Nonnegative matrix factorization is a linear dimensionality reduction technique used for decomposing high-dimensional nonnegative data matrices for extracting basic and latent features. This technique plays fundamental roles in music analysis, signal processing, sound separation, and spectral data analysis. Given a time-varying objective function or a nonnegative time-dependent data matrix Y(t), the nonnegative factors of Y(t) can be obtained by taking the limit points of the trajectories of the corresponding ordinary differential equations. When the data are time dependent, it is natural to devise factorization techniques that capture the time dependency. To achieve this, one needs to solve continuous-time dynamical systems derived from iterative optimization schemes and construct nonnegative matrix factorization algorithms based on the solution curves. This article presents continuous nonnegative matrix factorization methods based on the solution of systems of ordinary differential equations associated with time-dependent data. In particular, we propose two new continuous-time algorithms based on the Kullback–Leibler divergence and the Amari α -divergence.

We propose a system of first-order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes. The proposed model is based on Hill and step functions which are used to simulate the switch-like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached. A simplified piece-wise linear system is also proposed to study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin.

In this paper we consider numerical techniques to locate the event points of the differential system x′=f(x), where f is a discontinuous vector field along an event surface splitting the state space into two different regions R1 and R2 and f(x)=fi(x) when x∈Ri, for i=1,2 while f1(x)≠f2(x) when x∈Σ. Methods based on Adams multistep schemes which approach the event surface Σ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f1 (respectively, f2) in the region R2 (respectively R1) and are based on the computation–at each step– of a new time step

Gene regulatory networks can be described by continuous models in which genes are acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modeled as a binary on-off response function around a certain threshold concentration. Different thresholds can regulate the behaviors of genes, so that the combined effect on a gene is generally assumed to obey Boolean-like composition rules. Analyzing the behavior of such network model is a challenging task in mathematical simulation, particularly when at least one variable is close to one of its thresholds, called switching domains. In this paper, we briefly review a particular class model for gene regulation networks, namely, the piece-wise linear model and we present an event-driven method to analyze the motion in switching domains.

In questo articolo proponiamo l’impiego delle fattorizzazioni matriciali non negative per l’analisi dei dati nell’Educational Data Mining. Il metodo si basa su un processo di decomposizione di un dataset per l’estrazione di informazioni latenti di immediata interpretazione. In particolare, l’applicazione delle fattorizzazioni non negative a score matrix consente di generare in modo automatico le cosiddette question matrix (Q-matrix), che descrivono le abilità necessarie affinché uno studente possa rispondere adeguatamente a questionari di valutazione. Un esempio su dati real-world illustra l’efficacia del metodo.

BACKGROUND: Multiple myeloma (MM) is a cancer of terminally differentiated plasma that is part of a spectrum of blood diseases. The role of the micro-environment is crucial for MM clonal evolution. METHODS: This paper describes the analysis carried out on a limited number of genes automatically extracted by a nonnegative matrix factorization (NMF) based approach from gene expression profiles of bone marrow fibroblasts of patients with monoclonal gammopathy of undetermined significance (MGUS) and MM. RESULTS: Automatic exploration through NMF, combined with a motivated post-processing procedure and a pathways analysis of extracted genes, allowed to infer that a functional switch is required to lead fibroblasts to acquire pro-tumorigenic activity in the progression of the disease from MGUS to MM. CONCLUSION: The extracted biologically relevant genes may be representative of the considered clinical conditions and may contribute to a deeper understanding of tumor behavior.

In this paper we face the problem of intelligently analyze Twitter data. We propose a novel workflow based on Nonnegative Matrix Factorization (NMF) to collect, organize and analyze Twitter data. The proposed workflow firstly fetches tweets from Twitter (according to some search criteria) and processes them using text mining techniques; then it is able to extract latent features from tweets by using NMF, and finally it clusters tweets and extracts human-interpretable topics. We report some preliminary experiments demonstrating the effectiveness of the proposed workflow as a tool for Intelligent Data Analysis (IDA), indeed it is able to extract and visualize interpretable topics from some newly collected Twitter datasets, that are automatically grouped together according to these topics. Furthermore, we numerically investigate the influence of different initializations mechanisms for NMF algorithms on the factorization results when very sparse Twitter data are considered. The numerical comparisons confirm that NMF algorithms can be used as clustering method in place of the well known k-means.

Nonnegative Matrix Factorization (NMF) is an unsupervised learning method for extracting latent features in high-dimensional nonnegative data. This report presents NMF strategies based on the solution of the systems of ordinary differen- tial equations associated with both static and time dependent data. Here the NMF minimization problem is treated in two ways: we start off by deriving dynami- cal systems using the gradient descent approach and finish off with quasi-Newton optimization. The steepest descent approach by itself is divided into two cases: con- strained and unconstrained minimization. Given a time-varying objective function or a nonnegative time dependent (static) data matrix Y , the approximate nonneg- ative factors of Y can be obtained by taking the limit points of the trajectories of the corresponding ODEs. In the case when the data to deal with is time dependent, it is a good practice to devise an NMF strategy that captures the time dependency. In either of the above cases the natural thing to do is solve the continuous-time dy- namical systems derived from iterative optimization schemes and construct NMF algorithms based on the solution curves.

We discuss Non-negative Matrix Factorization (NMF) techniques from the point of view of Intelligent Data Analysis (IDA), i.e. the intelligent application of human expertise and computational models for advanced data analysis. As IDA requires human involvement in the analysis process, the understandability of the results coming from computational models has a prominent importance. We therefore review the latest developments of NMF that try to fulfill the understandability requirement in several ways. We also describe a novel method to decompose data into user-defined --- hence understandable --- parts by means of a mask on the feature matrix, and show the method's effectiveness through some numerical examples.

Non-negative dyadic data, that is data representing observations which relate two finite sets of objects, appear in several domain applications, such as text-mining-based information retrieval, collaborative filtering and recom- mender systems, micro-array analysis and computer vision. Discovering la- tent subgroups among data is a fundamental task to be performed on dyadic data. In this context, clustering and co-clustering techniques are relevant tools for extracting and representing latent information in high dimensional data. Recently, Non-negative Matrix Factorizations attracted a great interest as clustering methods, due to their capability of performing a parts-based de- composition of data. In this paper, we focus our attention on how NMF with additional constraints can be properly applied for co-clustering non-negative dyadic data. In particular, we present a process which aims at enhancing the performance of 3-factors NMF as a co-clustering method, by identifying a clearer correlation structure represented by the block matrix. Experimental evaluation performed on some common datasets, by applying the proposed approach on two different NMF algorithms, shows that, in most cases, the quality of the obtained clustering increases, especially in terms of average inter-cluster similarity.

We study the problem of detecting and localizing objects in still, gray-scale images making use of the part-based representation provided by non-negative matrix factorizations. Non-negative matrix factorization represents an emerging example of subspace methods which is able to extract interpretable parts from a set of template image objects and then to additively use them for describing individual objects. In this paper, we present a prototype system based on some non-negative factorization algorithms, which differ in the additional properties added to the non-negative representation of data, in order to investigate if any additional constraint produces better results in general object detection via non-negative matrix factorizations.

Nonnegative Matrix Factorization (NMF) is a class of low-rank dimensionality reduction methods which approximates a given nonnegative data matrix into the product of two nonnegative matrices of proper dimensions performing the so called additive partbased decomposition of data. Due to the peculiar representation of information through purely additive linear combinations and the preservation of data nonnegativity, NMF has been recognized as one of the most promising method to analyse gene expression data coming from any microarray experiment. This paper briefly reviews some aspects and practical issues related to NMF when this technique is applied to microarray data. In particular, issues such as interpretation of factorization results, mechanisms of information visualization and extraction techniques used in the context of microarray data analysis and their relationship with some concepts usually appearing in information theory are discussed.

We face the problem of interpreting parts of a dataset as small selections of features. Particularly, we propose a novel masked non- negative matrix factorization algorithm which is used to explain data as a composition of interpretable parts which are actually hidden in them and to introduce knowledge in the factorization process. Numerical ex- amples prove the effectiveness of the proposed MNMF algorithm as a useful tool for Intelligent Data Analysis.

In this paper we illustrate the use of Nonnegative Matrix Factorization (NMF) to analyze real data derived from an e-learning context. NMF is a matrix decomposition method which extracts latent information from data in such a way that it can be easily interpreted by humans. Particularly, the NMF of a score matrix can automatically generate the so called Q-matrix. In an e-learning scenario, the Q-matrix describes the abilities to be acquired by students to correctly answer evaluation exams. An example on real response data illustrates the effectiveness of this factorization method as a tool for EDM.

Non-negative matrix factorization is a multivariate analysis method which is proven to be useful in many areas such as bio-informatics, molecular pattern discovery, pattern recognition, document clustering and so on. It seeks a reduced representation of a multivariate data matrix into the product of basis and encoding matrices possessing only non-negative elements, in order to learn the so called part-based representations of data. All algorithms for computing non-negative matrix factorization are iterative, therefore particular emphasis must be placed on a proper initialization of NMF because of its local convergence. The problem of selecting appropriate starting matrices becomes more complex when data possess special meaning as in document clustering. In this paper, we propose the adoption of the subtractive clustering algorithm as a scheme to generate initial matrices for non-negative matrix factorization algorithms. Comparisons with other commonly adopted initializations of non-negative matrix factorization algorithms have been performed and the proposed scheme reveals to be a good trade-off between effectiveness and speed. Moreover, the effectiveness of the proposed initialization to suggest a number of basis for NMF, when data distances are estimated, is illustrated when NMF is used for solving clustering problems where the number of groups in which the data are grouped is not known a-priori. The influence of a proper rank factor on the interpretability and the effectiveness of the results are also discussed.