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In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$…

Optimization and Control · Mathematics 2014-07-08 Lorenzo Finesso , Peter Spreij

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$…

Optimization and Control · Mathematics 2007-05-23 Lorenzo Finesso , Peter Spreij

Nonnegative matrix factorization (NMF) is a popular method used to reduce dimensionality in data sets whose elements are nonnegative. It does so by decomposing the data set of interest, $\mathbf{X}$, into two lower rank nonnegative matrices…

Methodology · Statistics 2021-07-05 Phillip Shreeves , Jeffrey L. Andrews , Xinchen Deng , Ramie Ali-Adeeb , Andrew Jirasek

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open…

Computational Complexity · Computer Science 2017-03-24 Dmitry Chistikov , Stefan Kiefer , Ines Marušić , Mahsa Shirmohammadi , James Worrell

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. Restricted NMF requires in…

Formal Languages and Automata Theory · Computer Science 2016-05-24 Dmitry Chistikov , Stefan Kiefer , Ines Marušić , Mahsa Shirmohammadi , James Worrell

Nonnegative Matrix Factorization (NMF) is the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices and has been shown to be particularly useful in many applications, e.g., in text mining, image…

Optimization and Control · Mathematics 2012-08-13 Nicolas Gillis , François Glineur

Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this…

Machine Learning · Statistics 2014-12-10 Nicolas Gillis

Nonnegative Matrix Factorization (NMF) has been a popular representation method for pattern classification problem. It tries to decompose a nonnegative matrix of data samples as the product of a nonnegative basic matrix and a nonnegative…

Machine Learning · Statistics 2013-12-06 Jim Jing-Yan Wang

Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data with applications such as image analysis, text mining, audio source separation and hyperspectral unmixing. Given a data matrix $M$ and a…

Machine Learning · Computer Science 2021-04-14 Junjun Pan , Nicolas Gillis

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its…

Numerical Analysis · Mathematics 2025-11-11 Junjun Pan , Valentin Leplat , Michael Ng , Nicolas Gillis

In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF). Several aspects of NMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions,…

Numerical Analysis · Computer Science 2017-03-03 Nicolas Gillis

In this paper we explore avenues for improving the reliability of dimensionality reduction methods such as Non-Negative Matrix Factorization (NMF) as interpretive exploratory data analysis tools. We first explore the difficulties of the…

Artificial Intelligence · Computer Science 2009-04-22 Nikolaos Vasiloglou , Alexander G. Gray , David V. Anderson

In the Nonnegative Matrix Factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$ where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r…

Data Structures and Algorithms · Computer Science 2011-11-04 Sanjeev Arora , Rong Ge , Ravi Kannan , Ankur Moitra

Traditional nonnegative matrix factorization (NMF) learns a new feature representation on the whole data space, which means treating all features equally. However, a subspace is often sufficient for accurate representation in practical…

Computer Vision and Pattern Recognition · Computer Science 2022-04-19 Junhang Li , Jiao Wei , Can Tong , Tingting Shen , Yuchen Liu , Chen Li , Shouliang Qi , Yudong Yao , Yueyang Teng

Non-negative matrix factorization (NMF) is a common method for generating topic models from text data. NMF is widely accepted for producing good results despite its relative simplicity of implementation and ease of computation. One…

Machine Learning · Computer Science 2016-08-09 Brendan Gavin , Vijay Gadepally , Jeremy Kepner

Non-negative matrix factorization (NMF) is a matrix decomposition problem with applications in unsupervised learning. The general form of this problem (along with many of its variants) is NP-hard in nature. In our work, we explore how this…

Non-negative Matrix Factorization (NMF) is an intensively used technique for obtaining parts-based, lower dimensional and non-negative representation. Researchers in biology, medicine, pharmacy and other fields often prefer NMF over other…

Machine Learning · Computer Science 2025-02-04 Matej Mihelčić , Pauli Miettinen

Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains…

Distributed, Parallel, and Cluster Computing · Computer Science 2015-10-01 Ramakrishnan Kannan , Grey Ballard , Haesun Park

Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in…

Machine Learning · Computer Science 2007-05-23 Patrik O. Hoyer

Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g.,…

Optimization and Control · Mathematics 2012-08-13 Nicolas Gillis , François Glineur
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