Sparse Non-negative Matrix Factorization Algorithm Research Articles

Topological analysis of overlapping dynamic brain networks under naturalistic stimuli based on sparse non-negative matrix factorization algorithm

Based on the overlapping hypothesis of human brain networks, the sparse non-negative matrix factorization algorithm was used to decompose the dynamic brain functional networks under naturalistic stimuli into a set of spatiotemporal overlapping subgraphs and weight coefficients. This is followed by an analysis of how these subgraphs characterize the topographies in dynamic brain networks. It is also found that specific local topology is modulated by emotional activities. Default Mode Network, Ventral Attention Network, Auditory Network, Visual Network, and some brain regions in other cognitive networks are mainly involved. Our results are expected to provide solutions for future brain function detection and diagnosis of brain diseases based on naturalistic stimuli.

Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing Based on Endmember Independence and Spatial Weighted Abundance

Hyperspectral image unmixing is an important task for remote sensing image processing. It aims at decomposing the mixed pixel of the image to identify a set of constituent materials called endmembers and to obtain their proportions named abundances. Recently, number of algorithms based on sparse nonnegative matrix factorization (NMF) have been widely used in hyperspectral unmixing with good performance. However, these sparse NMF algorithms only consider the correlation characteristics of abundance and usually just take the Euclidean structure of data into account, which can make the extracted endmembers become inaccurate. Therefore, with the aim of addressing this problem, we present a sparse NMF algorithm based on endmember independence and spatial weighted abundance in this paper. Firstly, it is assumed that the extracted endmembers should be independent from each other. Thus, by utilizing the autocorrelation matrix of endmembers, the constraint based on endmember independence is to be constructed in the model. In addition, two spatial weights for abundance by neighborhood pixels and correlation coefficient are proposed to make the estimated abundance smoother so as to further explore the underlying structure of hyperspectral data. The proposed algorithm not only considers the relevant characteristics of endmembers and abundances simultaneously, but also makes full use of the spatial-spectral information in the image, achieving a more desired unmixing performance. The experiment results on several data sets further verify the effectiveness of the proposed algorithm.

A Novel Signal Separation Method Based on Improved Sparse Non-Negative Matrix Factorization.

In order to separate and extract compound fault features of a vibration signal from a single channel, a novel signal separation method is proposed based on improved sparse non-negative matrix factorization (SNMF). In view of the traditional SNMF failure to perform well in the underdetermined blind source separation, a constraint reference vector is introduced in the SNMF algorithm, which can be generated by the pulse method. The square wave sequences are constructed as the constraint reference vector. The output separated signal is constrained by the vector, and the vector will update according to the feedback of the separated signal. The redundancy of the mixture signal will be reduced during the constantly updating of the vector. The time–frequency distribution is firstly applied to capture the local fault features of the vibration signal. Then the high dimension feature matrix of time–frequency distribution is factorized to select local fault features with the improved SNMF method. Meanwhile, the compound fault features can be separated and extracted automatically by using the sparse property of the improved SNMF method. Finally, envelope analysis is used to identify the feature of the output separated signal and realize compound faults diagnosis. The simulation and test results show that the proposed method can effectively solve the separation of compound faults for rotating machinery, which can reduce the dimension and improve the efficiency of algorithm. It is also confirmed that the feature extraction and separation capability of proposed method is superior to the traditional SNMF algorithm.

Improvements in sparse non-negative matrix factorization for hyperspectral unmixing algorithms

The hyperspectral unmixing technique is of great significance for hyperspectral data analysis. The non-negative matrix factorization (NMF)-based method is one of the most popular unmixing methods. However, the sparse constraints commonly used in NMF methods are not differentiable at zero, which affects the stability and accuracy of the results. In addition, sparse constraints are supposed to have endmember departure limitations. An improved sparse NMF algorithm is proposed. First, a biased L1 / 2 sparse constraint is used to effectively explore the sparsity of abundances, which can also have a sparse effect under the condition of the abundance sum-to-one constraint. Second, to alleviate the endmember departure effect, a minimum energy difference constraint is introduced. Simulated and real-data experiments verify the advantages of the proposed method against state-of-the-art sparse NMF methods.

A unified framework for sparse non-negative least squares using multiplicative updates and the non-negative matrix factorization problem

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the sparse codes. We show that the proposed framework encompasses a large class of S-NNLS algorithms and provide a computationally efficient inference procedure based on multiplicative update rules. Such update rules are convenient for solving large sets of S-NNLS problems simultaneously, which is required in contexts like sparse non-negative matrix factorization (S-NMF). We provide theoretical justification for the proposed approach by showing that the local minima of the objective function being optimized are sparse and the S-NNLS algorithms presented are guaranteed to converge to a set of stationary points of the objective function. We then extend our framework to S-NMF, showing that our framework leads to many well known S-NMF algorithms under specific choices of prior and providing a guarantee that a popular subclass of the proposed algorithms converges to a set of stationary points of the objective function. Finally, we study the performance of the proposed approaches on synthetic and real-world data.

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

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.

Smooth nonnegative matrix and tensor factorizations for robust multi-way data analysis

In this paper, we discuss new efficient algorithms for nonnegative matrix factorization (NMF) with smoothness constraints imposed on nonnegative components or factors. Such constraints allow us to alleviate certain ambiguity problems, which facilitates better physical interpretation or meaning. In our approach, various basis functions are exploited to flexibly and efficiently represent the smooth nonnegative components. For noisy input data, the proposed algorithms are more robust than the existing smooth and sparse NMF algorithms. Moreover, we extend the proposed approach to the smooth nonnegative Tucker decomposition and smooth nonnegative canonical polyadic decomposition (also called smooth nonnegative tensor factorization). Finally, we conduct extensive experiments on synthetic and real-world multi-way array data to demonstrate the advantages of the proposed algorithms.

Fast and Efficient Estimation of Individual Ancestry Coefficients

Inference of individual ancestry coefficients, which is important for population genetic and association studies, is commonly performed using computer-intensive likelihood algorithms. With the availability of large population genomic data sets, fast versions of likelihood algorithms have attracted considerable attention. Reducing the computational burden of estimation algorithms remains, however, a major challenge. Here, we present a fast and efficient method for estimating individual ancestry coefficients based on sparse nonnegative matrix factorization algorithms. We implemented our method in the computer program sNMF and applied it to human and plant data sets. The performances of sNMF were then compared to the likelihood algorithm implemented in the computer program ADMIXTURE. Without loss of accuracy, sNMF computed estimates of ancestry coefficients with runtimes ∼10-30 times shorter than those of ADMIXTURE.

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

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.

NMF with Sparse Regularizations in Transformed Domains

Non-negative blind source separation (non-negative BSS), which is also referred to as non-negative matrix factorization (NMF), is a very active field in domains as different as astrophysics, audio processing or biomedical signal processing. In this context, the efficient retrieval of the sources requires the use of signal priors such as sparsity. If NMF has now been well studied with sparse constraints in the direct domain, only very few algorithms can encompass non-negativity together with sparsity in a transformed domain since simultaneously dealing with two priors in two different domains is challenging. In this article, we show how a sparse NMF algorithm coined non-negative generalized morphological component analysis (nGMCA) can be extended to impose non-negativity in the direct domain along with sparsity in a transformed domain, with both analysis and synthesis formulations. To our knowledge, this work presents the first comparison of analysis and synthesis priors ---as well as their reweighted versions--- in the context of blind source separation. Comparisons with state-of-the-art NMF algorithms on realistic data show the efficiency as well as the robustness of the proposed algorithms.

Manifold Regularized Sparse NMF for Hyperspectral Unmixing

Hyperspectral unmixing is one of the most important techniques in analyzing hyperspectral images, which decomposes a mixed pixel into a collection of constituent materials weighted by their proportions. Recently, many sparse nonnegative matrix factorization (NMF) algorithms have achieved advanced performance for hyperspectral unmixing because they overcome the difficulty of absence of pure pixels and sufficiently utilize the sparse characteristic of the data. However, most existing sparse NMF algorithms for hyperspectral unmixing only consider the Euclidean structure of the hyperspectral data space. In fact, hyperspectral data are more likely to lie on a low-dimensional submanifold embedded in the high-dimensional ambient space. Thus, it is necessary to consider the intrinsic manifold structure for hyperspectral unmixing. In order to exploit the latent manifold structure of the data during the decomposition, manifold regularization is incorporated into sparsity-constrained NMF for unmixing in this paper. Since the additional manifold regularization term can keep the close link between the original image and the material abundance maps, the proposed approach leads to a more desired unmixing performance. The experimental results on synthetic and real hyperspectral data both illustrate the superiority of the proposed method compared with other state-of-the-art approaches.

Tumor Classification Based on Non-Negative Matrix Factorization Using Gene Expression Data

This paper presents a new method for tumor classification using gene expression data. In the proposed method, we first select genes using nonnegative matrix factorization (NMF) or sparse NMF (SNMF), and then we extract features from the selected genes by virtue of NMF or SNMF. At last, we apply support vector machines (SVM) to classify the tumor samples using the extracted features. In order for a better classification, a modified SNMF algorithm is also proposed. The experimental results on benchmark three microarray data sets validate that the proposed method is efficient. Moreover, the biological meaning of the selected genes are also analyzed.

Blind Spectral Unmixing Based on Sparse Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a widely used method for blind spectral unmixing (SU), which aims at obtaining the endmembers and corresponding fractional abundances, knowing only the collected mixing spectral data. It is noted that the abundance may be sparse (i.e., the endmembers may be with sparse distributions) and sparse NMF tends to lead to a unique result, so it is intuitive and meaningful to constrain NMF with sparseness for solving SU. However, due to the abundance sum-to-one constraint in SU, the traditional sparseness measured by L0/L1-norm is not an effective constraint any more. A novel measure (termed as S-measure) of sparseness using higher order norms of the signal vector is proposed in this paper. It features the physical significance. By using the S-measure constraint (SMC), a gradient-based sparse NMF algorithm (termed as NMF-SMC) is proposed for solving the SU problem, where the learning rate is adaptively selected, and the endmembers and abundances are simultaneously estimated. In the proposed NMF-SMC, there is no pure index assumption and no need to know the exact sparseness degree of the abundance in prior. Yet, it does not require the preprocessing of dimension reduction in which some useful information may be lost. Experiments based on synthetic mixtures and real-world images collected by AVIRIS and HYDICE sensors are performed to evaluate the validity of the proposed method.