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To address the issues of poor anti-noise performance of second-order statistics and low estimation accuracy in previous joint underdetermined blind source separation (JUBSS) methods, we propose a novel JUBSS method based on the dependence between different data sets and the advantages of cross third-order cumulant in resisting distributed noise. The method involves several steps. Firstly, we calculate the cross third-order cumulant of multiple whitening data sets with different delays. Then, we stack several third-order cumulants into fourth-order tensors. Next, we decompose the fourth-order tensor using Canonical Polyadic through weight nonlinear least squares, which allows us to estimate the mixed matrix. Finally, depending on the independence of source signals, we propose a matrix diagonalization method to recover the source signal. Experiments demonstrate that the method effectively suppresses the influence of Gaussian noise and performs well in underdetermined, positive and overdetermined cases and produces a better performance than various common approaches. Specifically, for the 3 × 4 mixed model with signal-to-noise ratio of 20 dB, the average relative error is − 14.48 dB, the average similarity coefficient is 0.92 and the signal-to-interference ratio is 24.84 dB. [ABSTRACT FROM AUTHOR]

There is growing interest to extend low-rank matrix decompositions to multi-way arrays, or tensors. One fundamental low-rank tensor decomposition is the canonical polyadic decomposition (CPD). The challenge of fitting a low-rank, nonnegative CPD model to Poisson-distributed count data is of particular interest. Several popular algorithms use local search methods to approximate the maximum likelihood estimator (MLE) of the Poisson CPD model. This work presents two new algorithms that extend state-of-the-art local methods for Poisson CPD. Hybrid GCP-CPAPR combines Generalized Canonical Decomposition (GCP) with stochastic optimization and CP Alternating Poisson Regression (CPAPR), a deterministic algorithm, to increase the probability of converging to the MLE over either method used alone. Restarted CPAPR with SVDrop uses a heuristic based on the singular values of the CPD model unfoldings to identify convergence toward optimizers that are not the MLE and restarts within the feasible domain of the optimization problem, thus reducing overall computational cost when using a multi-start strategy. We provide empirical evidence that indicates our approaches outperform existing methods with respect to converging to the Poisson CPD MLE. [ABSTRACT FROM AUTHOR]

The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at O 10 − 3 . [ABSTRACT FROM AUTHOR]

We advanced the practical development of compressive hyperspectral cameras for remote sensing scenarios with a design that simultaneously compresses and captures high-quality spectral information of a scene via configurable measurements. We built a prototype imaging system that is compatible with light-modulation devices that encode the incoming spectrum. The sensing approach enables a substantial reduction in the volume of data collected and transmitted, facilitating large-scale remote hyperspectral imaging. A main advantage of our sensing design is that it allows for adaptive sampling. When prior information of a survey region is available or gained, the modulation patterns can be re-programmed to efficiently sample and detect desired endmembers. Given target spectral signatures, we propose an optimization scheme that guides the encoding process. The approach severely reduces the number of required sampling patterns, with the ability to achieve image segmentation and correct distortions. Additionally, to decode the modulated data, we considered a novel reconstruction algorithm suited for large-scale images. The computational methodology leverages the multidimensional structure and redundant representation of hyperspectral images via the canonical polyadic decomposition of multiway arrays. Under realistic remote sensing scenarios, we demonstrated the efficiency of our approach with several data sets collected by our prototype camera and reconstructed by our low-rank tensor decoder. [ABSTRACT FROM AUTHOR]

Profiling immune responses across several dimensions, including time, patients, molecular features, and tissue sites, can deepen our understanding of immunity as an integrated system. These studies require new analytical approaches to realize their full potential. We highlight recent applications of tensor methods and discuss several future opportunities.

Background Back-projection has been used to correct the variance and polarity indeterminacies for the independent component analysis. The variance and polarity of the components are essential features of neuroscience studies.Objective This work extends the back-projection theory to canonical polyadic decomposition (CPD) for high-order tensors, aiming to correct the variance and polarity indeterminacies of the components extracted by CPD.Methods The tensor is reshaped into a matrix and decomposed using a suitable blind source separation algorithm. Subsequently, the coefficients are projected using back-projection theory, and other factor matrices are computed through a series of singular value decompositions of the back-projection matrix.Results By applying this method, the energy and polarity of each component are determined, effectively correcting the variance and polarity indeterminacies in CPD. The proposed method was validated using simulated tensor data and resting-state fMRI data.Conclusion Our proposed back-projection method for high-order tensors effectively corrects variance and polarity indeterminacies in CPD, offering a precise solution for calculating the energy and polarity required to extract meaningful features from neuroimaging data.

We consider the neural sparse representation to solve the Boltzmann equation with BGK and quadratic collision models, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and physical space so as to avoid the discretization in space and time. Different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision models, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All of these will significantly improve the efficiency of the network when solving the Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multiscale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod tube problems and a 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods. [ABSTRACT FROM AUTHOR]

A joint blind source separation (JBSS) algorithm based on decomposition of high ‐ order cumulant tensors is proposed. The algorithm recovers the source signals from the observation signals of multiset data. Firstly, the higher ‐ order cross ‐ cumulant tensors of observation signals of multiset data are calculated. Due to the potential diagonal structures of cumulant tensors, the JBSS problem can be transformed into canonical polyadic decomposition (CPD) of a higher‐order tensor. Next, by tensor train decomposition (TTD）, the higher‐order tensor is decomposed into a simple tensor network composed of a set of interconnected core tensors of orders not higher than 3. The CPD of a higher‐order tensor thereby is transformed into a set of CPDs of order‐3 tensors. Finally, according to the links between TTD and CPD, after several CPDs of order ‐ 3 tensors, the mixed matrices of multi ‐ dataset can be obtained by reordering and rescaling the factor matrices sequentially, resulting in the separation of the source signals. Simulation results show that the proposed algorithm operated at a faster speed [ABSTRACT FROM AUTHOR]

The canonical polyadic (CP) decomposition of tensors is one of the most important tensor decompositions. While the well‐known alternating least squares (ALS) algorithm is often considered the workhorse algorithm for computing the CP decomposition, it is known to suffer from slow convergence in many cases and various algorithms have been proposed to accelerate it. In this article, we propose a new accelerated ALS algorithm that accelerates ALS in a blockwise manner using a simple momentum‐based extrapolation technique and a random perturbation technique. Specifically, our algorithm updates one factor matrix (i.e., block) at a time, as in ALS, with each update consisting of a minimization step that directly reduces the reconstruction error, an extrapolation step that moves the factor matrix along the previous update direction, and a random perturbation step for breaking convergence bottlenecks. Our extrapolation strategy takes a simpler form than the state‐of‐the‐art extrapolation strategies and is easier to implement. Our algorithm has negligible computational overheads relative to ALS and is simple to apply. Empirically, our proposed algorithm shows strong performance as compared to the state‐of‐the‐art acceleration techniques on both simulated and real tensors. [ABSTRACT FROM AUTHOR]

The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at O10−3.

Collecting neuroimaging data in the form of tensors (i.e. multidimensional arrays) has become more common in mental health studies, driven by an increasing interest in studying the associations between neuroimaging phenotypes and clinical disease manifestation. Motivated by a neuroimaging study of post‐traumatic stress disorder (PTSD) from the Grady Trauma Project, we study a tensor response quantile regression framework, which enables novel analyses that confer a detailed view of the potentially heterogeneous association between a neuroimaging phenotype and relevant clinical predictors. We adopt a sensible low‐rank structure to represent the association of interest, and propose a simple two‐step estimation procedure which is easy to implement with existing software. We provide rigorous theoretical justifications for the intuitive two‐step procedure. Simulation studies demonstrate good performance of the proposed method with realistic sample sizes in neuroimaging studies. We conduct the proposed tensor response quantile regression analysis of the motivating PTSD study to investigate the association between fMRI resting‐state functional connectivity and PTSD symptom severity. Our results uncover non‐homogeneous effects of PTSD symptoms on brain functional connectivity, which cannot be captured by existing tensor response methods. [ABSTRACT FROM AUTHOR]

The analysis of functional near-infrared spectroscopy (fNIRS) signals has not kept pace with the increased use of fNIRS in the behavioral and brain sciences. The popular grand averaging method collapses the oxygenated hemoglobin data within a predefined time of interest window and across multiple channels within a region of interest, potentially leading to a loss of important temporal and spatial information. On the other hand, the tensor decomposition method can reveal patterns in the data without making prior assumptions of the hemodynamic response and without losing temporal and spatial information. The aim of the current study was to examine whether the tensor decomposition method could identify significant effects and novel patterns compared to the commonly used grand averaging method for fNIRS signal analysis. We used two infant fNIRS datasets and applied tensor decomposition (i.e., canonical polyadic and Tucker decompositions) to analyze the significant differences in the hemodynamic response patterns across conditions. The codes are publicly available on GitHub. Bayesian analyses were performed to understand interaction effects. The results from the tensor decomposition method replicated the findings from the grand averaging method and uncovered additional patterns not detected by the grand averaging method. Our findings demonstrate that tensor decomposition is a feasible alternative method for analyzing fNIRS signals, offering a more comprehensive understanding of the data and its underlying patterns. [ABSTRACT FROM AUTHOR]

The canonical polyadic decomposition (CPD) of higher-order tensors, a.k.a. PARAFAC, has shown excellent performance in two-dimensional direction of arrival (DOA) estimation using the acoustic vector sensor array (AVSA). However, most existing studies pay little attention to the manifold matrix structure of the AVSA during the CPD and are designed for uncorrelated sources. This paper presents a constrained CPD-based algorithm for DOA estimation using a uniform linear AVSA, whose manifold matrix is highly structured. Specifically, the manifold matrix equals to the Khatri-Rao product of a Vandermonde matrix and a proportional column-norm matrix. We show that DOA estimation accuracy is further improved by incorporating the prior structured information. Besides, we also extend the Toeplitz decorrelation technique to the AVSA to handle possibly correlated sources. The algorithm does not require iteration or peak searching and thus is computationally effective. Numerical simulations verify the effectiveness and superior performance of the algorithm. [ABSTRACT FROM AUTHOR]

We relate the condition numbers of computing three decompositions of symmetric tensors: the polyadic decomposition, the Waring decomposition, and a Tucker-compressed Waring decomposition. Based on this relation we can speed up the computation of these condition numbers by orders of magnitude through Tucker compression. [ABSTRACT FROM AUTHOR]

Canonical polyadic decomposition as one of the methods for link prediction of knowledge graphs, enabling link prediction complementation of some knowledge graphs containing regular data. However, when there is a large amount of sparse data and reversible relationships in the knowledge graph, the method cannot reflect the hidden connection between two entities and cannot process such data. To address the above issues, this paper proposed an enhanced canonical polyadic decomposition method to learn the two embedding vectors of the front entity and the back entity in the triad separately, and used probabilistic methods to generate higher quality negative example triads during the training process. It introduced ELU loss function and AMSGrad optimiser to effectively process reversible relationships. Experimental results on the generic dataset show that the proposed method can effectively improve link prediction accuracy, with a 5% performance improvement compared to the comparison model, and is also applied in the complementation of the automotive repair knowledge graph dataset, achieving 83. 2% correct entity complementation results. [ABSTRACT FROM AUTHOR]

Tensors, or multi-indexed arrays, play an important role in machine learning and signal processing. These higher-order generalizations of matrices allow for preservation of higher-order structure in data, and low rank decompositions of tensors allow for recovery of underlying information. In many cases, the underlying tensor of interest is positive (semi)definite, i.e., the homogeneous polynomial associated to the tensor has nonnegative evaluation on all inputs. An archetypal problem is that one has a noisy measurement of some low rank signal tensor of interest. This measurement itself does not have low rank, so one must compute a best low rank approximation of the measured tensor to recover component information. As it turns out, the set of tensors of rank less than or equal to R is, in general, not closed when R >1, and, as a consequence, best low rank approximations can fail to exist. In the case when a best low rank approximation does not exist, near optimal low rank approximations suffer numerical issues and cannot be used to reliably approximate underlying component information. As a consequence, existence guarantees for best low rank tensor approximations are of great practical and theoretical interest. This article develops deterministic guarantees for the existence of best low rank approximations of tensors which are positive semidefinite. In particular, we show that the set of low rank positive semidefinite tensors is relatively closed as a subset of the set of positive semidefinite tensors. We use this fact to give a deterministic bound which may be used to guarantee the existence of a best low rank approximation of a noisy low rank positive semidefinite tensor. Furthermore, we show that this condition guarantees uniqueness of the canonical polyadic decomposition for best low rank approximations. In addition, for order three tensors, we prove that our bound is sharp and that it can be computed using semidefinite programming. The main results of this article are illustrated through numerical experiments, which show that our bound is highly predictive of numerical issues when attempting to recover underlying component information from a noisy low rank positive semidefinite tensor. [ABSTRACT FROM AUTHOR]

Higher-order tensor canonical polyadic decomposition (CPD) with one or more of the latent factor matrices being columnwisely orthonormal has been well studied in recent years. However, most existing models penalize the noises, if occurring, by employing the least squares loss, which may be sensitive to non-Gaussian noise or outliers, leading to bias estimates of the latent factors. In this paper, we derive a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise such as the Cauchy noise, or outliers. By exploring the half-quadratic property of the model, we develop the so-called half-quadratic alternating direction method of multipliers (HQ-ADMM) to solve the model. Each subproblem involved in HQ-ADMM admits a closed-form solution. Thanks to some nice properties of the Cauchy loss, we show that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration. Numerical experiments on synthetic and real data demonstrate the effectiveness of the proposed model and algorithm. [ABSTRACT FROM AUTHOR]

Modern neural networks have revolutionized the fields of computer vision (CV) and Natural Language Processing (NLP). They are widely used for solving complex CV tasks and NLP tasks such as image classification, image generation, and machine translation. Most state-of-the-art neural networks are over-parameterized and require a high computational cost. One straightforward solution is to replace the layers of the networks with their low-rank tensor approximations using different tensor decomposition methods. This article reviews six tensor decomposition methods and illustrates their ability to compress model parameters of convolutional neural networks (CNNs), recurrent neural networks (RNNs) and Transformers. The accuracy of some compressed models can be higher than the original versions. Evaluations indicate that tensor decompositions can achieve significant reductions in model size, run-time and energy consumption, and are well suited for implementing neural networks on edge devices. [ABSTRACT FROM AUTHOR]

Alternating least squares is a classic, easily implemented, yet widely used method for tensor canonical polyadic approximation. Its subsequential and global convergence is ensured if the partial Hessians of the blocks during the whole sequence are uniformly positive definite. This paper shows that this positive definiteness assumption can be weakened in two ways. Firstly, if the smallest positive eigenvalues of the partial Hessians are uniformly positive, and the solutions of the subproblems are properly chosen, then global convergence holds. This allows the partial Hessians to be only positive semidefinite. Next, if at a limit point, the partial Hessians are positive definite, then global convergence also holds. We also discuss the connection of such an assumption to the uniqueness of exact CP decomposition. [ABSTRACT FROM AUTHOR]

The analysis of functional near-infrared spectroscopy (fNIRS) signals has not kept pace with the increased use of fNIRS in the behavioral and brain sciences. The popular grand averaging method collapses the oxygenated hemoglobin data within a predefined time of interest window and across multiple channels within a region of interest, potentially leading to a loss of important temporal and spatial information. On the other hand, the tensor decomposition method can reveal patterns in the data without making prior assumptions of the hemodynamic response and without losing temporal and spatial information. The aim of the current study was to examine whether the tensor decomposition method could identify significant effects and novel patterns compared to the commonly used grand averaging method for fNIRS signal analysis. We used two infant fNIRS datasets and applied tensor decomposition (i.e., canonical polyadic and Tucker decompositions) to analyze the significant differences in the hemodynamic response patterns across conditions. The codes are publicly available on GitHub. Bayesian analyses were performed to understand interaction effects. The results from the tensor decomposition method replicated the findings from the grand averaging method and uncovered additional patterns not detected by the grand averaging method. Our findings demonstrate that tensor decomposition is a feasible alternative method for analyzing fNIRS signals, offering a more comprehensive understanding of the data and its underlying patterns.

Identification of informative signatures from electrophysiological signals is important for understanding brain developmental patterns, where techniques such as magnetoencephalography (MEG) are particularly useful. However, less attention has been given to fully utilizing the multidimensional nature of MEG data for extracting components that describe these patterns. Tensor factorizations of MEG yield components that encapsulate the data's multidimensional nature, providing parsimonious models identifying latent brain patterns for meaningful summarization of neural processes. To address the need for meaningful MEG signatures for studies of pediatric cohorts, we propose a tensor-based approach for extracting developmental signatures of multi-subject MEG data. We employ the canonical polyadic (CP) decomposition for estimating latent spatiotemporal components of the data, and use these components for group level statistical inference. Using CP decomposition along with hierarchical clustering, we were able to extract typical early and late latency event-related field (ERF) components that were discriminative of high and low performance groups ( p < 0.05 ) and significantly correlated with major cognitive domains such as attention, episodic memory, executive function, and language comprehension. We demonstrate that tensor-based group level statistical inference of MEG can produce signatures descriptive of the multidimensional MEG data. Furthermore, these features can be used to study group differences in brain patterns and cognitive function of healthy children. We provide an effective tool that may be useful for assessing child developmental status and brain function directly from electrophysiological measurements and facilitate the prospective assessment of cognitive processes. [ABSTRACT FROM AUTHOR]

We propose a Riemannian conjugate gradient algorithm for approximating incomplete tensors by canonical polyadic decompositions of low rank. Our main contribution consists of an explicit expression for an almost everywhere complete set of geodesics of the Segre manifold of rank‐1 tensors. These are leveraged in our Riemannian optimization algorithm over a geometrically convenient parametrization of rank‐r$$ r $$ tensors to move in the direction of a tangent vector over the r$$ r $$‐fold product of Segre manifolds. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and for identifying pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we can recover the tensor decomposition from only 6.5%$$ 6.5\% $$ of the data. In our numerical experiments, the proposed Riemannian conjugate gradient algorithm was competitive with a state‐of‐the‐art quasi‐Newton method with truncated conjugate gradient inner solves from Tensorlab in terms of accuracy and could reduce the computation time by half. [ABSTRACT FROM AUTHOR]

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system—the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption. [ABSTRACT FROM AUTHOR]

Estimating the minimal number of rank-1 tensors in the Canonical Polyadic Decomposition (CPD), known as the canonical rank, is a challenging area of research. To address this problem, we propose a method based on convex optimization to jointly estimate the CP factors and the canonical rank called FARAC for joint FActors and RAnk canonical estimation for the PolyadiC Decomposition. We formulate the FARAC method as a convex optimization problem in which a sparse promoting constraint is added to the superdiagonal of the core tensor of the CPD, whereas the Frobenius norm of the offdiagonal terms is constrained to be bounded. We propose an alternated minimization strategy for the Lagrangien to solve the optimization problem. The FARAC method has been validated on synthetic data with varying levels of noise, as well as on three real data sets. Compared to state-of-the-art methods, FARAC exhibits very good performance in terms of rank estimation accuracy for a large range of SNR values. Additionally, FARAC can handle the case in which the canonical rank exceeds one of the dimensions of the input tensor.

Latency overhead in distributed-memory parallel CPD-ALS scales with the number of processors, limiting the scalability of computing CPD of large irregularly sparse tensors. This overhead comes in the form of sparse reduce and expand operations performed on factor-matrix rows via point-to-point messages. We propose to hide the latency overhead through embedding all of the point-to-point messages incurred by the sparse reduce and expand into dense collective operations which already exist in the CPD-ALS. The conventional parallel CPD-ALS algorithm is not amenable for embedding so we propose a computation/communication rearrangement to enable the embedding. We embed the sparse expand and reduce into a hypercube-based ALL-REDUCE operation to limit the latency overhead to $O(\log _2 K)$ O (log 2 K) for a $K$ K -processor system. The embedding comes with the cost of increased bandwidth overhead due to the multi-hop routing of factor-matrix rows during the embedded-ALL-REDUCE. We propose an embedding scheme that takes advantage of the expand/reduce properties to reduce this overhead. Furthermore, we propose a novel recursive bipartitioning framework that enables simultaneous hypergraph partitioning and subhypergraph-to-subhypercube mapping to achieve subtensor-to-processor assignment with the objective of reducing the bandwidth overhead during the embedded-ALL-REDUCE. We also propose a bin-packing-based algorithm for factor-matrix row to processor assignment aiming at reducing processors’ maximum send and receive volumes during the embedded-ALL-REDUCE. Experiments on up to 4096 processors show that the proposed framework scales significantly better than the state-of-the-art point-to-point method. [ABSTRACT FROM AUTHOR]

This paper proposes a model-free distribution system state estimation method based on tensor completion using canonical polyadic decomposition. In particular, we consider a setting where the network is divided into multiple areas. The measured physical quantities at buses located in the same area are processed by an area controller. A three-way tensor is constructed to collect these measured quantities. The measurements are analyzed locally to recover the full state information of the network. A distributed closed-form iterative algorithm based on the alternating direction method of multipliers is developed to obtain the low-rank factors of the whole network state tensor where information exchange happens only between neighboring areas. The convergence properties of the distributed algorithm and the sufficient conditions on the number of samples for each smaller network that guarantee the identifiability of the factors of the state tensor are presented. To demonstrate the efficacy of the proposed algorithm and to check the identifiability conditions, numerical simulations are carried out using the IEEE 123-bus system and a large-scale real utility feeder. [ABSTRACT FROM AUTHOR]

In order to avoid searching the peak value in space domain, when estimating nested array’s the direction of direction of arrival(DOA), the canonical polyadic decomposition(CPD) was applied into the nested array, namely using the one time singular value decomposition(SVD), bilinear mapping and tensor decomposition to obtain the steering vector matrix and arrival angle.However, the existing CPD algorithm only can be applied in noiseless environment, the algorithm was improved by utilizing SVD two times, and was made to be applied in both noiseless and noisy environments.The simulation results demonstrate that in the same signal to noise ratio(SNR) and snapshot, the DOA estimation algorithm of nested array based on the improved CPD has better performances and less running time than the MUSIC and space smoothing algorithms.

Video hashing is an efficient technique for tasks like copy detection and retrieval. This paper utilizes canonical polyadic (CP) decomposition and Hahn moments to design a robust video hashing. The first significant contribution is the secondary frame construction. It uses three weighted techniques to generate three secondary frames for each video group, which can effectively capture features of video frames from different aspects and thus improves discrimination. Another contribution is the deep feature extraction via the ResNet50 and CP decomposition. The use of the ResNet50 can provide rich features and the CP decomposition can learn a compact and discriminative representation from the rich features. In addition, the Hahn moments of secondary frames are taken to construct hash elements. Extensive experiments on the open video dataset demonstrate that the proposed algorithm surpasses several state-of-the-art algorithms in balancing discrimination and robustness. [ABSTRACT FROM AUTHOR]

The Canonical Polyadic (CP) tensor decomposition is frequently used as a model in applications in a variety of different fields. Using jackknife resampling to estimate parameter uncertainties is often desirable but results in an increase of the already high computational cost. Upon observation that the resampled tensors, though different, are nearly identical, we show that it is possible to extend the recently proposed Concurrent ALS (CALS) technique to a jackknife resampling scenario. This extension gives access to the computational efficiency advantage of CALS for the price of a modest increase (typically a few percent) in the number of floating point operations. Numerical experiments on both synthetic and real-world datasets demonstrate that the new workflow based on a CALS extension can be several times faster than a straightforward workflow where the jackknife submodels are processed individually.

In this paper, we propose to extend the standard Convolutional Dictionary Learning problem to a tensor representation where the activations are constrained to be “low-rank” through a Canonical Polyadic decomposition. We show that this additional constraint increases the robustness of the CDL with respect to noise and improve the interpretability of the final results. In addition, we discuss in detail the advantages of this representation and introduce two algorithms, based on ADMM or FISTA, that efficiently solve this problem. We show that by exploiting the low rank property of activations, they achieve lower complexity than the main CDL algorithms. Finally, we evaluate our approach on a wide range of experiments, highlighting the modularity and the advantages of this tensorial low-rank formulation. [ABSTRACT FROM AUTHOR]

A hierarchical iterative algorithm for the canonical polyadic decomposition (CPD) of tensors is proposed by improving the traditional conjugate gradient least squares (CGLS) method. Methods based on algebraic operations are investigated with the objective of estimating the direction of arrival (DoA) and polarization parameters of signals impinging on an array with electromagnetic (EM) vector‐sensors. The proposed algorithm adopts a hierarchical iterative strategy, which enables the algorithm to obtain a fast recovery for the highly collinear factor matrix. Moreover, considering the same accuracy threshold, the proposed algorithm can achieve faster convergence compared with the alternating least squares (ALS) algorithm wherein the highly collinear factor matrix is absent. The results reveal that the proposed algorithm can achieve better performance under the condition of fewer snapshots, compared with the ALS‐based algorithm and the algorithm based on generalized eigenvalue decomposition (GEVD). Furthermore, with regard to an array with a small number of sensors, the observed advantage in estimating the DoA and polarization parameters of the signal is notable.

We introduce a supervised learning framework for target functions that are well approximated by a sum of (few) separable terms. The framework proposes to approximate each component function by a B-spline, resulting in an approximant where the underlying coefficient tensor of the tensor product expansion has a low-rank polyadic decomposition parametrization. By exploiting the multilinear structure, as well as the sparsity pattern of the compactly supported B-spline basis terms, we demonstrate how such an approximant is well-suited for regression and classification tasks by using the Gauss–Newton algorithm to train the parameters. Various numerical examples are provided analyzing the effectiveness of the approach.

Alkali–surfactant–polymer (ASP) flooding holds a critical position in oil development. This paper is concerned with the distributed parameter system (DPS) modeling and development scheme optimization of ASP flooding. Owing to the neglect of coupling among three-dimensional (3-D) spatial information, most of the existing data-driven modeling methods fail to provide a high-precision model for this type of strongly nonlinear and spatiotemporal coupling 3-D DPS. To overcome this shortcoming, an integrated spatiotemporal modeling method (ISM) is proposed, which reconstructs temporal–spatial relationship to avoid spatial mapping distortion based on canonical polyadic decomposition (CPD), and a spatiotemporal MSSO-LS-SVM algorithm is presented to compensate the nonlinear dynamics and decrease truncation error. Moreover, the synchronous optimization of ASP injection strategy and oil wells switching scheduling is an urgent industry requirement of intelligent oilfield decision system, which can be described a mixed-integer optimal control/dynamic programming problem. This paper first puts forward a mixed-integer approximate dynamic programming (MIADP) algorithm, dedicated for solving this class of optimal control problem with integer control subset requirements. In the MIADP, a parallel mixed-integer policy improvement framework is developed, where (i) the continuous control weight updates by the temporal difference (TD) error, (ii) the quantum evolution mechanism is designed to learn the integer control, and (iii) the two parts share information interactively and iterate to approach the optimal mixed-integer policy. At last, a case study on ASP flooding is taken, and the superior performance metrics illustrate the effectiveness of the proposed ISM method and MIADP algorithm. • An integrated spatiotemporal modeling method for a complex industrial 3-D DPS is proposed. • 3-D spatial mapping distortion is avoided to reach a high modeling accuracy. • A novel mixed-integer approximate dynamic programming solution framework is designed. • MIADP realizes synchronous optimization for continuous policy and integer policy. • Effectiveness and superiority are tested by a real-world ASP flooding case study. [ABSTRACT FROM AUTHOR]

In order to avoid searching the peak value in space domain, when estimating nested array's the direction of direction of arrival(DOA), the canonical polyadic decomposition(CPD) was applied into the nested array, namely using the one time singular value decomposition(SVD), bilinear mapping and tensor decomposition to obtain the steering vector matrix and arrival angle. However, the existing CPD algorithm only can be applied in noiseless environment, the algorithm was improved by utilizing SVD two times, and was made to be applied in both noiseless and noisy environments. The simulation results demonstrate that in the same signal to noise ratio(SNR) and snapshot, the DOA estimation algorithm of nested array based on the improved CPD has better performances and less running time than the MUSIC and space smoothing algorithms. [ABSTRACT FROM AUTHOR]

The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraints. This includes extensions of multilinear algebraic uniqueness conditions originally developed for the CPD. We obtain a deterministic uniqueness condition that admits a constructive interpretation. Computationally, we reduce the bilinear factorization problem into a CPD problem, which can be solved via a matrix EigenValue Decomposition (EVD). Under the given conditions, the discussed EVD-based algorithms are guaranteed to return the exact bilinear factorization. Finally, we make a connection between bilinear factorizations subject to monomial equality constraints and the coupled block term decomposition, which allows us to translate monomial structures into low-rank structures. [ABSTRACT FROM AUTHOR]

Given that the mixing matrix of underdetermined blind source separation (UBSS) changes with the recording environment, offline UBSS methods encounter difficulty in satisfying the time-varying estimation demand. Therefore, in this work, an online tensor algorithm has been proposed to estimate the time-varying mixing matrix for separating an instantaneous linear underdetermined mixture. First, we construct a canonical polyadic tensor model by assuming individually correlated sources. Second, an online tensor algorithm is applied to decompose the canonical polyadic tensor model to ensure the accuracy of the time-varying mixing matrix. Finally, two types of data, including speech and biomedical signals, have been used to substantiate the effectiveness of the proposed algorithm in estimating the time-varying mixing matrix for UBSS. The results show that the developed online tensor algorithm is significantly superior to the conventional offline UBSS methods in terms of time consumption and accuracy. [ABSTRACT FROM AUTHOR]

We consider the joint estimation of direction-of-arrival (DOA) and polarization of constant modulus (CM) signals based on an electromagnetic vector-sensor (EMVS) array. Two algebraic algorithms for canonical polyadic decomposition (CPD) with CM constraint are proposed in two scenarios, in which the source signals are fully and partially CM, respectively. The proposed algorithms use the analytic CM algorithm in the first step to calculate the source matrix, and then exploit the CPD structure of the data tensor to compute the remaining factor matrices, from which the DOA and polarization parameters can finally be obtained. Due to the algebraic nature, the proposed algorithms are faster and more stable than the optimization based algorithms, and can be used to effectively initialize the latter. We have shown that the proposed algorithms have more relaxed uniqueness conditions than unconstrained CPD, and thus can be applied in highly underdetermined cases where the number of source signals greatly exceeds that of the EMVSs. Simulation results are provided to illustrate the performance of the proposed algorithms.