Multi-way Data Research Articles

Image Completion Using Low Tensor Tree Rank and Total Variation Minimization

Tensor completion recovers missing entries of multiway data. Most of the current methods exploit the low-rank tensor structure for image completion applications. In this paper, we simultaneously exploit the globally multidimensional structure and locally piecewise smoothness to further enhance the performance. In the proposed optimization model, the low tensor tree rank minimization is used for the global data structure, and the total variation minimization is used for the local structure. Two kinds of total variation functions are discussed. The optimization problem is transformed into several subproblems by alternating direction method of multipliers. The subproblem on low tensor tree rank minimization is solved by singular value thresholding, and the subproblem on total variation minimization can be solved by soft thresholding. Numerical experiments on color images and light field images demonstrate that the proposed method outperforms most of the state-of-the-art methods in terms of recovery accuracy and computational complexity.

Multi-way chromatographic calibration—A review

This review discusses recent advances in both theory and experiment regarding multi-way chromatographic calibration. The focus is directed towards the chemometric processing of multi-way data arrays in which one or more instrumental modes describe the elution time behavior across liquid or gas chromatographic columns, and the remaining ones are of spectroscopic nature, e.g. UV–vis absorption, fluorescence emission (either as vectors or matrices) or mass spectrometry. Successful multi-way data decomposition into profiles for the contributing components allows one to perform quantitative analysis of complex samples. Quantitative applications usually make appropriate use of the second-order advantage which is inherent to multi-way data processing, and reduce the complexity of the data to a virtual univariate calibration which is typical of classical analytical chemistry. A number of recent experimental works will be discussed to illustrate the above concepts.

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The informational basis of the completed research was the annual sociological survey of the New Economic Association and the NEA Journal, conducted in the years 2018-2019, in order to clarify the attitude of the economic community to the 27 (twenty seven) currently published Russian journals, included in the data base of RSCI (Russian Science Citations Index) (section Economics. Economic Sciences). As a result of the obtained sociological information, the size of the readership of the analyzed journals was determined and their homogeneous groups were identified with similar characteristics of the readership within each of them and with significant differences between the groups. The ranking of economic journals was presented on the basis of the Rating-2019 criterion, constructed, using the Multiway data analysis, which provides a quantitative determination of the weight function of aggregating of private ratings, as well as the aggregate rating vector of journals, that reflect hidden relationships between their characteristics; by clustering journals, statistically homogeneous groups of journals were identified. The paper presented a comparative analysis of the ranking of journals, based on Rating-2019 and RSCI ratings, censorious remarks (criticisms) were formulated, regarding the criteria Scince Index and Public Expertise, formed by RSCI.

Differentially Expressed Genes Extracted by the Tensor Robust Principal Component Analysis (TRPCA) Method

In the big data era, sequencing technology has produced a large number of biological sequencing data. Different views of the cancer genome data provide sufficient complementary information to explore genetic activity. The identification of differentially expressed genes from multiview cancer gene data is of great importance in cancer diagnosis and treatment. In this paper, we propose a novel method for identifying differentially expressed genes based on tensor robust principal component analysis (TRPCA), which extends the matrix method to the processing of multiway data. To identify differentially expressed genes, the plan is carried out as follows. First, multiview data containing cancer gene expression data from different sources are prepared. Second, the original tensor is decomposed into a sum of a low‐rank tensor and a sparse tensor using TRPCA. Third, the differentially expressed genes are considered to be sparse perturbed signals and then identified based on the sparse tensor. Fourth, the differentially expressed genes are evaluated using Gene Ontology and Gene Cards tools. The validity of the TRPCA method was tested using two sets of multiview data. The experimental results showed that our method is superior to the representative methods in efficiency and accuracy aspects.

Measuring uncertainties: a theoretical approach

When our aim is to draw the possible developments of future events, we are faced with a practical obstacle. Indeed, we cannot have any empirical experience of the future. Have we, therefore, to be inferred that forecasting, exploring future or, better: exploring futures, or anticipating futures have not to be considered activities of a scientific kind? Answer to such a difficult question requires a multidisciplinary approach, where statistical models, methodology of social science and of course statistics and sociology as a whole - are enhanced in their ability to express the change - and sometimes the risk that the change itself implies. A great help in understanding complexity, and trends, comes from a method for multi-way data, based on the joint application of a factorial analysis and regression over time, called dynamic factor analysis (DFA).

Improved Robust Tensor Principal Component Analysis via Low-Rank Core Matrix

Robust principal component analysis (RPCA) has been widely used for many data analysis problems in matrix data. Robust tensor principal component analysis (RTPCA) aims to extract the low rank and sparse components of multidimensional data, which is a generation of RPCA. The current RTPCA methods are directly based on tensor singular value decomposition (t-SVD), which is a new tensor decomposition method similar to singular value decomposition (SVD) in matrices. These methods focus on utilizing different sparse constraints for real applications and make less analysis for tensor nuclear norm (TNN) defined in t-SVD. However, we find low-rank structure still exists in the core tensor and existing methods can not fully extract the low-rank structure of tensor data. To further exploit the low-rank structures in multiway data, we extract low-rank component for the core matrix whose entries are from the diagonal elements of the core tensor. Based on this idea, we have defined a new TNN that extends TNN with core matrix and propose a creative algorithm to deal with RTPCA problems. The results of numerical experiments show that the proposed method outperforms state-of-the-art methods in terms of both accuracy and computational complexity.

Unsupervised learning low-rank tensor from incomplete and grossly corrupted data

Low-rank tensor completion and recovery have received considerable attention in the recent literature. The existing algorithms, however, are prone to suffer a failure when the multiway data are simultaneously contaminated by arbitrary outliers and missing values. In this paper, we study the unsupervised tensor learning problem, in which a low-rank tensor is recovered from an incomplete and grossly corrupted multidimensional array. We introduce a unified framework for this problem by using a simple equation to replace the linear projection operator constraint, and further reformulate it as two convex optimization problems through different approximations of the tensor rank. Two globally convergent algorithms, derived from the alternating direction augmented Lagrangian (ADAL) and linearized proximal ADAL methods, respectively, are proposed for solving these problems. Experimental results on synthetic and real-world data validate the effectiveness and superiority of our methods.

Enhanced Principal Tensor Analysis as a tool for 3-way geological data reconstructions

Principal tensor analysis (PTA) is considered as a potentially useful tool in geosciences, particularly for reconstructions of multi-way (time (depth), space, and proxies) datasets. We introduce an extension of PTA: PTA enhanced with Singular Spectrum Analysis (SSA). Using this enhancement, we were able to isolate clear patterns in noisy multi-way data. As an illustrative example, the method has been applied to 4-way data tensor (time, space, proxies, and delay-time) constructed from marine sediment proxies. Possible restrictions and main reasons for the method's limitations are discussed. The algorithm has been implemented as an R function which is provided in the supplementary data and appendix.

Low rank tensor completion for multiway visual data

Tensor completion recovers missing entries of multiway data. Teh missing of entries could often be caused during teh data acquisition and transformation. In dis paper, we provide an overview of recent development in low rank tensor completion for estimating teh missing components of visual data, e. g. , color images and videos. First, we categorize these methods into two groups based on teh different optimization models. One optimizes factors of tensor decompositions wif predefined tensor rank. Teh other iteratively updates teh estimated tensor via minimizing teh tensor rank. Besides, we summarize teh corresponding algorithms to solve those optimization problems in details. Numerical experiments are given to demonstrate teh performance comparison when different methods are applied to color image and video processing.

A test of weak separability for multi-way functional data, with application to brain connectivity studies

This paper concerns the modeling of multi-way functional data where double or multiple indices are involved. We introduce a concept of weak separability. The weakly separable structure supports the use of factorization methods that decompose the signal into its spatial and temporal components. The analysis reveals interesting connections to the usual strongly separable covariance structure, and provides insights into tensor methods for multi-way functional data. We propose a formal test for the weak separability hypothesis, where the asymptotic null distribution of the test statistic is a chi-square type mixture. The method is applied to study brain functional connectivity derived from source localized magnetoencephalography signals during motor tasks.

Learning with Similarity Functions: a Tensor-Based Framework

Machine learning algorithms are typically designed to deal with data represented as vectors. Several major applications, however, involve multi-way data, such as video sequences and multi-sensory arrays. In those cases, tensors endow a more consistent way to capture multi-modal relations, which may be lost by a conventional remapping of original data into a vector representation. This paper presents a tensor-oriented machine learning framework, and shows that the theory of learning with similarity functions provides an effective paradigm to support this framework. The proposed approach adopts a specific similarity function, which defines a measure of similarity between a pair of tensors. The performance of the tensor-based framework is evaluated on a set of complex, real-world, pattern-recognition problems. Experimental results confirm the effectiveness of the framework, which compares favorably with state-of-the-art machine learning methodologies that can accept tensors as inputs. Indeed, a formal analysis proves that the framework is more efficient than state-of-the-art methodologies also in terms of computational cost. The paper thus provides two main outcomes: (1) a theoretical framework that enables the use of tensor-oriented similarity notions and (2) a cognitively inspired notion of similarity that leads to computationally efficient predictors.

A multi-way data analysis approach for structural health monitoring of a cable-stayed bridge

A large-scale cable-stayed bridge in the state of New South Wales, Australia, has been extensively instrumented with an array of accelerometer, strain gauge, and environmental sensors. The real-time continuous response of the bridge has been collected since July 2016. This study aims at condition assessment of this bridge by investigating three aspects of structural health monitoring including damage detection, damage localization, and damage severity assessment. A novel data analysis algorithm based on incremental multi-way data analysis is proposed to analyze the dynamic response of the bridge. This method applies incremental tensor analysis for data fusion and feature extraction, and further uses one-class support vector machine on this feature to detect anomalies. A total of 15 different damage scenarios were investigated; damage was physically simulated by locating stationary vehicles with different masses at various locations along the span of the bridge to change the condition of the bridge. The effect of damage on the fundamental frequency of the bridge was investigated and a maximum change of 4.4% between the intact and damage states was observed which corresponds to a small severity damage. Our extensive investigations illustrate that the proposed technique can provide reliable characterization of damage in this cable-stayed bridge in terms of detection, localization and assessment. The contribution of the work is threefold; first, an extensive structural health monitoring system was deployed on a cable-stayed bridge in operation; second, an incremental tensor analysis was proposed to analyze time series responses from multiple sensors for online damage identification; and finally, the robustness of the proposed method was validated using extensive field test data by considering various damage scenarios in the presence of environmental variabilities.

NeuralCP: Bayesian Multiway Data Analysis with Neural Tensor Decomposition

Multiway data are widely observed in neuroscience, health informatics, food science, etc. Tensor decomposition is an important technique for capturing high-order interactions among such multiway data. Classical tensor decomposition methods, such as the Tucker decomposition and the CANDECOMP/PARAFAC (CP), assume that the complex interactions among objects are multi-linear and thus insufficient to represent nonlinear relationships in data. To effectively model the complex nonlinear relationships of a tensor, we design a neural model joining neural networks with the Bayesian tensor decomposition, in which the high-order interactions are captured by neural networks. By taking advantages of the nonlinear modeling provided by the neural networks and the uncertainty modeling provided by Bayesian models, we replace the multi-linear product in traditional Bayesian tensor decomposition with a more flexible neural function (i.e., a multi-layer perceptron) whose parameters can be learned from data. Our model can be efficiently optimized with stochastic gradient descent. Accordingly, it is scalable to large real-world tensor. We conducted experiments on both synthetic data and real-world chemometrics tensor data. Experimental results have demonstrated that the proposed model can achieve significantly higher prediction performance than the state-of-the-art tensor decomposition approaches. The proposed nonlinear tensor decomposition method, i.e., NeuralCP, has been demonstrated to obtain promising prediction results on many multi-way data.

Extracting multi-mode ERP features using fifth-order nonnegative tensor decomposition

Background:Preprocessed Event-related potential (ERP) data are usually organized in multi-way tensor, in which tensor decomposition serves as a powerful tool for data processing. Due to the limitation of computation burden for multi-way data and the low algorithm performance of stability and efficiency, multi-way ERP data are conventionally reorganized into low-order tensor or matrix before further analysis. However, the reorganization may hamper mode specification and spoil the interaction information among different modes. New Method:In this study, we applied a fifth-order tensor decomposition to a set of fifth-order ERP data collected by exerting proprioceptive stimulus on left and right hand. One of the latest nonnegative CANDECOMP/PARAFAC (NCP) decomposition methods implemented by alternating proximal gradient (APG) was employed. We also proposed an improved DIFFIT method to select the optimal component number for the fifth-order tensor decomposition. Results:By the fifth-order NCP model with a proper component number, the ERP data were fully decomposed into spatial, spectral, temporal, subject and condition factors in each component. The results showed more pairs of components with symmetric activation region in left and right hemisphere elicited by contralateral stimuli on hand. Comparison with Existing Method(s):In our experiment, more interesting components and coherent brain activities were extracted, compared with previous studies. Conclusions:The discovered activities elicited by proprioceptive stimulus are in line with those in relevant cognitive neuroscience studies. Our proposed method has proved to be appropriate and viable for processing high-order EEG data with well-preserved interaction information among all modes.

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA

The widespread use of multisensor technology and the emergence of big data sets have brought the necessity to develop more versatile tools to represent higher order data with multiple aspects and high dimensionality. Data in the form of multidimensional arrays, also referred to as tensors, arise in a variety of applications including chemometrics, hyperspectral imaging, high-resolution videos, neuroimaging, biometrics, and social network analysis. Early multiway data analysis approaches reformatted such tensor data as large vectors or matrices and then resorted to dimensionality reduction methods developed for classical two-way analysis such as principal component analysis (PCA). However, one cannot discover hidden components within multiway data using conventional PCA. To this end, tensor decomposition methods which are flexible in the choice of the constraints and that extract more general latent components have been proposed. In this paper, we review the major tensor decomposition methods with a focus on problems targeted by classical PCA. In particular, we present tensor methods that aim to solve three important challenges typically addressed by PCA: dimensionality reduction, i.e., low-rank tensor approximation; supervised learning, i.e., learning linear subspaces for feature extraction; and robust low-rank tensor recovery. We also provide experimental results to compare different tensor models for both dimensionality reduction and supervised learning applications.

A Fused CP Factorization Method for Incomplete Tensors.

Low-rank tensor completion methods have been advanced recently for modeling sparsely observed data with a multimode structure. However, low-rank priors may fail to interpret the model factors of general tensor objects. The most common method to address this drawback is to use regularizations together with the low-rank priors. However, due to the complex nature and diverse characteristics of real-world multiway data, the use of a single or a few regularizations remains far from efficient, and there are limited systematic experimental reports on the advantages of these regularizations for tensor completion. To fill these gaps, we propose a modified CP tensor factorization framework that fuses the l2 norm constraint, sparseness ( l1 norm), manifold, and smooth information simultaneously. The factorization problem is addressed through a combination of Nesterov's optimal gradient descent method and block coordinate descent. Here, we construct a smooth approximation to the l1 norm and TV norm regularizations, and then, the tensor factor is updated using the projected gradient method, where the step size is determined by the Lipschitz constant. Extensive experiments on simulation data, visual data completion, intelligent transportation systems, and GPS data of user involvement are conducted, and the efficiency of our method is confirmed by the results. Moreover, the obtained results reveal the characteristics of these commonly used regularizations for tensor completion in a certain sense and give experimental guidance concerning how to use them.

Multi-way feature selection for ECoG-based Brain-Computer Interface

The paper addresses the problem of designing Brain-Computer Interfaces. It investigates feature selection methods in regression, applied to ECoG-based motion decoding. The problem is to predict hand trajectories from the voltage time series of cortical activity. A special characteristic of this problem is the inherently multi-way structure of feature description. The feature description resides in spatial-spectra-temporal domain and includes the voltage time series and their spectral representation. Since electrocorticographic data is highly correlated in temporal, spectral and spatial domains, redundancy of the feature space as well as its dimensionality become a major obstacle for robust solution of the regression problem both in multi-way and flat cases. Feature selection reduces dimensionality and increases model robustness. It plays the crucial role in obtaining adequate predictions.The main contribution of this paper is the following. We propose a filtering feature selection for multi-way data. The proposed method extends quadratic programming feature selection (QPFS) approach. QPFS selects a subset of features by solving a quadratic problem. It incorporates estimates of similarity between features and their relevance to the regression problem. QPFS offers an effective way to leverage similarities between features and their importance. Our modification allows to apply QPFS to multi-way data. By taking the multi-way structure of features into account, the proposed modification reduces computational costs of optimization problem in QPFS. Experimental outcomes demonstrate that the proposed modification improves prediction quality of resultant models. The proposed method is model-free and provides interpretable results, which makes it relevant for knowledge extraction and domain analysis.

PARAFAC HPLC-DAD metabolomic fingerprint investigation of reference and crossed coffees

In this study two cultivars of Coffea arabica L., Bourbon (reference) and IPR101 (crossing) were analyzed. The extracts were prepared according to a simplex centroid design with four components, ethanol, ethyl acetate, dichloromethane, and hexane. Multiway data were obtained by HPLC-DAD analysis of the fifteen different mixtures for each cultivar. The PARAFAC methodology was used to investigate the chromatographic fingerprint. For both cultivars, Factor 1 was able to discriminate mixtures containing ethyl acetate as solvent. Factor 2 indicated that mixtures in pure ethanol and binary mixtures containing ethanol were the most efficient in extracting chlorogenic acids and factor 3 identified methylxanthines through spectrophotometric profile in all mixtures. Higher concentrations were obtained by the ethanol, dichloromethane and hexane ternary mixture for the Bourbon cultivar and by the quaternary mixture of these solvents with ethyl acetate for the IPR101 cultivar. Trigonelline and cafestol were extracted in both cultivars. The reference coffee showed higher relative abundances of cafestol ester, chlorogenic acids and trigonelline whereas the crossed coffee showed higher levels of caffeine. To confirm these results, UPLC-MS was used as a complementary method to confirm the presence of the metabolites in these extracts. The three way PARAFAC strategy determines correlations of HPLC-DAD chromatographic and spectral data simultaneously with samples permitting a more unambiguous assignment of metabolic groups than can be obtained treating chromatographic and spectral data separately by two way methods. This can provide higher quality chromatographic fingerprints for food chemistry.

Tensor Completion Based on Triple Tubal Nuclear Norm

Many tasks in computer vision suffer from missing values in tensor data, i.e., multi-way data array. The recently proposed tensor tubal nuclear norm (TNN) has shown superiority in imputing missing values in 3D visual data, like color images and videos. However, by interpreting in a circulant way, TNN only exploits tube (often carrying temporal/channel information) redundancy in a circulant way while preserving the row and column (often carrying spatial information) relationship. In this paper, a new tensor norm named the triple tubal nuclear norm (TriTNN) is proposed to simultaneously exploit tube, row and column redundancy in a circulant way by using a weighted sum of three TNNs. Thus, more spatial-temporal information can be mined. Further, a TriTNN-based tensor completion model with an ADMM solver is developed. Experiments on color images, videos and LiDAR datasets show the superiority of the proposed TriTNN against state-of-the-art nuclear norm-based tensor norms.

Application of Substituent Electronic Descriptors QSAR Model of 2-amino-6-arylsulfonylbenzonitriles as HIV-1 Reverse Transcriptase Inhibitors based on the MOLMAP Approach

The HIV-1 reverse transcriptase (RT) is a major target for drug development. Inhibition of this enzyme has been one of the primary therapeutic strategies in suppressing the replication of HIV-1. A series of 2-amino-6-arylsulfonylbenzonitrile derivatives were subjected to quantitative structure-activity relationship (QSAR) analysis. Very recently, we proposed the use of substituent electronic descriptors (SED) instead of the electronic descriptors of whole molecules as new and expedite source of electronic descriptors. In this study, we used SED parameters in QSAR modeling of anti HIV-1 activity of 6-arylsulfonylbenzonitrile derivatives. In SED methodology produces a vector of electronic descriptors for each substituent and thus a matrix of SED is generated for each molecule. Consequently, a three-dimensional array is obtained by staking the data matrices of different molecules beside each other. As a novel multiway data analysis method, molecular maps of atom-level properties (MOLMAP) approach was also used to transfer a three-dimensional array of SED descriptors into new two-dimensional parameters using Kohonen network, following by genetic algorithm-based partial least square(GA-PLS) to connect a quantitative relationship between the Kohonen scores and biological activity.In unfolding data, HOMO1, HOMOB1, SOFB1 and EPHA4 represent the most important indices on QSAR equation derived by PLS analysis. Accurate QSAR models were obtained by both approaches. The resulted GA-PLS model of MOLMAP approach possessed high statistical quality r2= 0.83 and q2=0.70. It could explain and predict about 70% of variances in the anti-HIV1 inhibitory activity of the studied molecules. However, the superiority of three-way analysis of SED parameters based on MOLMAP approach with respect to simple unfolding was obtained.