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This study presents a comprehensive exploration of topic modeling methods tailored for large language model (LLM) using data obtained from Web of Science and LexisNexis from June 1, 2020, to December 31, 2023. The data collection process involved queries focusing on LLMs, including "Large language model," "LLM," and "ChatGPT." Various topic modeling approaches were evaluated based on performance metrics, including diversity and coherence. latent Dirichlet allocation (LDA), nonnegative matrix factorization (NMF), combined topic models (CTM), and bidirectional encoder representations from Transformers topic (BERTopic) were employed for performance evaluation. Evaluation metrics were computed across platforms, with BERTopic demonstrating superior performance in diversity and coherence across both LexisNexis and Web of Science. The experiment result reveals that news articles maintain a balanced coverage across various topics and mainly focus on efforts to utilize LLM in specialized domains. Conversely, research papers are more concise and concentrated on the technology itself, emphasizing technical aspects. Through the insights gained in this study, it becomes possible to investigate the future path and the challenges that LLMs should tackle. Additionally, they could offer considerable value to enterprises that utilize LLMs to deliver services. [ABSTRACT FROM AUTHOR]

Spectral line confocal imaging (LCI) technology enables high-resolution, rapid 3D surface morphology analysis of transparent materials and other samples, positioning it as a leading optical measurement tool. Nevertheless, where the sensor measures samples with significant variations in surface reflectance, the peak signals of the acquired images exhibit low signal-to-noise ratios or become distorted, hampering the peak extraction algorithm from accurately locating their peak positions. Consequently, it's arduous to reconstruct their precise surface topography. To enhance the sensor's dynamic measurement range and adaptability, this paper introduces an adaptive correction approach. It utilizes a regularized non-negative matrix to decompose images with various grey levels. This is followed by weighted frequency filtering and Gaussian enhancement of all the grey layers. Eventually, these layers are fused into one image with high contrast and uniformity. Using this approach, the sensor is capable of achieving precise 3D reconstruction with high-quality peak signals. Additionally, this study experimentally demonstrates the efficacy of the suggested approach by measuring the interdigital electrode and printed circuit boards (PCB). It should be noted that the technique presented in this paper is not limited to spectral line confocal instruments and is also effective for line laser and line structured light instruments. • A spectral line confocal system for high-speed 3D surface topography measurements was developed and built in-house. • A proposed adaptive signal correction algorithm to address the problem of reflectance difference surface measurement. • Enhances the dynamic measurement range and adaptability of the instrument. • The method is both simple and effective, as it requires only a single measurement followed by processing single image. • The method is also applicable to line laser and line structured light instruments. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words F which correspond to the non-existence of an edge between two vertices, and a collection of repeated words R with multiplicities which correspond to multiple edges between two vertices. In general, for given collections F of forbidden words and R of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift. [ABSTRACT FROM AUTHOR]

Heart failure (HF) is a life-threatening disease affecting at least 64 million people worldwide. Hence, it places great stresses on patients and healthcare systems. Accordingly, providing a computerized model for HF prediction will help in enhancing diagnosis, treatment, and long-term management of HF. In this paper, we introduce a new guided attentive HF prediction approach. In this method, a sparse-guided feature ranking method is proposed. Firstly, a Gauss–Seidel strategy is applied to the preprocessed feature pool for low-rank approximation procedure with a trace-norm regularization. The resultant sparse attributes, after a Spearman ranking elimination, are employed to guide the original feature pool through linear translation-variant model. Then, a fast Newton-based method is employed for a non-negative matrix factorization for the guided feature pool. The resultant bases of the factorization process are finally utilized in the adopted deep attentive predictive model. For the final prediction stage, instead of the commonly used machine learning approaches, we introduce an attentive-based classifier. It employs sequential attention to choose the most proper salient features for efficient interpretability and learning process. For the evaluation of the proposed HF prediction model, three different datasets are employed, i.e., UCI, Faisalabad, and Framingham datasets. Compared to state-of-the-art techniques, the proposed approach outperforms their performance on all datasets with even small feature sizes. With only four feature bases, the proposed method achieves an average accuracy of 98%, while, with full feature bases, full accuracy is gained. [ABSTRACT FROM AUTHOR]

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Given a matrix X ∈ R + m × n with non-negative entries, the cone factorization problem concerns computing { a 1 , ... , a m } ⊆ K belonging to a convex cone K ⊆ R k , and { b 1 , ... , b n } ⊆ K ∗ belonging to its dual so that X ij = ⟨ a i , b j ⟩ for all i ∈ [ m ] , j ∈ [ n ] . Cone factorizations are fundamental to mathematical optimization as they allow us to express convex bodies as feasible regions of linear conic programs. In this paper, we introduce the symmetric-cone multiplicative update (SCMU) algorithm for computing cone factorizations in settings where K is symmetric; i.e., the cone is self-dual and homogeneous. Our proposed SCMU algorithm is multiplicative in the sense that each iterate is updated by applying a suitably chosen automorphism of the cone, which is is obtained via a generalization of the geometric mean to symmetric cones. In particular, the SCMU update ensures that iterates remain in the interior of the cone by design. When applied to direct sums of simpler symmetric cones, the SCMU algorithm preserves the direct sum structure, and hence specifies an algorithm for computing cone factorizations over such cones. By using an extension of the Lieb's concavity theorem to symmetric cones, we show that the squared-loss objective is non-decreasing under the SCMU algorithm. When specialized to the nonnegative orthant, the SCMU algorithm recovers the seminal multiplicative update algorithm for computing Non-negative Matrix Factorizations developed by Lee and Seung. Last, we apply the SCMU algorithm to investigate hybrid lifts (combining SDP- and SOCP-based descriptions) of regular polygons from a numerical perspective. [ABSTRACT FROM AUTHOR]

Sparse linear prediction methods suffer from decreased prediction accuracy when the predictor variables have cluster structure (e.g., highly correlated groups of variables). To improve prediction accuracy, various methods have been proposed to identify variable clusters from the data and integrate cluster information into a sparse modeling process. But none of these methods achieve satisfactory performance for prediction, variable selection and variable clustering performed simultaneously. This paper presents Variable Cluster Principal Component Regression (VC‐PCR), a prediction method that uses variable selection and variable clustering in order to solve this problem. Experiments with real and simulated data demonstrate that, compared to competitor methods, VC‐PCR is the only method that achieves simultaneously good prediction, variable selection, and clustering performance when cluster structure is present. [ABSTRACT FROM AUTHOR]

Data-driven fault diagnosis, identifying abnormality causes using collected industrial data, is one of the challenging tasks for intelligent industry safety management. It is worth noting that practical industrial data are usually related to a mixture of several physical attributes, such as the operating environment, product quality and working conditions. However, the traditional models may not be sufficient to leverage the coherent information for diagnostic performance enhancement, due to their shallow architecture. This paper presents a hierarchical matrix factorization (HMF) that relies on a succession of matrix factoring to find an efficient representation of industrial data for fault diagnosis. Specifically, HMF consecutively decomposes data into several hierarchies. The intermediate hierarchies play the role of analysis operators which automatically learn implicit characteristics of industrial data; the final hierarchy outputs high-level and discriminative features. Furthermore, HMF is also extended in a nonlinear manner by introducing activation functions, referred as NHMF, to deal with nonlinearities in practical industrial processes. The applications of HMF and NHMF to fault diagnosis are evaluated by the multiple-phase flow process. The experimental results show that our models achieve competitive performance against the considered shallow and deep models, consuming less computing time than deep models. [ABSTRACT FROM AUTHOR]

The detection of gas leaks using acoustic signals is often compromised by environmental noise, which significantly impacts the accuracy of subsequent leak identification. Current noise reduction algorithms based on non-negative matrix factorization (NMF) typically utilize the Euclidean distance as their objective function, which can exacerbate noise anomalies. Moreover, these algorithms predominantly rely on simple techniques like Wiener filtering to estimate the amplitude spectrum of pure signals. This approach, however, falls short in accurately estimating the amplitude spectrum of non-stationary signals. Consequently, this paper proposes an improved non-negative matrix factorization (INMF) noise reduction algorithm that enhances the traditional NMF by refining both the objective function and the amplitude spectrum estimation process for reconstructed signals. The improved algorithm replaces the conventional Euclidean distance with the Kullback–Leibler (KL) divergence and incorporates noise and sparse constraint terms into the objective function to mitigate the adverse effects of signal amplification. Unlike traditional methods such as Wiener filtering, the proposed algorithm employs an adaptive Minimum Mean-Square Error-Log Spectral Amplitude (MMSE-LSA) method to estimate the amplitude spectrum of non-stationary signals adaptively across varying signal-to-noise ratios. Comparative experiments demonstrate that the INMF algorithm significantly outperforms existing methods in denoising leakage acoustic signals. [ABSTRACT FROM AUTHOR]

Specular highlight removal is a challenging task in the field of image enhancement, while it can significantly improve the quality of image in highlight regions. Recently, deep learning-based methods have been widely adopted in this task, demonstrating excellent performance by training on either massive paired data, wherein both the highlighted and highlight-free versions of the same image are available, or unpaired datasets where the one-to-one correspondence is inapplicable. However, it is difficult to obtain the corresponding highlight-free version of a highlight image, as the latter has already been produced under specific lighting conditions. In this paper, we propose a method for weakly supervised specular highlight removal that only requires highlight images. This method involves generating highlight-free images from highlight images with the guidance of masks estimated using non-negative matrix factorization (NMF). These highlight-free images are then fed consecutively into a series of modules derived from a Cycle Generative Adversarial Network (Cycle-GAN)-style network, namely the highlight generation, highlight removal, and reconstruction modules in sequential order. These modules are trained jointly, resulting in a highly effective highlight removal module during the verification. On the specular highlight image quadruples (SHIQ) and the LIME datasets, our method achieves an accuracy of 0.90 and a balance error rate (BER) of 8.6 on SHIQ, and an accuracy of 0.89 and a BER of 9.1 on LIME, outperforming existing methods and demonstrating its potential for improving image quality in various applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors. [ABSTRACT FROM AUTHOR]

In this paper, we propose an alternating block variant of the linearized Bregman iterations for a class of regularized nonnegative matrix factorization (NMF) problems. The proposed method exploits the block structure of NMF, utilizes the smooth adaptable property of the loss function based on the Bregman distance, and at the same time follows the iterative regularization idea of the linearized Bregman iterations method. Theoretically, we show that the proposed method is a descent method by adjusting the involved parameters. Finally, we end with several illustrative numerical experiments. [ABSTRACT FROM AUTHOR]

Initialization plays an important role in the accuracy of endmember extraction algorithms (EEAs) in linear hyperspectral unmixing (LHU). Random initialization can lead to varying endmembers generated by EEAs. To address this challenge, an initialization strategy has been introduced, encompassing vertex component analysis (VCA), automatic target generation process (ATGP), among others. These techniques significantly contribute to enhancing the accuracy of EEAs. However, complex initialization is sometimes less preferable, prompting the unexplored question of whether there exists an EEA robust to initialization. This paper focuses on analyzing this issue within the context of minimum simplex volume-based (MV) methods, which have received considerable attention in the past two decades due to their robustness against the absence of pure pixels. MV methods typically formulate LHU as an optimization problem, most of which includes a non-convex volume term. Additionally, many MV methods use VCA as an initialization strategy. Firstly, this paper demonstrates that the variable splitting augmented Lagrangian approach (SISAL), as a representative non-convex MV method, heavily depends on initialization. To our knowledge, the impact of initialization for MV methods has not been thoroughly analyzed before. Furthermore, this paper proposes an initially robust MV method by introducing a new convex MV term. Numerical experiments conducted on simulated and real datasets demonstrate its outstanding performance in accuracy and robustness to initialization. Throughout the experiments the proposed method proves to be the most stable, which is crucial in real scene where the ground truth is unknown beforehand. [ABSTRACT FROM AUTHOR]

In this paper, we consider the graph regularization Q-weighted nonnegative matrix factorization problem in multi-view clustering. Based on the Q-weighted norm property, this problem is transformed into the minimization problem of the trace function. The necessary condition for the existence of a solution is given. The proximal alternating nonnegative least squares method and its acceleration method are designed to solve it. The convergence theorem is also given. The feasibility and effectiveness of the proposed methods are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Hyperspectral unmixing (HU) has been a vital technique in hyperspectral image processing. Over the past years, many algorithms have been developed based on the nonnegative matrix factorization (NMF) framework. However, the conventional methods based on the NMF need prior information of the number of endmembers, which is usually estimated by a separate procedure. Therefore, the precision of estimating the number of endmembers will impact the subsequent unmixing effect. Thus, collaborating between the two processes is desired to avoid the propagation of errors. This paper proposes a new NMF-based method, which can estimate the number of endmembers and extract the endmembers and the abundance maps. Firstly, we introduce the weighted group sparsity regularization to adaptively eliminate the abundances corresponding to the redundant endmembers, while the abundances of the true endmembers are preserved. Then the number of endmembers can be determined by the reconstruction error function and the ${l_2}$ l 2 -norm of the abundance vectors. Secondly, since the simplex determined by the endmembers has the minimum volume among all simplexes that circumscribe the data points, we impose the total variational regularization to force the endmembers to be close to each other to minimize the simplex volume. And because the total variational regularization avoids the use of noisy pixels, it can decrease the influence of noise and improve the robustness of the solution. For the proposed model, we design an effective algorithm to solve it. Also, the experimental results based on the synthetic data and real data demonstrate that the proposed method can identify the endmembers number well and achieve robust unmixing results. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability. [ABSTRACT FROM AUTHOR]

In this paper we consider the vector-valued Schrödinger operator − Δ + V , where the potential term V is a matrix-valued function whose entries belong to L loc 1 (R d) and, for every x ∈ R d , V (x) is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in L 1 (R d , R m). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q ∈ (1 , ∞) ∪ { ∞ } , we obtain maximal inequality in L p (R d , R m) , for p in between 1 and some q , and generation results. [ABSTRACT FROM AUTHOR]

Marine mucilage, also known as sea-snot, is an environmental disaster with ecological, economical, and public health-related impacts, which is occurring with increasing frequency and severity worldwide. One of the latest mucilage occurrences, the mucilage outbreak of Spring 2021 in the Sea of Marmara, lasted for well over three months and covered more than area of 1160 $k{m^2}$ k m 2 , and was repeated in 2022, although to a smaller extent in terms of duration, coverage area, and severity. Remotely sensed images acquired by Earth observation sensors provide a convenient and promising alternative for environmental monitoring of marine mucilage, with respect to field and laboratory work which are costly, require expert knowledge, and which have to be conducted in a small time-window and have limited generalizability due to the under-sampling of a large medium with a highly dynamic character. This work presents an unmixing-based approach using nonnegative matrix factorization for marine mucilage monitoring from remotely sensed hyperspectral data. The proposed method enables to identify the spectral signature of mucilage, and its variants where applicable, and the precise detection and estimation of mucilage concentrations in affected regions. Additionally, the proposed approach facilitates monitoring the temporal variations of mucilage presence, distribution, and accumulation, between data acquired from the same region at different dates or times. This enables quantitative spatio-temporal mapping of distributions and the potential of assessing the severity of outbreaks, without the requirement of manually prepared ground truth maps or carefully selected thresholds. Real hyperspectral data acquired by the PRISMA mission are used to validate the proposed approach qualitatively and quantitatively. The paper also discusses future directions and recommendations to enhance the performance and applicability of the proposed approach for marine mucilage analysis, and environmental monitoring in general. [ABSTRACT FROM AUTHOR]

A list Λ = { λ 1 , ... , λ n } of complex numbers is said to be realizable if there exists an n × n nonnegative matrix A whose spectrum is Λ. In this case A is called a realizing matrix for Λ. The problem of characterizing all realizable lists Λ is known as the Nonnegative Inverse Eigenvalue Problem (NIEP). If A is required to be persymmetric , the Persymmetric Nonnegative Inverse Eigenvalue Problem (PNIEP) arises. In this paper, we show that the NIEP and the PNIEP are equivalent for any realizable list of three complex numbers. Furthermore, we give a new sufficient condition for a list of three complex numbers to be the spectrum of a persymmetric nonnegative matrix with prescribed diagonal entries. For some realizable lists of four complex numbers, we show that the NIEP and the PNIEP are equivalent. In general, we show that these problems are different. Finally, we found a sufficient condition for a trace zero list of five complex numbers to be the spectrum of a persymmetric nonnegative matrix. [ABSTRACT FROM AUTHOR]

During robot-assisted rehabilitation, failure to recognize lower limb movement may efficiently limit the development of exoskeleton robots, especially for individuals with knee pathology. A major challenge encountered with surface electromyography (sEMG) signals generated by lower limb movements is variability between subjects, such as motion patterns and muscle structure. To this end, this paper proposes an sEMG-based lower limb motion recognition using an improved support vector machine (SVM). Firstly, non-negative matrix factorization (NMF) is leveraged to analyze muscle synergy for multi-channel sEMG signals. Secondly, the multi-nonlinear sEMG features are extracted, which reflect the complexity of muscle status change during various lower limb movements. The Fisher discriminant function method is utilized to perform feature selection and reduce feature dimension. Then, a hybrid genetic algorithm-particle swarm optimization (GA-PSO) method is leveraged to determine the best parameters for SVM. Finally, the experiments are carried out to distinguish 11 healthy and 11 knee pathological subjects by performing three different lower limb movements. Results demonstrate the effectiveness and feasibility of the proposed approach in three different lower limb movements with an average accuracy of 96.03% in healthy subjects and 93.65% in knee pathological subjects, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if A T = B for a nonnegative integer matrix T , then we say A ≤ B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k -Catalan arrays, k -Schröder arrays, and k -Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur. [ABSTRACT FROM AUTHOR]

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops A G S will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix A G S − σ n I , where σ denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph G S is explored. In addition, the existence of a graph with self-loops for every odd energy is proved. [ABSTRACT FROM AUTHOR]

The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

This paper introduces a new lower-limb rehabilitation machine that meets the rehabilitation needs of hemiplegic patients. First, a left–right independent rotary pedal mechanism was selected to facilitate rehabilitation and adapt to the user's physical condition. Then, a half model of the lower-limb rehabilitation machine is designed and manufactured with ergonomics in mind. As analytical tools, we combine non-negative matrix factorization and non-negative double singular value decomposition to calculate muscle synergy of the walking muscle surface electromyography (sEMG) signal, and use cosine similarity to evaluate the similarity between walking and pedaling activities. By comparing the results of the walking and pedaling experiments, the effectiveness of pedaling in gait rehabilitation is revealed. To further improve the similarity between walking and pedaling, double integration of the sEMG signal is introduced, and the relationship between load input and rotation angle is described for the first time using Fourier series. The results of the experiment confirmed that more than half of the 10 subjects performed pedaling exercises similar to walking using Fourier series loading compared to pedaling exercises with normal constant loading. This loading parameter may have the potential to improve rehabilitation efficiency for many subjects compared to the usual exercise. [ABSTRACT FROM AUTHOR]

Recently, a spectral Favard theorem was presented for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The paper establishes conditions, expressed in terms of continued fractions, under which an oscillatory tetradiagonal Hessenberg matrix can have such a positive bidiagonal factorization. Oscillatory tetradiagonal Toeplitz matrices are examined as a case study of matrices that admit a positive bidiagonal factorization. Furthermore, the paper proves that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a positive bidiagonal factorization, but all the interior points of the ray do have such a positive bidiagonal factorization. [ABSTRACT FROM AUTHOR]

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B (H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B (H) and the finite rank operators F (H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A ⁎ = X A Y which we believe is a new connection relating the semigroup theory with the theory of operator equations. We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators. In this paper, we focus on a separate M n (C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A ⁎ = X A Y in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries. [ABSTRACT FROM AUTHOR]

Community structure over social media (SM) is the collaborative group of globally spread users with identical characteristics and ideologies. The collective features of SM are inherent with both the implicit and explicit nature of end-users. This paper presents an analytical and methodological community detection framework to bind passive users' implicit and explicit nature after scrutinizing graphical data to identify seed nodes and communities. Moreover, this work provides the concept of the unsupervised machine learning approach over the graphical perspective of SM to identify the trade-off between similarity of nodes attributes and density of connections for social theories. Subsequently, this paper evaluates a comprehensive analysis of the benchmark community detection algorithm (CDA) Label Propagation Algorithm (LPA), Clique Percolation Method (CPM), Democratic Estimate of the Modular Organization of a Network (DEMON) and Non-Negative Matrix Factorization (NMF). The evaluation has been carried out over modularity and normalized mutual information of resultant structured community on six real-time SM data set. The performance of benchmark CDAs is significantly increased after incorporating social theories. NMF, DEMON, CPM and LPA gained the highest improvement over Zachary's Karate Club data sets, i.e. approximate 26.91%, 21.68%, 18.79%, 19.96%, respectively. [ABSTRACT FROM AUTHOR]

The multispectral (MS) and the panchromatic (PAN) images represent complementary and synergistic spatial spectral information, how to make optimal use of the advantages of them has become a hot research topic. This paper proposes a selectable Transformer and Gist CNN network (STGC-Net). It designs a subspace similar recombination module (SSR-Module) based on non-negative matrix factorization (NMF) and the self-attention mechanism for feature decomposition. This can alleviate the redundant information of multi-modal data and extract their own singular and common features. Considering that the MS and the PAN images exhibit different advantageous properties, a selectable self-attention spectral feature extraction module (S 3 FE-Module) and a multi-stream Gist spatial feature extraction module (MGSFE-Module) are proposed for the different singular features. The former can refine the Transformer's input and simultaneously characterize the sequence information between channels for the MS image. The latter introduces the positional relationship between local features while extracting spatial features for the PAN image, thereby improving the accuracy of scene classification. Experimental results indicate that the proposed method performs better than the other methods. The relevant code of this paper is provided at: https://github.com/ru-willow/ST-GC-Net. • A subspace similar recombination module is proposed. • The selectable self-attention spectral feature extraction module is designed. • The multi-stream Gist spatial feature extraction module is introduced. [ABSTRACT FROM AUTHOR]

Neuroblastoma is the most common extracranial solid tumor in children. It is a highly heterogeneous tumor consisting of different subcellular types and genetic abnormalities. Literature data confirm the biological and clinical complexity of this cancer, which requires a wider availability of gene targets for the implementation of personalized therapy. This paper presents a study of neuroblastoma samples from primary tumors of untreated patients. The focus of this analysis is to evaluate the impact that the inflammatory process may have on the pathogenesis of neuroblastoma. Eighty-eight gene profiles were selected and analyzed using a non-negative matrix factorization framework to extract a subset of genes relevant to the identification of an inflammatory phenotype, whose targets (PIK3CG, NFATC2, PIK3R2, VAV1, RAC2, COL6A2, COL6A3, COL12A1, COL14A1, ITGAL, ITGB7, FOS, PTGS2, PTPRC, ITPR3) allow further investigation. Based on the genetic signals automatically derived from the data used, neuroblastoma could be classified according to stage rather than as a "cold" or "poorly immunogenic" tumor. [ABSTRACT FROM AUTHOR]

This paper proposes a dual-channel speech separation method based on spatial estimation via sparse Bayesian inference (SBI) and nonnegative matrix factorization (NMF). The spatial information estimated by traditional compressed sensing (CS) models is insufficient when two microphones receive limited columns of mixed signals. Considering the sparsity of peak values in the cross-correlation spectrum between two received signals, the proposed method builds a new CS model based on cross-correlation spectrum and applies SBI algorithm to solve this model to improve the estimation accuracy of spatial information. Combined the spatial information with the spectral features decomposed by NMF, NMF coefficient matrix masks belonging to individual source are generated for pre-separation. To mitigate retained potential interference components, a post-separation processing stage is designed using an expectation maximization (EM) algorithm based on a Gaussian mixture model (GMM). The estimated spatial information and binary time–frequency masks are used for parameter initialization of the EM algorithm. The experimental results using real-world speech data show that the proposed method can achieve better separation performance compared to various existing methods. [ABSTRACT FROM AUTHOR]

The A α -matrix of a digraph G is defined as A α (G) = α D + (G) + (1 − α) A (G) , where α ∈ [ 0 , 1) , D + (G) is the diagonal outdegree matrix and A (G) is the adjacency matrix. The k -th A α spectral moment of G is defined as ∑ i = 1 n λ α i k , where λ α i are the eigenvalues of the A α -matrix of G , and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second A α spectral moment (also known as the A α energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third A α spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs. [ABSTRACT FROM AUTHOR]

Hyperspectral imaging captures detailed spectral data for remote sensing. However, due to the limited spatial resolution of hyperspectral sensors, each pixel of a hyperspectral image (HSI) may contain information from multiple materials. Although the hyperspectral unmixing (HU) process involves estimating endmembers, identifying pure spectral components, and estimating pixel abundances, existing algorithms mostly focus on just one or two tasks. Blind source separation (BSS) based on nonnegative matrix factorization (NMF) algorithms identify endmembers and their abundances at each pixel of HSI simultaneously. Although they perform well, the factorization results are unstable, require high computational costs, and are difficult to interpret from the original HSI. CUR matrix decomposition selects specific columns and rows from a dataset to represent it as a product of three small submatrices, resulting in interpretable low-rank factorization. In this paper, we propose a new blind HU framework based on CUR factorization called CUR-HU that performs the entire HU process by exploiting the low-rank structure of given HSIs. CUR-HU incorporates several techniques to perform the HU process with a performance comparable to state-of-the-art methods but with higher computational efficiency. We adopt a deterministic sampling method to select the most informative pixels and spectrum components in HSIs. We use an incremental QR decomposition method to reduce computation complexity and estimate the number of endmembers. Various experiments on synthetic and real HSIs are conducted to evaluate the performance of CUR-HU. CUR-HU performs comparably to state-of-the-art methods for estimating the number of endmembers and abundance maps, but it outperforms other methods for estimating the endmembers and the computational efficiency. It has a 9.4 to 249.5 times speedup over different methods for different real HSIs. [ABSTRACT FROM AUTHOR]

Nowadays, deep representations have gained significant attention due to their outstanding performance in a wide range of tasks. However, the interpretability of deep representations in specific applications poses a significant challenge. For instances where the generated quantity matrices exhibit symmetry, this paper introduces a variant of deep matrix factorization (deep MF) called deep L p smooth symmetric non-negative matrix factorization (DSSNMF), which aims to improve the extraction of clustering structures inherent in complex hierarchical and graphical representations in high-dimensional datasets by improving the sparsity of the factor matrices. We successfully applied DSSNMF to synthetic datasets as well as datasets related to post-traumatic stress disorder (PTSD) to extract several hierarchical communities. Specifically, we identified non-disjoint communities within the partial correlation networks of PTSD psychiatric symptoms, resulting in highly meaningful clinical interpretations. Numerical experiments demonstrate the promising applications of DSSNMF in fields like network analysis and medicine. [ABSTRACT FROM AUTHOR]

The probabilistic topic model imposes a low-rank structure on the expectation of the corpus matrix. Therefore, singular value decomposition (SVD) is a natural tool of dimension reduction. We propose an SVD-based method for estimating a topic model. Our method constructs an estimate of the topic matrix from only a few leading singular vectors of the data matrix, and has a great advantage in memory use and computational cost for large-scale corpora. The core ideas behind our method include a pre-SVD normalization to tackle severe word frequency heterogeneity, a post-SVD normalization to create a low-dimensional word embedding that manifests a simplex geometry, and a post-SVD procedure to construct an estimate of the topic matrix directly from the embedded word cloud. We provide the explicit rate of convergence of our method. We show that our method attains the optimal rate in the case of long and moderately long documents, and it improves the rates of existing methods in the case of short documents. The key of our analysis is a sharp row-wise large-deviation bound for empirical singular vectors, which is technically demanding to derive and potentially useful for other problems. We apply our method to a corpus of Associated Press news articles and a corpus of abstracts of statistical papers. for this article are available online. [ABSTRACT FROM AUTHOR]

Community structure is a significant characteristic of complex networks, and community detection has valuable applications in network structure analysis. Non-negative matrix factorization (NMF) is a key set of algorithms used to solve the community detection issue. Nevertheless, the localization of feature vectors in the adjacency matrix, which represents the characteristics of complex network structures, frequently leads to the failure of NMF-based approaches when the data matrix has a low density. This paper presents a novel algorithm for detecting sparse network communities using non-negative matrix factorization (NMF). The algorithm utilizes local feature vectors to represent the original network topological features and learns regularization matrices. The resulting feature matrices effectively reveal the global structure of the data matrix, demonstrating enhanced feature expression capabilities. The regularized data matrix resolves the issue of localized feature vectors caused by sparsity or noise, in contrast to the adjacency matrix. The approach has superior accuracy in detecting community structures compared to standard NMF-based community detection algorithms, as evidenced by experimental findings on both simulated and real-world networks. [ABSTRACT FROM AUTHOR]

This paper considers the proportional–integral–derivative (PID) control for continuous-time positive systems. A three-stage strategy is introduced to design the PID controller. In the first stage, the proportional and integral components of the PID control are designed. A matrix decomposition approach is used to describe the gain matrices of the proportional and integral components. The positivity and stability of the closed-loop systems without the derivative component of PID control are achieved by the properties of a Metzler and Hurwitz matrix. In the second stage, a non-negative inverse matrix is constructed to maintain the Metzler and Hurwitz properties of the closed-loop system matrix in the first stage. To deal with the inverse of the derivative component of PID control, a matrix decomposition approach is further utilized to design a non-negative inverse matrix. Then, the derivative component is obtained by virtue of the designed inverse matrix. All the presented conditions can be solved by virtue of a linear programming approach. Furthermore, the three-stage PID design is developed for a state observer-based PID controller. Finally, a simulation example is provided to verify the effectiveness and validity of the proposed design. [ABSTRACT FROM AUTHOR]

Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called the discrete maximum principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convectiondominated regime. In fact, in this case it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with the main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time both satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated scenario. [ABSTRACT FROM AUTHOR]

In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-step Noda iteration for computing the Perron root and Perron vector of an irreducible nonnegative matrix by successively solving two linear systems with the same coefficient matrix. For every positive initial vector, the two-step Noda iteration always converges and has a cubic asymptotic convergence rate. As an application, the two-step Noda iteration is applied to compute the smallest eigenpair of irreducible nonsingular M-matrices. Numerical examples are provided for testing the proposed method. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. The aim is to find a low rank approximation for nonnegative matrix M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under a separability assumption, which requires that the columns of the factor matrix are some columns of the input matrix M. In this paper, we generalize the separability assumption based on 3-factor NMF (M = P1SP2), and require that S is a submatrix of the input matrix M. We refer to this NMF as a Coseparable NMF (CoS-NMF). In the paper, we discuss and study mathematical properties of CoS-NMF, and present its relationships with other matrix factorizations such as generalized separable NMF, tri-symNMF, biorthogonal trifactorization and CUR decomposition. An optimization method for CoS-NMF is proposed, and an alternating fast gradient method is employed to determine the rows and the columns of M for the submatrix S. Numerical experiments on synthetic data sets, document data sets, and facial data sets are conducted to verify the effectiveness of the proposed CoS-NMF model. By comparison with state-of-the-art methods, the CoS-NMF model performs very well in a coclustering task by finding useful features, and keeps a good approximation to the input data matrix as well. [ABSTRACT FROM AUTHOR]

• Reviewed studies using finite mixture modeling in transportation and related domains. • Identified six subdomains in transportation with the aid of nonnegative matrix factorization. • Presented a typology of eight types of heterogeneity. • Examined how the transportation literature has treated each of the key elements of the mixture modeling framework. • Discussed several methodological avenues and raised some issues that are important to mixture modeling applications. Accounting for some types of heterogeneity has been an important pathway to improving our models in the transportation domain, specifically in travel behavior research. This study examines the finite mixture modeling (latent class modeling) framework, which has been an appealing approach to that end. Through a comprehensive and systematic review, the paper aims to provide a broader understanding of the usage landscape and also insights into detailed elements. We firstly set up the mixture modeling framework; outline an arena of various relevant research fields; and explain how it is connected to transportation analyses. Then, by using the Scopus database, we explore relevant papers to investigate macroscopic trends in usage of the methodology (yearly trends and research topics). We identify six subdomains in transportation with the aid of nonnegative matrix factorization. We examine several components of the mixture modeling framework in detail. Each subsection covers certain elements of the framework and thus illuminates the landscape of usage and related issues: eight types of heterogeneity; two modeling approaches (exploratory and confirmatory); types of problems (supervised and unsupervised learning); membership model; outcome model; selecting the number of classes; comparisons with competing models; and software and estimation. At the end, we present a few current frontiers and potential directions for future research, and offer further discussion on several issues that arise in the context of mixture models. [ABSTRACT FROM AUTHOR]

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω -monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C 2 , δ functions for some δ > 0 , namely that f is ω -monotone for some Hölder modulus of continuity ω. Counterexamples were provided for any larger modulus of continuity. In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M = [ a k j ] k , j = 1 n can be written as a finite sum of squares of C 2 , δ vector fields if the diagonal entries a k k are ω -monotone for some Hölder modulus of continuity ω , and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed. Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series. [ABSTRACT FROM AUTHOR]

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available. [ABSTRACT FROM AUTHOR]

Community detection plays an important role in analyzing attributed networks. It attempts to find the optimal cluster structures to identify valuable information. Although deep nonnegative matrix factorization (DNMF) is widely used in community detection, it cannot be used to analyze attributed networks since only topology information is considered. Recent researches have taken attribute information into account, but we still need to face the following challenges. First, it is difficult to deal with topology noise and attribute noise in attributed networks at one stroke. Second, we need to balance the coupling between topology and node attributes with hyperparameters in most methods. However, with inappropriate hyperparameters, it is easy to cause interference and compromise between them. For the above challenges, in this paper, we propose a novel method, namely adaptive deep nonnegative matrix factorization. Specifically, we handle the inherent noise of attributed networks via dual-DNMF with autoencoder. And then, we use the attention mechanism to adaptively integrate topology information and attribute information without adjusting hyperparameters manually. Overall, our method not only handles the inherent noise in attributed networks, but also resolves the interference and compromise between topology and attributes in a generalized way. The results of comprehensive experiments support our conclusions and demonstrate that our method outperforms the state-of-the-art methods in most datasets. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to demonstrate common fixed point theorems in intuitionistic Mengerspace. In our main result we use the notion of CLR property for type (K) Compatible mappings. The importance of CLR property is that subspaces need not be closed for the survival of fixed points. Many authors utilize the concept of CLR property to prove theorems on fixed point in intuitionistic Mengerspaces for various type of compatible mappings.The results thus obtained, generalizes and extends some known results in intuitionistic menger space. [ABSTRACT FROM AUTHOR]

Surface-enhanced Raman spectroscopy (SERS) has wide diagnostic applications due to narrow spectral features that allow multiplex analysis. We have previously developed a multiplexed, SERS-based nanosensor for micro-RNA (miRNA) detection called the inverse molecular sentinel (iMS). Machine learning (ML) algorithms have been increasingly adopted for spectral analysis due to their ability to discover underlying patterns and relationships within large and complex data sets. However, the high dimensionality of SERS data poses a challenge for traditional ML techniques, which can be prone to overfitting and poor generalization. Non-negative matrix factorization (NMF) reduces the dimensionality of SERS data while preserving information content. In this paper, we compared the performance of ML methods including convolutional neural network (CNN), support vector regression, and extreme gradient boosting combined with and without NMF for spectral unmixing of four-way multiplexed SERS spectra from iMS assays used for miRNA detection. CNN achieved high accuracy in spectral unmixing. Incorporating NMF before CNN drastically decreased memory and training demands without sacrificing model performance on SERS spectral unmixing. Additionally, models were interpreted using gradient class activation maps and partial dependency plots to understand predictions. These models were used to analyze clinical SERS data from single-plexed iMS in RNA extracted from 17 endoscopic tissue biopsies. CNN and CNN-NMF, trained on multiplexed data, performed most accurately with RMSElabel = 0.101 and 9.68 × 10–2, respectively. We demonstrated that CNN-based ML shows great promise in spectral unmixing of multiplexed SERS spectra, and the effect of dimensionality reduction on performance and training speed. Graphical abstract This is a visual representation of the abstract. [ABSTRACT FROM AUTHOR]

NONNEGATIVE matrix factorization (NMF) is an effective technique for dimensionality reduction of high-dimensional data for tasks such as machine learning and data visualization. However, for practical clustering tasks, traditional NMF ignores the manifold information of both the data space and feature space, as well as the discriminative information of the data. In this paper, we propose a semisupervised NMF called dual-graph-regularization-constrained nonnegative matrix factorization with label discrimination (DCNMFLD). DCNMFLD combines dual graph regularization and prior label information as additional constraints, making full use of the intrinsic geometric and discriminative structures of the data, and can efficiently enhance the discriminative and exclusionary nature of clustering and improve the clustering performance. The evaluation of the clustering experimental results on four benchmark datasets demonstrates the effectiveness of our new algorithm. [ABSTRACT FROM AUTHOR]

Large-scale and high-dimensional time series data are widely generated in modern applications such as intelligent transportation and environmental monitoring. However, such data contains much noise, outliers, and missing values due to interference during measurement or transmission. Directly forecasting such types of data (i.e., anomalous data) can be extremely challenging. The traditional method to deal with anomalies is to cut out the time series with anomalous value entries or replace the data. Both methods may lose important knowledge from the original data. In this paper, we propose a multidimensional time series forecasting framework that can better handle anomalous values: the robust temporal nonnegative matrix factorization forecasting model (RTNMFFM) for multi-dimensional time series. RTNMFFM integrates the autoregressive regularizer into nonnegative matrix factorization (NMF) with the application of the L 2 , 1 norm in NMF. This approach improves robustness and alleviates overfitting compared to standard methods. In addition, to improve the accuracy of model forecasts on severely missing data, we propose a periodic smoothing penalty that keeps the sparse time slices as close as possible to the time slice with high confidence. Finally, we train the model using the alternating gradient descent algorithm. Numerous experiments demonstrate that RTNMFFM provides better robustness and better prediction accuracy. [ABSTRACT FROM AUTHOR]

We consider the dynamics of a collection of n > 1 populations in which each population has its own rate of growth or decay, fixed in continuous time, and migrants may flow from one population to another over a fixed network, at a rate, fixed over time, times the size of the sending population. This model is represented by an ordinary linear differential equation of dimension n with constant coefficients arrayed in an essentially nonnegative matrix. This paper identifies conditions on the parameters of the model (specifically, conditions on the eigenvalues and eigenvectors) under which the variance of the n population sizes at a given time is asymptotically (as time increases) proportional to a power of the mean of the population sizes at that given time. A power-law variance function is known in ecology as Taylor's Law and in physics as fluctuation scaling. Among other results, we show that Taylor's Law holds asymptotically, with variance asymptotically proportional to the mean squared, on an open dense subset of the class of models considered here. [ABSTRACT FROM AUTHOR]