scholarly journals A Symmetric Rank-One Quasi-Newton Method for Nonnegative Matrix Factorization

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Shu-Zhen Lai ◽  
Hou-Biao Li ◽  
Zu-Tao Zhang
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Hessian Matrix ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Matrix Approximation ◽  
Active Set ◽  
Rank One ◽  
Symmetric Rank One ◽  
Quasi Newton

As is well known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing, signal processing, and so forth. In this paper, an algorithm on nonnegative matrix approximation is proposed. This method is mainly based on a relaxed active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. The method improves some recent results. In addition, some numerical experiments are presented in the synthetic data, imaging processing, and text clustering. By comparing with the other six nonnegative matrix approximation methods, this method is more robust in almost all cases.

scholarly journals Active Set Type Algorithms for Nonnegative Matrix Factorization in Hyperspectral Unmixing

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Li Sun ◽  
Congying Han ◽  
Ziwen Liu
Keyword(s):  
Matrix Factorization ◽  
Quadratic Convergence ◽  
Nonnegative Matrix Factorization ◽  
Computational Cost ◽  
Nonnegative Matrix ◽  
Image Mining ◽  
Active Set ◽  
Hyperspectral Unmixing ◽  
Numerical Tests ◽  
Sparse Features

Hyperspectral unmixing is a powerful method of the remote sensing image mining that identifies the constituent materials and estimates the corresponding fractions from the mixture. We consider the application of nonnegative matrix factorization (NMF) for the mining and analysis of spectral data. In this paper, we develop two effective active set type NMF algorithms for hyperspectral unmixing. Because the factor matrices used in unmixing have sparse features, the active set strategy helps reduce the computational cost. These active set type algorithms for NMF is based on an alternating nonnegative constrained least squares (ANLS) and achieve a quadratic convergence rate under the reasonable assumptions. Finally, numerical tests demonstrate that these algorithms work well and that the function values decrease faster than those obtained with other algorithms.

scholarly journals An Inexact Update Method with Double Parameters for Nonnegative Matrix Factorization

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Xiangli Li ◽  
Wen Zhang ◽  
Xiaoliang Dong ◽  
Juanjuan Shi
Keyword(s):  
Matrix Factorization ◽  
Gradient Descent ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Optimal Solution ◽  
Active Set ◽  
Active Set Method ◽  
New Methods ◽  
Two Parameters ◽  
Multiplicative Update

Nonnegative matrix factorization (NMF) has been used as a powerful date representation tool in real world, because the nonnegativity of matrices is usually required. In recent years, many new methods are available to solve NMF in addition to multiplicative update algorithm, such as gradient descent algorithms, the active set method, and alternating nonnegative least squares (ANLS). In this paper, we propose an inexact update method, with two parameters, which can ensure that the objective function is always descent before the optimal solution is found. Experiment results show that the proposed method is effective.

scholarly journals Rank Selection in Nonnegative Matrix Factorization using Minimum Description Length

2017 ◽  
Vol 29 (8) ◽  
pp. 2164-2176 ◽  
Author(s):  
Steven Squires ◽  
Adam Prügel-Bennett ◽  
Mahesan Niranjan
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Minimum Description Length ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Real Data ◽  
Data Matrix ◽  
Constraint Forces ◽  
Data Points ◽  
Linear Dimensionality Reduction

Nonnegative matrix factorization (NMF) is primarily a linear dimensionality reduction technique that factorizes a nonnegative data matrix into two smaller nonnegative matrices: one that represents the basis of the new subspace and the second that holds the coefficients of all the data points in that new space. In principle, the nonnegativity constraint forces the representation to be sparse and parts based. Instead of extracting holistic features from the data, real parts are extracted that should be significantly easier to interpret and analyze. The size of the new subspace selects how many features will be extracted from the data. An effective choice should minimize the noise while extracting the key features. We propose a mechanism for selecting the subspace size by using a minimum description length technique. We demonstrate that our technique provides plausible estimates for real data as well as accurately predicting the known size of synthetic data. We provide an implementation of our code in a Matlab format.

Efficient Nonnegative Matrix Factorization by DC Programming and DCA

2016 ◽  
Vol 28 (6) ◽  
pp. 1163-1216 ◽  
Author(s):  
Hoai An Le Thi ◽  
Xuan Thanh Vo ◽  
Tao Pham Dinh
Keyword(s):  
Least Squares ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Dc Programming ◽  
Selection Strategy ◽  
Sparse Regularization ◽  
Dc Algorithm ◽  
Difference Of Convex Functions

In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms—one computing all variables and one deploying a variable selection strategy—are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.

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