Convex Nonnegative Matrix Factorization Research Articles

Nonnegative factorization of spectrogram for local damage detection of belt conveyor gearboxes

The problem of local damage detection for rotating machines is widely studied in the literature. In order to extract information of the damage, the approach based on the vibration signal analysis is applied very often. However, in many cases the analysis of the signal in the time domain is not sufficient. A good choice is the analysis of the signal into the time-frequency domain and extraction of the information on damage from this representation. In this paper, we propose to separate the time-frequency map (spectrogram) of the vibration signal into new partial spectrograms. We assume that one of the new time-frequency representation carries information about the damage, i.e. cyclic impulsive behavior. To perform the separation we apply the idea that is based on a very important topic in numerical algebra, which is matrix factorization. In the presented approach, Convex Non-negative Matrix Factorization (Convex NMF) is used. In the experiments, we analyze a real vibration signal from the heavy duty gearbox used in mining industry. The results obtained for real signal are compared with those obtained using the Spectral Kurtosis approach, one of the classical method used in the problem of local damage detection.

Semi-supervised convex nonnegative matrix factorizations with graph regularized for image representation

Non-negative matrix factorization (NMF) is a very effective method for high dimensional data analysis, which has been widely used in computer vision. It can capture the underlying structure of image in the low dimensional space using its parts-based representations. However, nonnegative entries are usually required for the data matrix in NMF, which limits its application. Besides, it is actually an unsupervised method without making use of prior information of data. In this paper, we propose a novel method called Pairwise constrained Graph Regularized Convex Nonnegative Matrix Factorization (PGCNMF), which not only allows the processing of mixed-sign data matrix but also incorporates pairwise constraints generated among all labeled data into Convex NMF framework. We expect that images which have the same class label will have very similar representations in the low dimensional space as much as possible, while images with different class labels will have dissimilar representations as much as possible. Clustering experiments on nonnegative and mixed-sign real-world image datasets are conducted to demonstrate the effectiveness of the proposed method.

Convex nonnegative matrix factorization with manifold regularization

Nonnegative Matrix Factorization (NMF) has been extensively applied in many areas, including computer vision, pattern recognition, text mining, and signal processing. However, nonnegative entries are usually required for the data matrix in NMF, which limits its application. Besides, while the basis and encoding vectors obtained by NMF can represent the original data in low dimension, the representations do not always reflect the intrinsic geometric structure embedded in the data. Motivated by manifold learning and Convex NMF (CNMF), we propose a novel matrix factorization method called Graph Regularized and Convex Nonnegative Matrix Factorization (GCNMF) by introducing a graph regularized term into CNMF. The proposed matrix factorization technique not only inherits the intrinsic low-dimensional manifold structure, but also allows the processing of mixed-sign data matrix. Clustering experiments on nonnegative and mixed-sign real-world data sets are conducted to demonstrate the effectiveness of the proposed method.

Neighborhood Preserving Convex Nonnegative Matrix Factorization

The convex nonnegative matrix factorization (CNMF) is a variation of nonnegative matrix factorization (NMF) in which each cluster is expressed by a linear combination of the data points and each data point is represented by a linear combination of the cluster centers. When there exists nonlinearity in the manifold structure, both NMF and CNMF are incapable of characterizing the geometric structure of the data. This paper introduces a neighborhood preserving convex nonnegative matrix factorization (NPCNMF), which imposes an additional constraint on CNMF that each data point can be represented as a linear combination of its neighbors. Thus our method is able to reap the benefits of both nonnegative data factorization and the purpose of manifold structure. An efficient multiplicative updating procedure is produced, and its convergence is guaranteed theoretically. The feasibility and effectiveness of NPCNMF are verified on several standard data sets with promising results.

Link Prediction via Convex Nonnegative Matrix Factorization on Multiscale Blocks

Low rank matrices approximations have been used in link prediction for networks, which are usually global optimal methods and lack of using the local information. The block structure is a significant local feature of matrices: entities in the same block have similar values, which implies that links are more likely to be found within dense blocks. We use this insight to give a probabilistic latent variable model for finding missing links by convex nonnegative matrix factorization with block detection. The experiments show that this method gives better prediction accuracy than original method alone. Different from the original low rank matrices approximations methods for link prediction, the sparseness of solutions is in accord with the sparse property for most real complex networks. Scaling to massive size network, we use the block information mapping matrices onto distributed architectures and give a divide-and-conquer prediction method. The experiments show that it gives better results than common neighbors method when the networks have a large number of missing links.

Discriminant Convex Non-negative Matrix Factorization for the classification of human brain tumours

The medical analysis of human brain tumours commonly relies on indirect measurements. Among these, magnetic resonance imaging (MRI) and spectroscopy (MRS) predominate in clinical settings as tools for diagnostic assistance. Pattern recognition (PR) methods have successfully been used in this task, usually interpreting diagnosis as a supervised classification problem. In MRS, the acquired spectral signal can be analyzed in an unsupervised manner to extract its constituent sources. Recently, this has been successfully accomplished using Non-negative Matrix Factorization (NMF) methods. In this paper, we present a method to introduce the available class information into the unsupervised source extraction process of a convex variant of NMF. Novel techniques to generate diagnostic predictions for new, unseen spectra using the proposed Discriminant Convex-NMF are also described and experimentally assessed.

Convex Non-Negative Matrix Factorization for Brain Tumor Delimitation from MRSI Data

BackgroundPattern Recognition techniques can provide invaluable insights in the field of neuro-oncology. This is because the clinical analysis of brain tumors requires the use of non-invasive methods that generate complex data in electronic format. Magnetic Resonance (MR), in the modalities of spectroscopy (MRS) and spectroscopic imaging (MRSI), has been widely applied to this purpose. The heterogeneity of the tissue in the brain volumes analyzed by MR remains a challenge in terms of pathological area delimitation.Methodology/Principal FindingsA pre-clinical study was carried out using seven brain tumor-bearing mice. Imaging and spectroscopy information was acquired from the brain tissue. A methodology is proposed to extract tissue type-specific sources from these signals by applying Convex Non-negative Matrix Factorization (Convex-NMF). Its suitability for the delimitation of pathological brain area from MRSI is experimentally confirmed by comparing the images obtained with its application to selected target regions, and to the gold standard of registered histopathology data. The former showed good accuracy for the solid tumor region (proliferation index (PI)>30%). The latter yielded (i) high sensitivity and specificity in most cases, (ii) acquisition conditions for safe thresholds in tumor and non-tumor regions (PI>30% for solid tumoral region; ≤5% for non-tumor), and (iii) fairly good results when borderline pixels were considered.Conclusions/SignificanceThe unsupervised nature of Convex-NMF, which does not use prior information regarding the tumor area for its delimitation, places this approach one step ahead of classical label-requiring supervised methods for discrimination between tissue types, minimizing the negative effect of using mislabeled voxels. Convex-NMF also relaxes the non-negativity constraints on the observed data, which allows for a natural representation of the MRSI signal. This should help radiologists to accurately tackle one of the main sources of uncertainty in the clinical management of brain tumors, which is the difficulty of appropriately delimiting the pathological area.

Convex non-negative matrix factorization for massive datasets

Non-negative matrix factorization (NMF) has become a standard tool in data mining, information retrieval, and signal processing. It is used to factorize a non-negative data matrix into two non-negative matrix factors that contain basis elements and linear coefficients, respectively. Often, the columns of the first resulting factor are interpreted as “cluster centroids” of the input data, and the columns of the second factor are understood to contain cluster membership indicators. When analyzing data such as collections of gene expressions, documents, or images, it is often beneficial to ensure that the resulting cluster centroids are meaningful, for instance, by restricting them to be convex combinations of data points. However, known approaches to convex-NMF suffer from high computational costs and therefore hardly apply to large-scale data analysis problems. This paper presents a new framework for convex-NMF that allows for an efficient factorization of data matrices of millions of data points. Triggered by the simple observation that each data point can be expressed as a convex combination of vertices of the data convex hull, we require the basic factors to be vertices of the data convex hull. The benefits of convex-hull NMF are twofold. First, for a growing number of data points the expected size of the convex hull, i.e. the number of its vertices, grows much slower than the dataset. Second, distance preserving low-dimensional embeddings allow us to efficiently sample the convex hull and hence to quickly determine candidate vertices. Our extensive experimental evaluation on large datasets shows that convex-hull NMF compares favorably to convex-NMF in terms of both speed and reconstruction quality. We demonstrate that our method can easily be applied to large-scale, real-world datasets, in our case consisting of 750,000 DBLP entries, 4,000,000 digital images, and 150,000,000 votes on World of Warcraft ®guilds, respectively.

Algorithms for Nonnegative Matrix Factorization with the β-Divergence

This letter describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parameterized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a β-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for β ∈ (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace).

Cultures of the Central Highlands, New Guinea

Cultures of the Central Highlands, New Guinea