Nonnegative Matrix Factorization models for knowledge extraction from biomedical and other real world data

Abstract

Inspect data for searching valuable information hidden in represents a key aspect in several fields. Fortunately, most of the available data presents an embedded mathematical structure which can be profitably exploited to better investigate latent patterns hidden in them. Dimensionality Reduction (DR) approaches represent one of the most suitable instrument to untangle latent information. These techniques aim to represent data under analysis onto a low-dimensional space allowing to consider most of all of intrinsic knowledge as ideal sources (namely basis) of the process under consideration. In this work we consider Nonnegative Matrix Factorizations (NMFs), which prove to be the most effective among DR approaches in analyzing real-life nonnegative data. NMF simulates the human part-based learning process which states that parts are combined additively to form a whole. Some variants of NMF will be also presented as minimization tasks to which regularization terms can be added in accordance to some additional characteristics (such as sparsity or orthogonality). We investigate significant computational and interpretative aspects related to NMF according to different application domains, with a specific attention to the analysis of biological data. Moreover we present a new NMF model designed for microarray data analysis that incorporates specific biological proprieties as different constraints. Since NMF and its variants are daily used in several application domains, we conclude stressing how NMF and its constrained variants work in some real life applications, showing some original works related to the analysis of data from engineering field.

Matrix Completion Based on Low-Rank and Local Features Applied to Images Recovery and Recommendation Systems

“…Tanner et al [17] proposed the Alternating Steepest Descent algorithm (ASD) which can replace the least square subproblem solution in LMaFit. The nonnegative matrix factorization [18] is another famous MFM, it has been widely used as it automatically extracts sparse and meaningful features from highdimensional data [19], [20]. However, the nonnegative matrix factorization does not capture the local features of the sparse observation matrix.…”

The axes of biology: a novel axes-based network embedding paradigm to decipher the functional mechanisms of the cell

“…Other approaches include matrix factorization-based ones, including e.g. : Singular Value Decomposition (SVD) [9], Principal Component Analysis (PCA) [10] with the sparse and probabilistic variants [11], Independent Component Analysis (ICA) [12] and Non-negative Matrix Factorizations (NMF) [13,14]. In particular, Non-negative Matrix Tri-Factorization (NMTF) is an extension of NMF and a well-known Machine Learning (ML) technique introduced for co-clustering and dimensionality reduction [15].…”

A Review on Initialization Methods for Nonnegative Matrix Factorization: Towards Omics Data Experiments

“…Various matrix factorizations have proven their effectiveness in handling omic data; the best known are: Singular Value Decomposition (SVD) [8,17], Principal Component Analysis (PCA) and its sparse and probabilistic variants [14], Independent Component Analysis (ICA) [18] and Nonnegative matrix factorizations (NMF) [3,[19][20][21][22][23]. Each of these techniques is based on different constraints that characterize the final properties of the matrix factors, leading to different optimization problems and numerical algorithms that must be used.…”

“…Note that each H kj describes the effect that the kth metagene has on the jth sample, so that a low value H kj indicates that the corresponding kth metagene has reduced importance in approximating the jth sample. NMF allows to reveal interpretable latent factors (unlike PCA or ICA, which also have negative entries without obvious biological significance) and to identify genes belonging to multiple pathways or biological processes [20,22,23,25]. As these reasons lead to a much more intuitive and interpretable representation, NMF is quite often preferred to other techniques.…”