Nicoletta Del Buono

A Continuous Technique for the Weighted Low-Rank Approximation Problem

This paper concerns with the problem of approximating a target matrix with a matrix of lower rank with respect to a weighted norm. Weighted norms can arise in several situations: when some of the entries of the matrix are not observed or need not to be treated equally. A gradient flow approach for solving weighted low rank approximation problems is provided. This approach allows the treatment of both real and complex matrices and exploits some important features of the approximation matrix that optimization techniques do not use. Finally, some numerical examples are provided.

Dynamical modeling of liver Aquaporin-9 expression and glycerol permeability in hepatic glucose metabolism

Liver is crucial in the homeostasis of glycerol, an important metabolic intermediate. Plasma glycerol is imported by hepatocytes mainly through Aquaporin-9 (AQP9), an aquaglyceroporin channel negatively regulated by insulin in rodents. AQP9 is of critical importance in glycerol metabolism since hepatic glycerol utilization is rate-limited at the hepatocyte membrane permeation step. Glycerol kinase catalyzes the initial step for the conversion of the imported glycerol into glycerol-3-phosphate, a major substrate for de novo synthesis of glucose (gluconeogenesis) and/or triacyglycerols (lipogenesis). A model addressing the glucose-insulin system to describe the hepatic glycerol import and metabolism and the correlation with the glucose homeostasis is lacking so far. Here we consider a system of first-order ordinary differential equations delineating the relevance of hepatocyte AQP9 in liver glycerol permeability. Assuming the hepatic glycerol permeability as depending on the protein levels of AQP9, a mathematical function is designed describing the time course of the involvement of AQP9 in mouse hepatic glycerol metabolism in different nutritional states. The resulting theoretical relationship is derived fitting experimental data obtained with murine models at the fed, fasted or re-fed condition. While providing useful insights into the dynamics of liver AQP9 involvement in male rodent glycerol homeostasis our model may be adapted to the human liver serving as an important module of a whole body-model of the glucose metabolism both in health and metabolic diseases.

Breast Cancer’s Microarray Data: Pattern Discovery Using Nonnegative Matrix Factorizations

One challenge in microarray analysis is to discover and capture valuable knowledge to understand biological processes and human disease mechanisms. Nonnegative Matrix Factorization (NMF) – a constrained optimization mechanism which decomposes a data matrix in terms of additive combination of non-negative factors– has been demonstrated to be a useful tool to reduce the dimension of gene expression data and to identify potentially interesting genes which explain latent structure hidden in microarray data.

Subtractive initialization of nonnegative matrix factorizations for document clustering

Nonnegative matrix factorizations (NMF) have recently assumed an important role in several fields, such as pattern recognition, automated image exploitation, data clustering and so on. They represent a peculiar tool adopted to obtain a reduced representation of multivariate data by using additive components only, in order to learn parts-based representations of data. All algorithms for computing the NMF 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 initialization matrices becomes more complex when data possess special meaning, and this is the case of document clustering. In this paper, we present a new initialization method which is based on the fuzzy subtractive scheme and used to generate initial matrices for NMF algorithms. A preliminary comparison of the proposed initialization with other commonly adopted initializations is presented by considering the application of NMF algorithms in the context of document clustering.

A Penalty Function for Computing Orthogonal Non-negative Matrix Factorizations

Nonnegative matrix factorization (NMF) is a widely-used method for multivariate analysis of nonnegative data to obtain a reduced representation of data matrix only using a basis matrix and a encoding variable matrix having non-negative elements. A NMF of a data matrix can be obtained by finding a solution of a nonlinear optimization problem over a specified cost function. In this paper we investigate the formulation and then the computational techniques to obtain orthogonal NMF, when the orthogonal constraint on the columns of the basis is added. We propose a penalty objective function to be minimized on the intersection of the set of non-negative matrices and the Stiefel manifold in order to derive a projected gradient flow whose solutions preserve both the orthogonality and the non-negativity.

Intelligent Twitter Data Analysis Based on Nonnegative Matrix Factorizations

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.

Robust embedded projective nonnegative matrix factorization for image analysis and feature extraction

Nonnegative matrix factorization (NMF) is an unsupervised learning method for decomposing high-dimensional nonnegative data matrices and extracting basic and intrinsic features. Since image data are described and stored as nonnegative matrices, the mining and analysis process usually involves the use of various NMF strategies. NMF methods have well-known applications in face recognition, image reconstruction, handwritten digit recognition, image denoising and feature extraction. Recently, several projective NMF (P-NMF) methods based on positively constrained projections have been proposed and were found to perform better than the standard NMF approach in some aspects. However, some drawbacks still affect the existing NMF and P-NMF algorithms; these include dense factors, slow convergence, learning poor local features, and low reconstruction accuracy. The aim of this paper is to design algorithms that address the aforementioned issues. In particular, we propose two embedded P-NMF algorithms: the first method combines the alternating least squares (ALS) algorithm with the P-NMF update rules of the Frobenius norm and the second one embeds ALS with the P-NMF update rule of the Kullback–Leibler divergence. To assess the performances of the proposed methods, we conducted various experiments on four well-known data sets of faces. The experimental results reveal that the proposed algorithms outperform other related methods by providing very sparse factors and extracting better localized features. In addition, the empirical studies show that the new methods provide highly orthogonal factors that possess small entropy values.

Part-Based Data Analysis with Masked Non-negative Matrix Factorization

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

Event Driven Approach for Simulating Gene Regulation Networks

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.

Improving knowledge on the activation of bone marrow fibroblasts in MGUS and MM disease through the automatic extraction of genes via a nonnegative matrix factorization approach on gene expression profiles.

BackgroundMultiple 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.MethodsThis 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.ResultsAutomatic 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.ConclusionThe extracted biologically relevant genes may be representative of the considered clinical conditions and may contribute to a deeper understanding of tumor behavior.