many domains such as probability, geoscience, medical imaging, computational geometry, combinatorial optimization, analytical chemistry, and machine learning; see [11,13] and the references therein. Many local optimization schemes have been developed to compute NMFs. They aim to identify local minima or stationary points of optimization problems that minimize the discrepancy between V and the approximation WH. Most of these iterative algorithms rely on a two-block coordinate descent scheme that consists in (approximatively) optimizing alternatively over W with H fixed, and vice versa; see [5,13] and the references therein. In this paper, we are interested in computing high-quality local minima for the NMF optimization problems without relying on the block coordinate descent (BCD) framework. We will perform the optimization over W and H jointly. Moreover, our focus is on finding exact NMFs, that is, computing nonnegative factors W and H such that V = WH, although our approaches can be used to find approximate NMFs as well.