PROBABILISTIC NON-NEGATIVE MATRIX FACTORIZATION: THEORY AND APPLICATION TO MICROARRAY DATA ANALYSIS

2014 ◽  
Vol 12 (01) ◽  
pp. 1450001 ◽  
Author(s):  
BELHASSEN BAYAR ◽  
NIDHAL BOUAYNAYA ◽  
ROMAN SHTERENBERG
Keyword(s):  
Physical Interpretation ◽  
Matrix Factorization ◽  
Dna Microarrays ◽  
Optimal Solution ◽  
Microarray Data Analysis ◽  
Factorization Problem ◽  
The Stability ◽  
Update Rules ◽  
Negative Factors ◽  
Non Negative Matrix Factorization

Non-negative matrix factorization (NMF) has proven to be a useful decomposition technique for multivariate data, where the non-negativity constraint is necessary to have a meaningful physical interpretation. NMF reduces the dimensionality of non-negative data by decomposing it into two smaller non-negative factors with physical interpretation for class discovery. The NMF algorithm, however, assumes a deterministic framework. In particular, the effect of the data noise on the stability of the factorization and the convergence of the algorithm are unknown. Collected data, on the other hand, is stochastic in nature due to measurement noise and sometimes inherent variability in the physical process. This paper presents new theoretical and applied developments to the problem of non-negative matrix factorization (NMF). First, we generalize the deterministic NMF algorithm to include a general class of update rules that converges towards an optimal non-negative factorization. Second, we extend the NMF framework to the probabilistic case (PNMF). We show that the Maximum a posteriori (MAP) estimate of the non-negative factors is the solution to a weighted regularized non-negative matrix factorization problem. We subsequently derive update rules that converge towards an optimal solution. Third, we apply the PNMF to cluster and classify DNA microarrays data. The proposed PNMF is shown to outperform the deterministic NMF and the sparse NMF algorithms in clustering stability and classification accuracy.

scholarly journals A Block Coordinate Descent-Based Projected Gradient Algorithm for Orthogonal Non-Negative Matrix Factorization

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 540
Author(s):  
Soodabeh Asadi ◽  
Janez Povh
Keyword(s):  
Matrix Factorization ◽  
Synthetic Data ◽  
Coordinate Descent ◽  
The Other ◽  
Gradient Algorithm ◽  
Block Coordinate Descent ◽  
Projected Gradient Method ◽  
Projected Gradient ◽  
Factorization Problem ◽  
Non Negative Matrix Factorization

This article uses the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalize the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalized objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods.

Blind Extraction of Correlated Fault Sources Based on Constrained Non-Negative Matrix Factorization

2012 ◽  
Vol 226-228 ◽  
pp. 760-764
Author(s):  
Ning Li ◽  
Hai Ting Chen
Keyword(s):  
Matrix Factorization ◽  
Matrix Decomposition ◽  
Separation Performance ◽  
Vibration Source ◽  
Practical Performance ◽  
Blind Extraction ◽  
The Stability ◽  
The Cost ◽  
Correlated Signals ◽  
Non Negative Matrix Factorization

Blind source separation (BSS) has been successfully used to extract undetected fault vibration sources from mixed observation signals by assuming that each unknown vibration source is mutually independent. However, conventional BSS algorithms cannot address the situation in which the fault source could be partially dependent on or correlated to other sources. For this, a new matrix decomposition method, called Non-negative Matrix Factorization (NMF), is introduced to separate these partially correlated signals. In this paper, the observed temporal signals are transformed into the frequency domain to satisfy the non-negative limit of NMF. The constraint of the least correlation between the separated sources is added into the cost function of NMF to enhance the stability of NMF, and the constrained non-negative matrix factorization (CNMF) is proposed. The simulation results show that the separation performance of CNMF is superior to the common BSS algorithms and the experiment result verifies the practical performance of CNMF.

A Generalized Divergence Measure for Nonnegative Matrix Factorization

2007 ◽  
Vol 19 (3) ◽  
pp. 780-791 ◽  
Author(s):  
Raul Kompass
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Auxiliary Function ◽  
Divergence Measure ◽  
Factorization Problem ◽  
Special Cases ◽  
Leibler Divergence ◽  
Update Rules ◽  
Quadratic Distance

This letter presents a general parametric divergence measure. The metric includes as special cases quadratic error and Kullback-Leibler divergence. A parametric generalization of the two different multiplicative update rules for nonnegative matrix factorization by Lee and Seung (2001) is shown to lead to locally optimal solutions of the nonnegative matrix factorization problem with this new cost function. Numeric simulations demonstrate that the new update rule may improve the quadratic distance convergence speed. A proof of convergence is given that, as in Lee and Seung, uses an auxiliary function known from the expectation-maximization theoretical framework.

Sparse Recovery Using Conjugate Gradient and Orthogonal Triangular Decomposition

2013 ◽  
Vol 756-759 ◽  
pp. 2479-2483
Author(s):  
Xi Hua Peng ◽  
Shan Xiong Chen ◽  
Xiao Yan Liu
Keyword(s):  
Combinatorial Optimization ◽  
Conjugate Gradient ◽  
Matrix Factorization ◽  
Matching Pursuit ◽  
Optimal Solution ◽  
Sparse Recovery ◽  
Triangular Decomposition ◽  
Solution Sequence ◽  
Matching Pursuit Algorithm ◽  
Non Negative Matrix Factorization

In this article, we propose the matching pursuit algorithm of combinatorial optimization based CGLS and LSQR. We use non-negative matrix factorization for measuring discrepancy of solution sequence between CGLS and LSQR, and represent combinatorial optimization based CGLS and LSQ to choose optimal solution sequences. The experiments indicate our method is extended to the case where target signal has been corrupted by noise, it demonstrate perfectly recovery ability of signal with noise.

scholarly journals Graph Sparse Nonnegative Matrix Factorization Algorithm Based on the Inertial Projection Neural Network

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Xiangguang Dai ◽  
Chuandong Li ◽  
Biqun Xiang
Keyword(s):  
Neural Network ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Optimal Solution ◽  
Factorization Method ◽  
Real World Data ◽  
Factorization Problem ◽  
Sparse Nonnegative Matrix Factorization ◽  
And Performance

We present a novel method, called graph sparse nonnegative matrix factorization, for dimensionality reduction. The affinity graph and sparse constraint are further taken into consideration in nonnegative matrix factorization and it is shown that the proposed matrix factorization method can respect the intrinsic graph structure and provide the sparse representation. Different from some existing traditional methods, the inertial neural network was developed, which can be used to optimize our proposed matrix factorization problem. By adopting one parameter in the neural network, the global optimal solution can be searched. Finally, simulations on numerical examples and clustering in real-world data illustrate the effectiveness and performance of the proposed method.

scholarly journals Informed Weighted Non-Negative Matrix Factorization Using αβ-Divergence Applied to Source Apportionment

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 253 ◽  
Author(s):  
Gilles Delmaire ◽  
Mahmoud Omidvar ◽  
Matthieu Puigt ◽  
Frédéric Ledoux ◽  
Abdelhakim Limem ◽  
...  
Keyword(s):  
Particulate Matter ◽  
Cost Function ◽  
Source Apportionment ◽  
Matrix Factorization ◽  
Expert Knowledge ◽  
Operating Conditions ◽  
Real Dataset ◽  
Update Rules ◽  
Available Information ◽  
Non Negative Matrix Factorization

In this paper, we propose informed weighted non-negative matrix factorization (NMF) methods using an α β -divergence cost function. The available information comes from the exact knowledge/boundedness of some components of the factorization—which are used to structure the NMF parameterization—together with the row sum-to-one property of one matrix factor. In this contribution, we extend our previous work which partly involved some of these aspects to α β -divergence cost functions. We derive new update rules which are extendthe previous ones and take into account the available information. Experiments conducted for several operating conditions on realistic simulated mixtures of particulate matter sources show the relevance of these approaches. Results from a real dataset campaign are also presented and validated with expert knowledge.

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