Survey on Probabilistic Models of Low-Rank Matrix Factorizations

Chapter 19 - Dimensionality Reduction and Latent Variable Modeling

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online.

The construction of literature-based networks of gene-gene interactions is one of the most important applications of text mining in bioinformatics. Extracting potential gene relationships from the biomedical literature may be helpful in building biological hypotheses that can be explored further experimentally. In this paper, we explore the utility of singular value decomposition (SVD) and non-negative matrix factorization (NMF) to extract unrecognized gene relationships from the biomedical literature by taking advantage of known gene relationships. We introduce a way to incorporate a priori knowledge of gene relationships into LSI/SVD and NMF. In addition, we propose a gene retrieval method based on NMF (GR/NMF), which shows comparable performance with latent semantic indexing based on SVD.

We review popular unsupervised learning methods for the analysis of high-dimensional data encountered in, for example, genomics, medical imaging, cohort studies, and biobanks. We show that four commonly used methods, principal component analysis, K-means clustering, nonnegative matrix factorization, and latent Dirichlet allocation, can be written as probabilistic models underpinned by a low-rank matrix factorization. In addition to highlighting their similarities, this formulation clarifies the various assumptions and restrictions of each approach, which eases identifying the appropriate method for specific applications for applied medical researchers. We also touch upon the most important aspects of inference and model selection for the application of these methods to health data.

Non-negative matrix factorization (NMF) has been one of the most popular methods for feature learning in the field of machine learning and computer vision. Most existing works directly apply NMF on high-dimensional image datasets for computing the effective representation of the raw images. However, in fact, the common essential information of a given class of images is hidden in their low rank parts. For obtaining an effective low-rank data representation, we in this paper propose a non-negative low-rank matrix factorization (NLMF) method for image clustering. For the purpose of improving its robustness for the data in a manifold structure, we further propose a graph regularized NLMF by incorporating the manifold structure information into our proposed objective function. Finally, we develop an efficient alternating iterative algorithm to learn the low-dimensional representation of low-rank parts of images for clustering. Alternatively, we also incorporate robust principal component analysis into our proposed scheme. Experimental results on four image datasets reveal that our proposed methods outperform four representative methods.

Low-rank matrix factorization problems such as non negative matrix factorization (NMF) can be categorized as a clustering or dimension reduction technique. The latter denotes techniques designed to find representations of some high dimensional dataset in a lower dimensional manifold without a significant loss of information. If such a representation exists, the features ought to contain the most relevant features of the dataset. Many linear dimensionality reduction techniques can be formulated as a matrix factorization. In this paper, we combine the conjugate gradient (CG) method with the Barzilai and Borwein (BB) gradient method, and propose a BB scaling CG method for NMF problems. The new method does not require to compute and store matrices associated with Hessian of the objective functions. Moreover, adopting a suitable BB step size along with a proper nonmonotone strategy which comes by the size of convex parameter ηk, results in a new algorithm that can significantly improve the CPU time, efficiency, the number of function evaluation. Convergence result is established and numerical comparisons of methods on both synthetic and real-world datasets show that the proposed method is efficient in comparison with existing methods and demonstrate the superiority of our algorithms.

Event Abstract Back to Event Comparing independent component analysis and non-negative matrix factorisation in the identification of event-related brain dynamics Sandra Ortega-Martorell1*, Ivan Olier2, Alfredo Vellido3, Paulo J. Lisboa4 and Wael El-Deredy2 1 Universitat Autònoma de Barcelona, Spain 2 University of Manchester, United Kingdom 3 Universitat Politècnica de Catalunya, Spain 4 Liverpool John Moores University, United Kingdom Matrix factorisation methods (such as principal and independent component analysis - PCA, ICA-, or singular value decomposition SVD) are commonly used for decomposing EEG data into factors from which the likely (latent) generating sources could be inferred or identified (Jung et al., 2000; Hoang et al., 2004). Since such factorisations are non-unique, different methods impose different constraints or priors in order to obtain a solution (e.g. components must be orthogonal in PCA, whereas independent in ICA). In non-negative matrix factorisation NMF (Paatero & Tapper,1994; Lee & Seung, 1999), the data matrix, X, is factorised into two non-negative matrices, W and H, whose product provides a good approximation of the original (X ≈ WH). Iterative methods are used to minimise the divergence between X and the product of W and H. In this study, a least squares method (Paatero & Tapper,1994 ) was used. We compared NMF and ICA in terms of: i. Their ability to identify eye movement artifacts in high density EEG electrode arrays. ii. Specificity in the separation of different event related brain dynamics from simultaneously activated sensory modalities. iii. Their computational cost, to assess their suitability for real time processing for the purposes of BCI applications. We used EEG recordings from 15 subjects as in (Brown et al., 2008). NMF was as good as ICA for identifying eye movement and separating different events. The former was much better, though, in the resolution of the components obtained. Computationally, NMF was marginally more efficient than FastICA. Keywords: Brain Signals, EEG Conference: XI International Conference on Cognitive Neuroscience (ICON XI), Palma, Mallorca, Spain, 25 Sep - 29 Sep, 2011. Presentation Type: Poster Presentation Topic: Poster Sessions: Modeling and Analysis of Brain Signals Citation: Ortega-Martorell S, Olier I, Vellido A, Lisboa PJ and El-Deredy W (2011). Comparing independent component analysis and non-negative matrix factorisation in the identification of event-related brain dynamics. Conference Abstract: XI International Conference on Cognitive Neuroscience (ICON XI). doi: 10.3389/conf.fnhum.2011.207.00082 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 16 Nov 2011; Published Online: 25 Nov 2011. * Correspondence: Dr. Sandra Ortega-Martorell, Universitat Autònoma de Barcelona, Barcelona, Spain, S.OrtegaMartorell@ljmu.ac.uk Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Sandra Ortega-Martorell Ivan Olier Alfredo Vellido Paulo J Lisboa Wael El-Deredy Google Sandra Ortega-Martorell Ivan Olier Alfredo Vellido Paulo J Lisboa Wael El-Deredy Google Scholar Sandra Ortega-Martorell Ivan Olier Alfredo Vellido Paulo J Lisboa Wael El-Deredy PubMed Sandra Ortega-Martorell Ivan Olier Alfredo Vellido Paulo J Lisboa Wael El-Deredy Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Matrix factorization plays an important role in scientific computation. The widely used one is singular value decomposition (SVD) which approximates the original data matrix with three lower rank matrices with orthogonality constraints. Recently nonnegative matrix factorization (NMF) considering the nonnegativity of data makes the results more interpretable than those of SVD. However NMF finds only two factor matrices and there is no significant index as singular values of SVD which can be used for sorting learned basis vectors. In this paper we take into account the nonnegativity for SVD and propose nonnegative SVD (NNSVD). The preliminary results on the microarray data of spermatogenesis show that NNSVD has advantages of both SVD and NMF.

RPCA (Robust Principal Component Analysis) recovers sparse and low rank components from the original observation data, RPCA commonly uses ADM (Alternate Direction Method) for solving, the efficiency of algorithm depends on the nuclear norm optimization solution, that is SVD. And the application of RPCA in computer vision, image and video and so on, large amounts of data, such as images and video, make it difficult for large-scale data SVD. In this paper, we used the random matrix algorithm to improve the SVD, respectively the Count Sketch algorithm, the Prototype Randomized k-SVD algorithm and the Faster Randomized k-SVD algorithm, The main idea is to reduce the size of the original large-scale data matrix and sample randomly. Using the random projection algorithm to obtain an approximation of the original matrix, and operate QR decomposition of the approximate matrix, get the unitary matrix corresponding to the approximate matrix, and do the corresponding SVD, finally we can get the results which is similar to the original matrix calculation. Obtaining approximation of the original data matrix. But the cost time and space have been greatly optimized. Simulation experiments based on single image and video foreground detection show that the proposed methods can greatly improve the efficiency of RPCA iterative optimization.

Analysis of two-mode network data using nonnegative matrix factorization

Matrix factorizations---where a given data matrix is approximated by a product of two or more factor matrices---are powerful data mining tools. Among other tasks, matrix factorizations are often used to separate global structure from noise. This, however, requires solving the `model order selection problem' of determining where fine-grained structure stops, and noise starts, i.e., what is the proper size of the factor matrices.

Matrix factorizations or matrix decompositions are methods that represent a matrix as a product of two or more matrices. There are various types of matrix factorizations such as LU factorization, Cholesky factorization, singular value decomposition etc. Matrix factorization is widely used in pattern recognition, image denoising, data clustering etc. Motivated by these applications, some properties and applications of various types of matrix factorizations are studied. One of the purposes of matrix factorization is to ease the computation. Thus, comparisons in term of computation time of various matrix factorizations in different areas are carried out.

This paper proposes methods for Machine Learning (ML)-based Beam Alignment (BA), using low-complexity ML models, and achieves a small pilot overhead. We assume a single-user massive mmWave MIMO, Uplink, using a fully analog architecture. Assuming large-dimension codebooks of possible beam patterns at UE and BS, this data-driven and model-based approach aims to partially and blindly sound a small subset of beams from these codebooks. The proposed BA is blind (no CSI), based on Received Signal Energies (RSEs), and circumvents the need for exhaustively sounding all possible beams. A sub-sampled subset of beams is then used to train several ML models such as low-rank Matrix Factorization (MF), non-negative MF (NMF), and shallow Multi-Layer Perceptron (MLP). We provide an extensive mathematical description of these models and the algorithms for each of them. Our extensive numerical results show that, by sounding only 10% of the beams from the UE and BS codebooks, the proposed ML tools are able to accurately predict the non-sounded beams through multiple transmitted power regimes. This observation holds as the codebook sizes at UE and BS vary from 128×128 to 1024×1024.

A recommendation system is a method that provides suggestions of items that might users like. There are many domains that can be recommended, one of the most demanded domains by users today is food. In the era of big data, food choices from the large amount of data make it difficult for users to choose the right food for them. The collaborative filtering (CF) approach is considered capable of providing accurate and high quality item suggestions. One of the algorithms that can provide good performance results from the CF approach is Matrix Factorization (MF). This study aims to test a dataset that contains product ratings of food items using three MF algorithms, which are Singular Value Decomposition (SVD), SVD with Implicit Ratings (SVD++), and Non-Negative Matrix Factorization (NMF). Different latent factors are also used for the purpose of improving the performance of the proposed recommendation system algorithm. The dataset used is Amazon Fine Food Reviews. The study shows NMF and SVD++ as the best algorithm for generating user rating predictions for items. NMF has the smallest average prediction error as measured by MAE which is 0.7311. While SVD++ obtains the smallest prediction error value of 1.0607 as measured using RMSE. In addition to these results, the top-n evaluation also shows that the proposed algorithm performs quite well. The hit ratio value for each different n-item always increases proportionally to the number of recommended n-items. The highest hit ratio value is generated from the SVD++ algorithm of 0.0025 on n-item recommendations of 25 items. Overall it can be said that the proposed algorithm has good performance in providing item recommendations.

Low-rank prior-based Fast-RPCA for clutter filtering and noise suppression in non-contrast ultrasound microvascular imaging