Farness preserving Non-negative matrix factorization

In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms-one computing all variables and one deploying a variable selection strategy-are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.

Microbiome datasets are often comprised of different representations or views which provide complementary information to understand microbial communities, such as metabolic pathways, taxonomic assignments, and gene families. Data integration methods including approaches based on nonnegative matrix factorization (NMF) combine multi-view data to create a comprehensive view of a given microbiome study by integrating multi-view information. In this paper, we proposed a novel variant of NMF which called Laplacian regularized joint non-negative matrix factorization (LJ-NMF) for integrating functional and phylogenetic profiles from HMP. We compare the performance of this method to other variants of NMF. The experimental results indicate that the proposed method offers an efficient framework for microbiome data analysis.

This paper presents a robust variant of nonnegative matrix factorization (NMF) based on complex Student's t distributions (t-NMF) for source separation of single-channel audio signals. The Itakura-Saito divergence NMF (Gaussian NMF) is justified for this purpose under an assumption that the complex spectra of source signals and those of the mixture signal are complex Gaussian distributed (the additiv-ity of power spectra holds). In fact, however, the source spectra are often heavy-tailed distributed. When the source spectra are complex Cauchy distributed, for example, the mixture spectra are also complex Cauchy distributed (the additivity of amplitude spectra holds). Using the complex t distribution that includes the complex Gaussian and Cauchy distributions as its special cases, we propose t-NMF as a unified extension of Gaussian NMF and Cauchy NMF. Furthermore, we propose the corresponding variant of positive semidefinite tensor factorization based on multivariate complex t distributions (t-PSDTF). The experimental results showed that while t-NMF and t-PSDTF were comparative to Gaussian counterparts in terms of peak performance, they worked much better on average because they are insensitive to initialization and tend to avoid local optima.

Discriminative concept factorization for data representation

Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences. To address this, in this work we propose to use structured random compression, that is, random projections that exploit the data structure, for two NMF variants: classical and separable. In separable NMF (SNMF) the left factors are a subset of the columns of the input matrix. We present suitable formulations for each problem, dealing with different representative algorithms within each one. We show that the resulting compressed techniques are faster than their uncompressed variants, vastly reduce memory demands, and do not encompass any significant deterioration in performance. The proposed structured random projections for SNMF allow to deal with arbitrarily shaped large matrices, beyond the standard limit of tall-and-skinny matrices, granting access to very efficient computations in this general setting. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples, showing the suitability of the proposed approaches.

In this paper, we present a novel method for obtaining a low dimensional representation of a complex brain network that: (1) can be interpreted in a neurobiologically meaningful way, (2) emphasizes group differences by accounting for label information, and (3) captures the variation in disease subtypes/severity by respecting the intrinsic manifold structure underlying the data. Our method is a supervised variant of non-negative matrix factorization (NMF), and achieves dimensionality reduction by extracting an orthogonal set of subnetworks that are interpretable, reconstructive of the original data, and also discriminative at the group level. In addition, the method includes a manifold regularizer that encourages the low dimensional representations to be smooth with respect to the intrinsic geometry of the data, allowing subjects with similar disease-severity to share similar network representations. While the method is generalizable to other types of non-negative network data, in this work we have used structural connectomes (SCs) derived from diffusion data to identify the cortical/subcortical connections that have been disrupted in abnormal neurological state. Experiments on a traumatic brain injury (TBI) dataset demonstrate that our method can identify subnetworks that can reliably classify TBI from controls and also reveal insightful connectivity patterns that may be indicative of a biomarker.

As a variant of nonnegative matrix factorization (NMF), symmetric NMF (SNMF) has shown to be effective for capturing the cluster structure embedded in the graph representation. In contrast to the existing SNMF-based clustering methods that empirically construct the similarity matrix and rigidly introduce the supervisory information to the assignment matrix, in this paper, we propose a novel SNMF-based semisupervised clustering method, namely, pairwise constraint propagation-induced SNMF (PCPSNMF). By formulating a single-constrained optimization problem, PCPSNMF is capable of learning the similarity and assignment matrices adaptively and simultaneously, in which a small amount of supervisory information in the form of pairwise constraints is introduced in a flexible way to guide the construction of the similarity matrix, and the two matrices communicate with each other to achieve mutual refinement until convergence. In addition, we propose an efficient alternating iterative algorithm to solve the optimization problem, whose convergence is theoretically proven. Experimental results over several benchmark image data sets demonstrate that PCPSNMF is less sensitive to initialization and produces higher clustering performance, compared with the state-of-the-art methods.

An analysis of the coherence of descriptors in topic modeling

Nonnegative matrix factorization (NMF), which can lead to nonsubtractive parts-based representation, has been demonstrated to be effective for dimensionality reduction of hyperspectral imagery (HSI). However, existing NMF methods applied to HSI use only a single spectral feature and do not take into consideration spatial information, such as texture or morphological features, while it has been widely acknowledged that exploiting multiple features can improve performance. Consequently, a variant of orthogonal NMF, which can not only achieve a nonnegative factorization but also exploit the complementary information that arises among heterogeneous features, is proposed for hyperspectral dimensionality reduction. The proposed method, which couples orthogonal NMF with a previous multiple-features-combining algorithm, yields a discriminative low-dimensional feature representation that matches the intuition that parts should sum to produce a whole. An efficient multiplicative updating procedure is derived, and its local convergence is guaranteed theoretically. Experimental results on two hyperspectral data sets demonstrate the effectiveness of the proposed method.

Non-negative matrix factorization (NMF) is a fundamental matrix decomposition technique that is used primarily for dimensionality reduction and is increasing in popularity in the biological domain. Although finding a unique NMF is generally not possible, there are various iterative algorithms for NMF optimization that converge to locally optimal solutions. Such techniques can also serve as a starting point for deep learning methods that unroll the algorithmic iterations into layers of a deep network. In this study, we develop unfolded deep networks for NMF and several regularized variants in both a supervised and an unsupervised setting. We apply our method to various mutation data sets to reconstruct their underlying mutational signatures and their exposures. We demonstrate the increased accuracy of our approach over standard formulations in analyzing simulated and real mutation 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.

The growing availability of hyperspectral data from planetary missions ranging from CubeSats to flagship programs has led to many studies aiming to infer surface mineralogy through remote sensing techniques. The Moon Mineralogical Mapper (M³) aboard Chandrayaan-1 provides visible and near-infrared data (430–3000nm) at 140 m/pixel resolution and has been widely used to infer lunar surface composition [1]. A common strategy involves applying unmixing models, often Non-Negative Matrix Factorization (NNMF), to M³ images with varying reflectance and compositional properties, using laboratory reference spectra from the RELAB database for comparison [2,3]. These spectra from standard materials represent typical lunar minerals such as olivine, pyroxenes, plagioclase, iron-titanium oxides, and volcanic glasses.This study first focuses on reproducing previous efforts to estimate the lunar mineralogy, applied to the Apollo 17 landing site (Figure 1). Then it shifts toward highlighting the methodological uncertainties involved in interpreting hyperspectral imagery using unmixing models in remote sensing applications.Figure. 1 (Top)Taurus-Littrow Valley. (Bottom) Mean (dark) reflectance spectrum and standard deviation (red) of the Taurus-Littrow Valley hyperspectral image.In the initial stage, a spatial binning with 8X8 window was applied to reduce the signal-to-noise ratio of the image. Principal Component Analysis (PCA) was then used to reduce data dimensionality and highlight surface variations. Initially applied on the binned image, PCA revealed that the first component explained 98% of the variance, mainly reflecting topography and illumination differences. To minimize these effects and enhance chemical information, the first derivative of the spectral data was applied (same preprocessing was applied to reference spectra, Figure.2). A second PCA was then performed to determine the chemical rank of the matrix, which suggested that NNMF should be carried out with 3–4 principal components.Figure. 2 ​First derivative of reference spectra from RELAB.NNMF was performed to unmix and extract surface components, possibly revealing features that may correspond to geochemical variations. In this final stage, it was assumed that three principal components could be present within the image. Figure. 3 shows the spatial distribution of the combination of these three components (top), along with the corresponding “pure” spectra (bottom) extracted by the model after applying NNMF using endmember initialization and non-negativity constraints [4] applied to the contributions. Commonly, when these results were compared with preprocessed spectra from RELAB, compounds such as pyroxenes, olivine, volcanic-glass, may be identified (Figure. 3), with acceptable lack of fit and a total variance explanation exceeding 99%.Figure. 3 False RGB showing the concentrations of the three main components after applying NNMF model using endmember initialization and non-negativity constraints.However, while spectral unmixing models have been widely applied in remote sensing for decades, limited attention has been paid to the intrinsic ambiguities they involve. Although simpler ambiguities (such as those related to scale or sign) can often be addressed using basic constraints like non-negativity, rotational ambiguity remains a significant and underexplored source of uncertainty. For these reasons, it is important to highlight that solutions derived from NNMF (commonly referred to as Multivariate Curve Resolution with non-negativity constraints) depend considerably on the degrees of freedom allowed by the optimization algorithm (e.g., Alternating Least Squares, ALS). To visualize and quantify this issue, feasibility plots were employed through MCRbands and FACpack (Figure. 4), which map the set of all admissible solutions within the NNMF framework [5-7].Figure. 4 Feasibility plots for the three components calculated. Right, the contributions and left, the spectral profiles obtained. Contributions: Blue and red represents the solution obtained and the plausible solution that could also be considered as optimal.Figure 4 shows that any space covered between those solutions could be considered feasible. On the left, the spectra calculated and the feasible spectra both indicate a strong lack of specificity and uniqueness in the solutions, clearly stating that the model is not stable enough to be deemed an optimal solution. The extent of solution variability reveals that differences between the acceptable solution and real solutions for the considered species are considerable due to rotational freedom. In conclusion, we emphasize the importance of a more rigorous consideration of ambiguity in remote sensing applications. Additionally, the presence of regolith and volcanic rocks on the lunar surface makes it unlikely that a 140 m/pixel represents a pure mineral spectrum [8].Keywords: Remote sensing, Chemometrics, Unmixing models, Near Infrared. Acknowledgements: Work supported through the PAMMAT project: “Alteration processes in Mars and Moon Meteorites, and Terrestrial Analogues at different environments: Mars2020, Rosalind Franklin and Returned Samples from Mars and Moon” (Grant No. PID2022-142750OB-I00), funded by the Spanish Agency for Research (through the Spanish Ministry of Science and Innovation, MCIN, and the European Regional Development Fund, FEDER).

Nonnegative matrix factorization (NMF) has been successfully used in different applications including computer vision, pattern recognition and text mining. NMF aims to decompose a data matrix into the product of two matrices (respectively denoted as the basis vectors and the encoding vectors), whose entries are constrained to be nonnegative. Unlike the ordinary NMF, we propose a novel NMF, denoted as MMNMF, which considers both geometrical information and discriminative information hidden in the data. The geometrical information is discovered by minimizing the distance among the encoding vectors, while the discriminative information is uncovered by maximizing the distance among base vectors. Clustering experiments are performed on the real-world data sets of faces, images, and documents to demonstrate the effectiveness of the proposed algorithm.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

Gene expression analysis of samples with mixed cell types only provides limited insight to the characteristics of specific tissues. In silico deconvolution can be applied to extract cell type specific expression, thus avoiding prohibitively expensive techniques such as cell sorting or single-cell sequencing. Non-negative matrix factorization (NMF) is a deconvolution method shown to be useful for gene expression data, in part due to its constraint of non-negativity. Unlike other methods, NMF provides the capability to deconvolve without prior knowledge of the components of the model. However, NMF is not guaranteed to provide a globally unique solution. In this work, we present FaStaNMF, a method that balances achieving global stability of the NMF results, which is essential for inter-experiment and inter-lab reproducibility, with accuracy and speed. Results: FaStaNMF was applied to four datasets with known ground truth, created based on publicly available data or by using our simulation infrastructure, RNAGinesis. We assessed FaStaNMF on three criteria - speed, accuracy, and stability, and it favorably compared to the standard approach of achieving reproduceable results with NMF. We expect that FaStaNMF can be applied successfully to a wide array of biological data, such as different tumor/immune and other disease microenvironments.