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fIn the domain of recommendation systems, matrix decomposition is an effective strategy for mitigating issues related to sparsity and low space utilization. The Alternating Least Squares (ALS) method, in particular, stands out for its ability to process data in parallel, thereby enhancing computational efficiency. However, when dealing with an original rating matrix, the ALS method may inadvertently sacrifice some information, leading to increased error rates. To address these challenges, this paper proposes a novel hybrid model that integrates matrix factorization with additional features. Furthermore, it leverages weighted similarity measures and employs advanced log-likelihood text mining techniques. These innovations are designed to tackle cold-start problems and sparsity issues while compensating for information loss to mitigate errors. Under the premise that our model employs consistent evaluation metrics and datasets, comparative analysis against existing models from related literature demonstrates superior performance. Specifically, our model achieves a lower Root Mean Square Error (RMSE) of 0.82 and 0.88, alongside a higher F1 score of 0.94 and 0.92 in two datasets. Our proposed hybrid approach effectively addresses sparsity and mitigates information loss in matrix factorization, as demonstrated by these results.

Tensor wheel (TW) decomposition combines the popular tensor ring and fully connected tensor network decompositions and has achieved excellent performance in tensor completion problem. A standard method to compute this decomposition is the alternating least squares (ALS). However, it usually suffers from slow convergence and numerical instability. In this work, the fast and robust SVD-based algorithms are investigated. Based on a result on TW-ranks, we first propose a deterministic algorithm that can estimate the TW decomposition of the target tensor under a controllable accuracy. Then, the randomized versions of this algorithm are presented, which can be divided into two categories and allow various types of sketching. Numerical results on synthetic and real data show that our algorithms have much better performance than the ALS-based method and are also quite robust. In addition, with one SVD-based algorithm, we also numerically explore the variability of TW decomposition with respect to TW-ranks and the comparisons between TW decomposition and other famous formats in terms of the performance on approximation and compression. [ABSTRACT FROM AUTHOR]

This study examines the psychometric properties of a screening protocol for dyslexia and demonstrates a special form of matrix factorization called Nous based on the Alternating Least Squares algorithm. Dyslexia presents an intrinsically multidimensional complex of cognitive loads. By building and enforcing a common 6-dimensional space, Nous extracts a multidimensional signal for each person and item from test data that increases the Shannon entropy of the dataset while at the same time being constrained to meet the special objectivity requirements of the Rasch model. The resulting Dyslexia Risk Scale (DRS) yields linear equal-interval measures that are comparable regardless of the subset of items taken by the examinee. Each measure and cell estimate is accompanied by an efficiently calculated standard error. By incorporating examinee age into the calibration process, the DRS can be generalized to all age groups to allow the tracking of individual dyslexia risk over time. The methodology was implemented using a 2019 calibration sample of 828 persons aged 7 to 82 with varying degrees of dyslexia risk. The analysis yielded high reliability (0.95) and excellent receiver operating characteristics (AUC = 0.96). The analysis is accompanied by a discussion of the information-theoretic properties of matrix factorization. [ABSTRACT FROM AUTHOR]

In this study, the problem of blind signal separation for coprime planar arrays is investigated. For coprime planar arrays comprising two uniform rectangular subarrays, we link the signal separation to the tensor-based model called coupled canonical polyadic decomposition (CPD) and propose an improved coupled trilinear decomposition approach. The output data of coprime planar arrays are modeled as a coupled tensor set that can be further interpreted as a coupled CPD model, allowing a signal separation to be achieved using coupled trilinear alternating least squares (TALS). Furthermore, in the procedure of the coupled TALS, a Vandermonde structure enforcing approach is explicitly applied, which is shown to ensure fast convergence. The results of Monto Carlo simulations show that our proposed algorithm has the same separation accuracy as the basic coupled TALS but with a faster convergence speed.

In this study, the problem of blind signal separation for coprime planar arrays is investigated. For coprime planar arrays comprising two uniform rectangular subarrays, we link the signal separation to the tensor‐based model called coupled canonical polyadic decomposition (CPD) and propose an improved coupled trilinear decomposition approach. The output data of coprime planar arrays are modeled as a coupled tensor set that can be further interpreted as a coupled CPD model, allowing a signal separation to be achieved using coupled trilinear alternating least squares (TALS). Furthermore, in the procedure of the coupled TALS, a Vandermonde structure enforcing approach is explicitly applied, which is shown to ensure fast convergence. The results of Monto Carlo simulations show that our proposed algorithm has the same separation accuracy as the basic coupled TALS but with a faster convergence speed. [ABSTRACT FROM AUTHOR]

In recent years, Raman spectroscopy has been used to study biological tissues. However, the analysis of experimental Raman spectra is still challenging, since the Raman spectra of most biological tissue components overlap significantly and it is difficult to separate individual components. New methods of analysis are needed that would allow for the decomposition of Raman spectra into components and the evaluation of their contribution. The aim of our work is to study the possibilities of the multivariate curve resolution alternating least squares (MCR-ALS) method for the analysis of skin tissues in vivo. We investigated the Raman spectra of human skin recorded using a portable conventional Raman spectroscopy setup. The MCR-ALS analysis was performed for the Raman spectra of normal skin, keratosis, basal cell carcinoma, malignant melanoma, and pigmented nevus. We obtained spectral profiles corresponding to the contribution of the optical system and skin components: melanin, proteins, lipids, water, etc. The obtained results show that the multivariate curve resolution alternating least squares analysis can provide new information on the biochemical profiles of skin tissues. Such information may be used in medical diagnostics to analyze Raman spectra with a low signal-to-noise ratio, as well as in various fields of science and industry for preprocessing Raman spectra to remove parasitic components. [ABSTRACT FROM AUTHOR]

Tensor models have been used extensively in signal processing applications to design different types of communication systems. This paper proposes, for the first time, the use of tensor models for optical communications. The signals of an optical dual-core coupler network are modeled as a multiway array (tensor), which satisfies a generalized Tucker train decomposition. This tensor model is then used to develop an estimation algorithm that enables the network parameters to be estimated from the input and output signals. The performance of this algorithm was evaluated by means of computer simulations, in terms of NMSE of the estimated parameters and convergence speed. For the tested configurations, good levels of NMSE with fast convergence were obtained, demonstrating the effectiveness of the proposed method as a promising tool for studying and designing optical devices, with a wide range of applications in the context of lightwave systems.

This paper describes an effcient reverse-mode differentiation algorithm for contraction operations for arbitrary and unconventional tensor network topologies. The approach leverages the tensor contraction tree of Evenbly and Pfeifer (2014), which provides an instruction set for the contraction sequence of a network. We show that this tree can be effciently leveraged for differentiation of a full tensor network contraction using a recursive scheme that exploits (1) the bilinear property of contraction and (2) the property that trees have a single path from root to leaves. While differentiation of tensor-tensor contraction is already possible in most automatic differentiation packages, we show that exploiting these two additional properties in the specific context of contraction sequences can improve effciency. Following a description of the algorithm and computational complexity analysis, we investigate its utility for gradient-based supervised learning for low-rank function recovery and for fitting real-world unstructured datasets. We demonstrate improved performance over alternating least-squares optimization approaches and the capability to handle heterogeneous and arbitrary tensor network formats. When compared to alternating minimization algorithms, we find that the gradient-based approach requires a smaller oversampling ratio (number of samples compared to number model parameters) for recovery. This increased effciency extends to fitting unstructured data of varying dimensionality and when employing a variety of tensor network formats. Here, we show improved learning using the hierarchical Tucker method over the tensor-train in high-dimensional settings on a number of benchmark problems. [ABSTRACT FROM AUTHOR]

In recent years, it has been increasingly clear that long non-coding RNAs (lncRNAs) are able to regulate their target genes at multi-levels, including transcriptional level, translational level, etc and play key regulatory roles in many important biological processes, such as cell differentiation, chromatin remodeling and more. Inferring potential lncRNA-disease associations is essential to reveal the secrets behind diseases, develop novel drugs, and optimize personalized treatments. However, biological experiments to validate lncRNA-disease associations are very time-consuming and costly. Thus, it is critical to develop effective computational models. In this study, we have proposed a method by alternating least squares based on matrix factorization to predict lncRNA-disease associations, referred to as ALSBMF. ALSBMF first decomposes the known lncRNA-disease correlation matrix into two characteristic matrices, then defines the optimization function using disease semantic similarity, lncRNA functional similarity and known lncRNA-disease associations and solves two optimal feature matrices by least squares method. The two optimal feature matrices are finally multiplied to reconstruct the scoring matrix, filling the missing values of the original matrix to predict lncRNA-disease associations. Compared to existing methods, ALSBMF has the same advantages as BPLLDA. It does not require negative samples and can predict associations related to novel lncRNAs or novel diseases. In addition, this study performs leave-one-out cross-validation (LOOCV) and five-fold cross-validation to evaluate the prediction performance of ALSBMF. The AUCs are 0.9501 and 0.9215, respectively, which are better than the existing methods. Furthermore colon cancer, kidney cancer, and liver cancer are selected as case studies. The predicted top three colon cancer, kidney cancer, and liver cancer-related lncRNAs were validated in the latest LncRNADisease database and related literature. In order to test the ability of ALSBMF to predict novel disease-associated lncRNAs and new lncRNA-associated diseases, all known associations of diseases and lncRNAs were eliminated, the predicted top five breast cancer, nasopharyngeal carcinoma cancer-related lncRNAs and top five H19, MALAT1 lncRNA-related cancers were validated in PubMed and dbSNP.

Alternating least squares (ALS) and its variations are the most commonly used algorithms for the PARAFAC decomposition of a tensor. However, it is still troubled for one how to accelerate the ALS algorithm with the reduced computational complexity. In this paper, a new acceleration method for the ALS with a matrix polynomial predictive model (MPPM) is proposed. In the MPPM, a matrix-valued function is first approximated by a matrix polynomial. It is shown that the future value of the function can be predicted by an FIR filter with the coefficients determined offline. By viewing each factor matrix of a tensor as a matrix-valued function, a new ALS algorithm, the ALS-MPPM algorithm, is then given. Analyses show that our ALS-MPPM algorithm is of low computational complexity and a close relation with the existing ALS algorithms. Moreover, to further accelerate the convergence of the proposed algorithm, a new technique called the multi-model (MM) prediction is also introduced. While the analytical results are verified by the numerical simulations, it is also shown that our ALS-MPPM outperforms the existing ALS-based algorithms in terms of the rate of convergence.

In recent years, the natural gas has displayed a growing significance in oil and gas exploration in the northwestern Junggar Basin (NWJB), although oil has been the main focus of exploration in the basin. Here, we systematically discuss the classification and origin of the natural gases from the NWJB based on the natural gas geochemistry and chemometric methods. The natural gases collected from the NWJB were chemometrically classified into three groups. Group A gases, defined as coal-derived gases, were likely generated from the mixing of the Jiamuhe Formation and Carboniferous strata. Group B gases, defined as the mixing of coal-derived and oil-associated gases, were restricted to the source rocks of group A and C gases. Group C gases, defined as oil-associated gases, were likely derived from both the Fengcheng and Wuerhe Formations, with a higher contribution from the latter strata. The result of this study suggests that the potential of oil generation in the Wuerhe Formation has been underestimated in the past. This is in accordance with geochemical and geological evidence. This study provides an effective chemometric method of natural gas classification and evaluation of hydrocarbon generation potential. This contributes to a better understanding of the origin of gases and distribution of oil and gas, assisting in exploration deployment in the basin. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of Block Coordinate Descent algorithms in valleys of the optimization landscape. Iterates converge linearly to a vicinity of the valley oor and then progress in a zig-zag fashion along the direction of the valley oor. When the valley sides are symmetric, the contraction factor to a vicinity of the valley oor appears to be no worse than 1/8, but without symmetry the contraction factor can approach 1. Progress along the direction of the valley oor is proportional to the gradient on the valley oor and inversely proportional to the narrowness" of the valley. We quantify narrowness using the eigenvalues of the Hessian on the valley oor and give explicit formulas for certain cases. Progress also depends on the direction of the valley with respect to the blocks of coordinates. When the valley sides are symmetric, we give an explicit formula for this dependence and use it to show that in higher dimensions nearly all directions give progress similar to the worst case direction. Finally, we observe that when starting the algorithm, the ordering of blocks in the first few steps can be important, but show that a greedy strategy with respect to objective function improvement can be a bad choice. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on the spotlight synthetic aperture radar (SAR) imaging for point scattering targets based on tensor modeling. In a real-world scenario, scatterers usually distribute in the block sparse pattern. Such a distribution feature has been scarcely utilized by the previous studies of SAR imaging. Our work takes advantage of this structure property of the target scene, constructing a multi-linear sparse reconstruction algorithm for SAR imaging. The multi-linear block sparsity is introduced into higher-order singular value decomposition (SVD) with a dictionary constructing procedure by this research. The simulation experiments for ideal point targets show the robustness of the proposed algorithm to the noise and sidelobe disturbance which always influence the imaging quality of the conventional methods. The computational resources requirement is further investigated in this paper. As a consequence of the algorithm complexity analysis, the present method possesses the superiority on resource consumption compared with the classic matching pursuit method. The imaging implementations for practical measured data also demonstrate the effectiveness of the algorithm developed in this paper.

Abstract This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank 23 decompositions for the 3 × 3 matrix multiplication tensor M 〈 3 〉. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12. We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of M 〈 n 〉 for larger n. [ABSTRACT FROM AUTHOR]

Tensor decomposition, the higher-order analogue to singular value decomposition, has emerged as a useful tool for finding relationships in large, sparse, multidimensional data. As this technique matures and is applied to increasingly larger data sets, the need for high performance implementations becomes critical. A better understanding of the performance characteristics of tensor decomposition on large and sparse tensors can help drive the development of such implementations. In this work, we perform an objective empirical evaluation of three state of the art parallel tools that implement the Canonical Decomposition/Parallel Factorization tensor decomposition algorithm using alternating least squares fitting (CP-ALS): SPLATT, DFacTo, and ENSIGN. We conduct performance studies across a variety of data sets and evaluate the tools with respect to total memory required, processor stall cycles, execution time, data distribution, and communication patterns. Furthermore, we investigate the performance of the implementations on tensors with up to 6 dimensions and when executing high rank decompositions. We find that tensor data structure layout and distribution choices can result in differences as large as 14.6x with respect to memory usage and 39.17x with respect to execution time. We provide an outline of a distributed heterogeneous CP-ALS implementation that addresses the performance issues we observe. • Performance study of parallel CP-ALS implementations for sparse tensors. • Measure memory usage, processor stall cycles, execution time and scalability. • Discuss MPI data distribution schemes and communication patterns. • Evaluate high rank decompositions and data sets with up to 6 dimensions. • Performance gaps are as large as 14.6x for memory usage and 39.17x for run time. [ABSTRACT FROM AUTHOR]

The Gifi system of analyzing categorical data through nonlinear varieties of classical multivariate analysis techniques is reviewed. The system is characterized by the optimal scaling of categorical variables which is implemented through alternating least squares algorithms. The main technique of homogeneity analysis is presented, along with its extensions and generalizations leading to nonmetric principal components analysis and canonical correlation analysis. A brief account of stability issues and areas of applications of the techniques is also given.

In the era of big data, people have to face information filtration problem. For those cases when users do not or cannot express their demands clearly, recommender system can analyse user’s information more proactive and intelligent to filter out something users want. This property makes recommender system play a very important role in the field of e-commerce, social network and so on. The collaborative filtering recommendation algorithm based on Alternating Least Squares (ALS) is one of common algorithms using matrix factorization technique of recommendation system. In this paper, we design the parallel implementation process of the recommendation algorithm based on Spark platform and the related technology research of recommendation systems. Because of the shortcomings of the recommendation algorithm based on ALS model, a new loss function is designed. Before the model is trained, the similarity information of users and items is fused. The experimental results show that the performance of the proposed algorithm is better than that of algorithm based on ALS.

A new approach to structural equation modeling based on so-called extended redundancy analysis has been recently proposed in the literature, enhanced with the added characteristic of generalizing redundancy analysis and reduced-rank regression models for more than two blocks. In this approach, the relationships between the observed exogenous variables and the observed endogenous variables are moderated by the presence of unobservable composites that were estimated as linear combinations of exogenous variables, permitting a great flexibility to specify and fit a variety of structural relationships. In this paper, we propose the SAS macro %ERA to specify and fit structural relationships in the extended redundancy analysis (ERA) framework. Two examples (simulation and real data) are provided in order to reproduce results appearing in the original article where ERA was proposed.

A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.

Abstract The regularization method could deal with the swamp effect of alternating least squares (ALS) algorithms for tensor decomposition. Usually, the regularization term is a norm of the difference between the solution and the current iterate. In this paper, we show that the norm could be weakened to a seminorm, so the selection of the regularization term could be more flexible. To overcome the swamp effect and avoid the drawback that the Hessian of the subproblem may get close to singular in the iterative process, we propose a seminorm regularized ALS algorithm for solving the canonical tensor decomposition. Moreover, in the new algorithm, we introduce a novel extrapolation in the update of each mode factor which makes an immediate impression on the update of subsequent ones. By assuming the boundness of the infinite sequence of iterates generated by the new algorithm, we establish the global convergence and the (weakly) linear convergence rate of the sequence of iterates Numerical experiments on synthetic and real-world problems illustrate that the new method is efficient and promising. [ABSTRACT FROM AUTHOR]

This paper focuses on the spotlight synthetic aperture radar (SAR) imaging for point scattering targets based on tensor modeling. In a real-world scenario, scatterers usually distribute in the block sparse pattern. Such a distribution feature has been scarcely utilized by the previous studies of SAR imaging. Our work takes advantage of this structure property of the target scene, constructing a multi-linear sparse reconstruction algorithm for SAR imaging. The multi-linear block sparsity is introduced into higher-order singular value decomposition (SVD) with a dictionary constructing procedure by this research. The simulation experiments for ideal point targets show the robustness of the proposed algorithm to the noise and sidelobe disturbance which always influence the imaging quality of the conventional methods. The computational resources requirement is further investigated in this paper. As a consequence of the algorithm complexity analysis, the present method possesses the superiority on resource consumption compared with the classic matching pursuit method. The imaging implementations for practical measured data also demonstrate the effectiveness of the algorithm developed in this paper. [ABSTRACT FROM AUTHOR]

Most of the existing procedures for sparse principal component analysis (PCA) use a penalty function to obtain a sparse matrix of weights by which a data matrix is post-multiplied to produce PC scores. In this paper, we propose a new sparse PCA procedure which differs from the existing ones in two ways. First, the new procedure does not sparsify the weight matrix. Instead, the so-called loadings matrix is sparsified by which the score matrix is post-multiplied to approximate the data matrix. Second, the cardinality of the loading matrix i.e., the total number of nonzero loadings, is pre-specified to be an integer without using penalty functions. The procedure is called unpenalized sparse loading PCA (USLPCA). A desirable property of USLPCA is that the indices for the percentages of explained variances can be defined in the same form as in the standard PCA. We develop an alternate least squares algorithm for USLPCA which uses the fact that the PCA loss function can be decomposed as a sum of a term irrelevant to the loadings, and another one being easily minimized under cardinality constraints. A procedure is also presented for selecting the best cardinality using information criteria. The procedures are assessed in a simulation study and illustrated with real data examples. [ABSTRACT FROM AUTHOR]

The current work investigates the capability of a tailored multivariate curve resolution–alternating least squares (MCR-ALS) algorithm to analyse glucose, phosphate, ammonium and acetate dynamics simultaneously in an E. coli BL21 fed-batch fermentation. The high-cell-density (HCDC) process is monitored by ex situ online attenuated total reflection (ATR) Fourier transform infrared (FTIR) spectroscopy and several in situ online process sensors. This approach efficiently utilises automatically generated process data to reduce the time and cost consuming reference measurement effort for multivariate calibration. To determine metabolite concentrations with accuracies between ±0.19 and ±0.96·gL−l, the presented utilisation needs primarily—besides online sensor measurements—single FTIR measurements for each of the components of interest. The ambiguities in alternating least squares solutions for concentration estimation are reduced by the insertion of analytical process knowledge primarily in the form of elementary carbon mass balances. Thus, in this way, the established idea of mass balance constraints in MCR combines with the consistency check of measured data by carbon balances, as commonly applied in bioprocess engineering. The constraints are calculated based on online process data and theoretical assumptions. This increased calculation effort is able to replace, to a large extent, the need for manually conducted quantitative chemical analysis, leads to good estimations of concentration profiles and a better process understanding.