Nonnegative singular value decomposition for microarray data analysis of spermatogenesis

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.

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.

Low-rank matrix factorizations such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) are a large class of methods for pursuing the low-rank approximation of a given data matrix. The conventional factorization models are based on the assumption that the data matrices are contaminated stochastically by some type of noise. Thus the point estimations of low-rank components can be obtained by Maximum Likelihood (ML) estimation or Maximum a posteriori (MAP). In the past decade, a variety of probabilistic models of low-rank matrix factorizations have emerged. The most significant difference between low-rank matrix factorizations and their corresponding probabilistic models is that the latter treat the low-rank components as random variables. This paper makes a survey of the probabilistic models of low-rank matrix factorizations. Firstly, we review some probability distributions commonly-used in probabilistic models of low-rank matrix factorizations and introduce the conjugate priors of some probability distributions to simplify the Bayesian inference. Then we provide two main inference methods for probabilistic low-rank matrix factorizations, i.e., Gibbs sampling and variational Bayesian inference. Next, we classify roughly the important probabilistic models of low-rank matrix factorizations into several categories and review them respectively. The categories are performed via different matrix factorizations formulations, which mainly include PCA, matrix factorizations, robust PCA, NMF and tensor factorizations. Finally, we discuss the research issues needed to be studied in the future.

Non-negative Matrix Factorization (NMF) and Singular Value Decomposition (SVD) are widely recognized as pivotal dimensionality reduction techniques in the literature, particularly for deep learning applications involving large and high-dimensional datasets like SEM images. This study systematically evaluates the impact of SVD and NMF on the performance, efficiency, and energy consumption of four deep learning architectures: GoogleNet, AlexNet, ResNet, and SqueezeNet. By applying these techniques to reduce dataset dimensions, we observed that SVD excelled in computational efficiency, achieving up to 35% faster processing times compared to raw datasets. NMF, on the other hand, provided superior feature interpretability, which proved beneficial for tasks requiring meaningful pattern extraction. Energy consumption analysis revealed that SVD led to a 28% reduction in computational energy cost on average, making it a practical choice for resource-constrained environments. Among the evaluated models, ResNet consistently delivered the highest classification accuracy after dimensionality reduction, showing an improvement of 4-6% over models trained on non-reduced data. These findings underscore the critical role of dimensionality reduction in enhancing the scalability, energy efficiency, and classification accuracy of deep learning models, offering valuable insights for optimizing high-dimensional data applications in both academic and industrial contexts.

This paper presents the transform domain image steganography schemes using three popular matrix factorization techniques and contourlet transform. It is known that security of image steganography is mainly evaluated using undetectability of stego image when steganalyzer examines it in order to detect the presence of hidden secret information. Good imperceptibility only suggests eavesdropper’s inability to suspect about the hidden information; however stego image may be analyzed by applying certain statistical checks when it is being transmit- ted through the channel. This work focusses on improving undetectability by employing ma- trix decomposition techniques along with transform domain image steganography. Singular value decomposition (SVD), QR factorization, Nonnegative matrix factorization (NMF) are employed to decompose contourlet coefficients of cover image and secret is embedded into its matrix factorized coefficients. The variety of investigations include the effect of matrix decomposition techniques on major attributes of image steganography like imperceptibility, robustness to a variety of image processing operations, and universal steganalysis perfor- mance. Better imperceptibility, large capacity, and poor detection accuracy compared to existing work validate the efficacy of the proposed image steganography algorithm. Compa- rative analysis amongst three matrix factorization methods is also presented and analyzed.

Dimension reduction techniques, such as singular value decomposition (SVD) and nonnegative matrix factorization (NMF), have been successfully applied in image hashing by retaining the essential features of the original image matrix. However, a concern of great importance in image hashing is that no single solution is optimal and robust against all types of attacks. The contribution of this paper is threefold. First, we introduce a recently proposed dimension reduction technique, referred as Fast Johnson-Lindenstrauss Transform (FJLT), and propose the use of FJLT for image hashing. FJLT shares the low distortion characteristics of a random projection, but requires much lower computational complexity. Secondly, we incorporate Fourier-Mellin transform into FJLT hashing to improve its performance under rotation attacks. Thirdly, we propose a new concept, namely, content-based fingerprint, as an extension of image hashing by combining different hashes. Such a combined approach is capable of tackling all types of attacks and thus can yield a better overall performance in multimedia identification. To demonstrate the superior performance of the proposed schemes, receiver operating characteristics analysis over a large image database and a large class of distortions is performed and compared with the state-of-the-art image hashing using NMF.

Non-negative matrix factorization (NMF) provides a lower rank approximation of a matrix. Due to nonnegativity imposed on the factors, it gives a latent structure that is often more physically meaningful than other lower rank approximations such as singular value decomposition (SVD). Most of the algorithms proposed in literature for NMF have been based on minimizing the Frobenius norm. This is partly due to the fact that the minimization problem based on the Frobenius norm provides much more flexibility in algebraic manipulation than other divergences. In this paper we propose a fast NMF algorithm that is applicable to general Bregman divergences. Through Taylor series expansion of the Bregman divergences, we reveal a relationship between Bregman divergences and Euclidean distance. This key relationship provides a new direction for NMF algorithms with general Bregman divergences when combined with the scalar block coordinate descent method. The proposed algorithm generalizes several recently proposed methods for computation of NMF with Bregman divergences and is computationally faster than existing alternatives. We demonstrate the effectiveness of our approach with experiments conducted on artificial as well as real world data.

Maintaining data mining accuracy on distorted datasets is an important issue in privacy preserving data mining. Using matrix approximation, we propose several efficient and flexible techniques to address this issue, and utilize unique characteristics of matrix factorization to maintain data pattern. We use the support vector machine classification to compare accuracy maintenance after data distortion by different methods. With better performance than some classical data perturbation approaches, nonnegative matrix factorization and singular value decomposition are considered to be promising techniques for privacy preserving data mining. Experimental results demonstrate that mining accuracy on the distorted data used these methods is almost as good as that on the original data, with added property of privacy preservation. It indicates that the matrix factorization-based data distortion schemes perturb only confidential attributes to meet privacy requirements while preserving general data pattern for knowledge extraction.

In the nonnegative matrix factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$, where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$, respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$, i.e. (approximately) minimize $\left\lVert{M - AW}_F\right\rVert$, where $\left\lVert\right\rVert_F$, denotes the Frobenius norm; we refer to this as approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where $A$ and $W$ are computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. (Without the restriction that $A$ and $W$ be nonnegative, both the exact and approximate problems can be solved optimally via the singular value decomposition.) We initiate a study of when this problem is solvable in polynomial time. Our results are the following: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant $r$. Indeed NMF is most interesting in applications precisely when $r$ is small. 2. We complement this with a hardness result, that if exact $NMF$ can be solved in time $(nm)^{o(r)}$, 3-SAT has a subexponential-time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in $n$, $m$, and $r$ under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.

Analysis of two-mode network data using nonnegative matrix factorization

The technology of low-field nuclear magnetic resonance (LF-NMR) is commonly used in food, agriculture, energy and chemical sectors due to its non-destructive, non-invasive, in situ, green and other advantages. Recently, this technology played an increasingly large role in the field of food-safety supervision especially. In oil product quality testing, conventional T2 spectrum inversion methods such as the non-negative singular value decomposition (SVD) algorithm can only reflect T2 spectrum in a smooth model. However, for a sparse model, the inversion result of non-negative SVD algorithm is expected to be very glossy, leading to low resolution of T2 spectrum and inaccurate analysis of sample property. To solve this problem, we propose a sparse T2 spectrum inversion algorithm based on the L1 norm minimization constraint. In this paper, we establish the sparse model expression of NMR echo curve, and obtain the T2 sparse spectrum inversion results based on the inner truncated Newton-point method. Furthermore, the effectiveness of L1 sparse inversion algorithm is examined by the synthetic data of the smooth model and the spare model which have different peak numbers and signaltonoise ratios (SNRs). Synthetic results show that compared with the non-negative SVD algorithm, the L1 sparse algorithm is appropriate for both the smooth model and the sparse model with higher inversion accuracy. When the number of T2 peaks in a sparse model changes from a single peak to a quad peak, the L1 sparse algorithm can still obtain accurate inversion results, while the SVD algorithm results in a gradual deterioration, and cannot even determine the peak number. Under the sparse model, when the SNR of the measured NMR curve is gradually changed from 5 dB to 50 dB, the L1 sparse algorithm at 20 dB or more can obtain accurate inversion results which have less than 10% peak error and less than 5% peak position error and amplitude average error. However, the non-negative SVD algorithm cannot obtain accurate results at each SNR. Finally, multiple sets of frying oil samples are utilized to prove the accuracy and robustness of L1 sparse inversion algorithm. Inversion results of seven sets of frying oil samples show that the L1 sparse algorithm prefers the non-negative SVD algorithm. The obtained T2 spectrum by the L1 sparse algorithm shows three peaks obviously, and the T21 peak area ratio S21 and the single component relaxation time T2w are higher linear with respect to frying time than the results by non-negative SVD algorithm, which is useful for detecting the frying oil quality change. The inversion results of the T2 spectrum at different SNRs are consistent with the synthetic results, i.e., when the SNR is reduced, the T2 spectrum inversion results from the L1 sparse algorithm are better than those from the non-negative SVD algorithm when SNR is greater than 20 dB.

Non-negative matrix factorization (NMF) is widely used in feature extraction and dimension reduction fields. Essentially, it is an optimization problem to determine two non-negative low rank matrices \(W_{m \times k}\) and \(H_{k \times n}\) for a given matrix \(A_{m \times n}\), satisfying \(A_{m \times n} \approx W_{m \times k}H_{k \times n}\). In this paper, a novel approach to improve the image decomposing and reconstruction effects by introducing the Singular Value Decomposing (SVD)-based initialization scheme of factor matrices W and H, and another measure called choosing rule to determine the optimum value of factor rank k, are proposed. The input image is first decomposed using SVD to get its singular values and corresponding eigenvectors. Then, the number of main components as the rank value k is extracted. Then, the singular values and corresponding eigenvectors are used to initialize W and H based on selected rank k. Finally, convergent results are obtained using multiplicative and additive update rules. However, iterative NMF algorithms’ convergence is very slow on most platforms limiting its practicality. To this end, a parallel implementation frame of described improved NMF algorithm using CUDA, a tool for algorithms parallelization on massively parallel processors, i.e., many-core graphics processors, is presented. Experimental results show that our approach can get better decomposing effect than traditional NMF implementations and dramatic accelerate rate comparing to serial schemes as well as existing distributed-system implementations.

This paper presents a comprehensive framework for addressing the inherent trade-offs among personalization, diversity, and coverage in hybrid advertisement recommendation systems. In response to the growing complexity of online advertising, the proposed system integrates collaborative filtering via Singular Value Decomposition (SVD) with content-based filtering using Non-negative Matrix Factorization (NMF), enhanced by a re-ranking mechanism based on coverage profiles. This hybrid design aims to deliver relevant, diverse, and widely distributed ads, thereby improving user engagement and fairness among content providers. To quantitatively evaluate these objectives, three novel metrics—Ad User Profile (AUP), Ad Candidate Set (ACS), and Ad Matching Degree (AMD)—are introduced. Through empirical analysis, including visualization and comparative experiments, the study demonstrates that the proposed system outperforms traditional models in achieving a more balanced recommendation outcome. Additionally, real-world case studies and optimization formulations are explored to support scalable deployment. This work contributes to the evolving landscape of recommender systems by proposing a scalable, user-centric approach that harmonizes personalization with fairness and discovery in digital advertising.

In order to detect radial velocity ship in complex sea state, a novel detection algorithm is proposed. Considering that the frequencies of the ship and first-order sea clutters meet the requirements of sparse in the Doppler domain, certain sparseness constraints are imposed on Non-negative Matrix Factorization (NMF) to improve the efficiency of extracting the frequency spectrum of ship from that of the sea clutter. The following two topics are discussed: (1) the detection algorithm based on NMF (NMFSCC); (2) the detection results of NMFSCC, compared with Singular Value Decomposition (SVD) algorithm. Simulation results demonstrate the usefulness of the proposed approach.

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.