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Tensor-ring (TR) decomposition was recently studied and applied for low-rank tensor completion due to its powerful representation ability of high-order tensors. However, most of the existing TR-based methods tend to suffer from deterioration when the selected rank is larger than the true one. To address this issue, this article proposes a new low-rank sparse TR completion method by imposing the Frobenius norm regularization on its latent space. Specifically, we theoretically establish that the proposed method is capable of exploiting the low rankness and Kronecker-basis-representation (KBR)-based sparsity of the target tensor using the Frobenius norm of latent TR-cores. We optimize the proposed TR completion by block coordinate descent (BCD) algorithm and design a modified TR decomposition for the initialization of this algorithm. Extensive experimental results on synthetic data and visual data have demonstrated that the proposed method is able to achieve better results compared to the conventional TR-based completion methods and other state-of-the-art methods and, meanwhile, is quite robust even if the selected TR-rank increases.

Fe/Ni NWs/NF catalysts were prepared by growing iron nanosheets on chain nickel nanowires. The Fe/Ni NWs/NF electrode has a 3D layered heterostructure, on which the Fe nanosheets are amorphous and show good oxygen evolution activity.

Accurate electroencephalogram (EEG) pattern decoding for specific mental tasks is one of the key steps for the development of brain-computer interface (BCI), which is quite challenging due to the considerably low signal-to-noise ratio of EEG collected at the brain scalp. Machine learning provides a promising technique to optimize EEG patterns toward better decoding accuracy. However, existing algorithms do not effectively explore the underlying data structure capturing the true EEG sample distribution and, hence, can only yield a suboptimal decoding accuracy. To uncover the intrinsic distribution structure of EEG data, we propose a clustering-based multitask feature learning algorithm for improved EEG pattern decoding. Specifically, we perform affinity propagation-based clustering to explore the subclasses (i.e., clusters) in each of the original classes and then assign each subclass a unique label based on a one-versus-all encoding strategy. With the encoded label matrix, we devise a novel multitask learning algorithm by exploiting the subclass relationship to jointly optimize the EEG pattern features from the uncovered subclasses. We then train a linear support vector machine with the optimized features for EEG pattern decoding. Extensive experimental studies are conducted on three EEG data sets to validate the effectiveness of our algorithm in comparison with other state-of-the-art approaches. The improved experimental results demonstrate the outstanding superiority of our algorithm, suggesting its prominent performance for EEG pattern decoding in BCI applications.

Non-negative Tucker decomposition (NTD) is one of the most popular techniques for tensor data representation. To enhance the representation ability of NTD by multiple intrinsic cues, that is, manifold structure and supervisory information, in this article, we propose a generalized graph regularized NTD (GNTD) framework for tensor data representation. We first develop the unsupervised GNTD (UGNTD) method by constructing the nearest neighbor graph to maintain the intrinsic manifold structure of tensor data. Then, when limited must-link and cannot-link constraints are given, unlike most existing semisupervised learning methods that only use the pregiven supervisory information, we propagate the constraints through the entire dataset and then build a semisupervised graph weight matrix by which we can formulate the semisupervised GNTD (SGNTD). Moreover, we develop a fast and efficient alternating proximal gradient-based algorithm to solve the optimization problem and show its convergence and correctness. The experimental results on unsupervised and semisupervised clustering tasks using four image datasets demonstrate the effectiveness and high efficiency of the proposed methods.

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawback, this article proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a nonasymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.

Non-negative Tucker decomposition (NTD) has been developed as a crucial method for non-negative tensor data representation. However, NTD is essentially an unsupervised method and cannot take advantage of label information. In this paper, we claim that the low-dimensional representation extracted by NTD can be treated as the predicted soft-clustering coefficient matrix and can therefore be learned jointly with label propagation in a unified framework. The proposed method can extract the physically-meaningful and parts-based representation of tensor data in their natural form while fully exploring the potential ability of the given labels with a nearest neighbors graph. In addition, an efficient accelerated proximal gradient (APG) algorithm is developed to solve the optimization problem. Finally, the experimental results on five benchmark image data sets for semi-supervised clustering and classification tasks demonstrate the superiority of this method over state-of-the-art methods.

High-order tensor data are prevalent in real-world applications, and multiway clustering is one of the most important techniques for exploratory data mining and compression of multiway data. However, existing multiway clustering is based on the K-means procedure and is incapable of addressing the issue of crossed membership degrees. To overcome this limitation, we propose a flexible multiway clustering model called approximately orthogonal nonnegative Tucker decomposition (AONTD). The new model provides extra flexibility to handle crossed memberships while fully exploiting the multilinear property of tensor data. The accelerated proximal gradient method and the low-rank compression tricks are adopted to optimize the cost function. The experimental results on both synthetic data and real-world cases illustrate that the proposed AONTD model outperforms the benchmark clustering methods by significantly improving the interpretability and robustness.

Epilepsy is a chronic disorder caused by excessive electrical discharges. Currently, clinical experts identify the seizure onset zone (SOZ) channel through visual judgment based on long-time intracranial electroencephalogram (iEEG), which is a very time-consuming, difficult and experience-based task. Therefore, there is a need for high-accuracy diagnostic aids to reduce the workload of clinical experts. In this article, we propose a method in which, the iEEG is split into the 20-s segment and for each patient, we ask clinical experts to label a part of the data, which is used to train a model and classify the remaining iEEG data. In recent years, machine learning methods have been successfully applied to solve some medical problems. Filtering, entropy and short-time Fourier transform (STFT) are used for extracting features. We compare them to wavelet transform (WT), empirical mode decomposition (EMD) and other traditional methods with the aim of obtaining the best possible discriminating features. Finally, we look for their medical interpretation, which is important for clinical experts. We achieve high-performance results for SOZ and non-SOZ data classification by using the labeled iEEG data and support vector machine (SVM), fully connected neural network (FCNN) and convolutional neural network (CNN) as classification models. In addition, we introduce the positive unlabeled (PU) learning to further reduce the workload of clinical experts. By using PU learning, we can learn a binary classifier with a small amount of labeled data and a large amount of unlabeled data. This can greatly reduce the amount and difficulty of annotation work by clinical experts. All together, we show that using 105 minutes of labeled data we achieve a classification result of 91.46% on average for multiple patients.

Unlike traditional tensor decompositions which model low-rankness in the original domain, the recently proposed tensor $*_L$ -Singular Value Decomposition ( $*_L$ –SVD) casts a new light on tensor analysis by exploiting low-rankness in the spectral domain. For convexized rank minimization within the framework of $*_L$ –SVD, the tensor $*_L$ –Tubal Nuclear Norm ( $*_L$ –TNN) outperforms classical tensorial extensions of matrix nuclear norm in many applications. In this paper, we first generalize $*_L$ –TNN to the $*_L$ -Spectral $k$ -Support Norm ( $*_L$ -SpSN- $k$ ) as a new low-rank regularizer, and then adopt it to formulate two estimators for tensor recovery from noisy linear observations. Further, statistical performance of the proposed estimators is analyzed by establishing both deterministic and non-asymptotic upper bounds on the estimation error, which indicates that the proposed tensor Dantzig selector enjoys near-optimal estimation error for tensor compressive sensing and tensor completion. Moreover, proximal operator of the proposed norm is derived and embeded in an efficient algorithm to compute the estimators. Experiments on synthetic datasets verify correctness of the proposed error bounds and image inpainting results demonstrate superiority of the proposed estimators.

Low-rank tensor completion (LRTC) aims to impute the missing entries from partially observed tensor data, among which low-rankness is of vital importance to get satisfactory results. In this letter, we propose a hierarchical low-rank factorization framework for high-order tensors. For the first layer, the low TR rank is exploited, and for the second layer the low-rankness of each TR core is further considered. With the hierarchical model, the low-rankness of the original tensor can be fully utilized and thus achieving better completion performance. Experimental results on synthetic data and on inpainting tasks using various datasets demonstrate the superior performance and efficiency of our proposed method as compared to the state-of-the-art algorithms.

Tensor-ring (TR) decomposition is a powerful tool for exploiting the low-rank property of multiway data and has been demonstrated great potential in a variety of important applications. In this article, non-negative TR (NTR) decomposition and graph-regularized NTR (GNTR) decomposition are proposed. The former equips TR decomposition with the ability to learn the parts-based representation by imposing non-negativity on the core tensors, and the latter additionally introduces a graph regularization to the NTR model to capture manifold geometry information from tensor data. Both of the proposed models extend TR decomposition and can be served as powerful representation learning tools for non-negative multiway data. The optimization algorithms based on an accelerated proximal gradient are derived for NTR and GNTR. We also empirically justified that the proposed methods can provide more interpretable and physically meaningful representations. For example, they are able to extract parts-based components with meaningful color and line patterns from objects. Extensive experimental results demonstrated that the proposed methods have better performance than state-of-the-art tensor-based methods in clustering and classification tasks.

Tensor completion has been widely used in computer vision and machine learning. Most existing tensor completion methods empirically assume the intrinsic tensor is simultaneous low-rank in all over modes. However, tensor data recorded from real-world applications may conflict with these assumptions, e.g., face images taken from different subjects often lie in a union of low-rank subspaces, which may result in a quite high rank or even full rank structure in its sample mode. To this aim, in this paper, we propose an imbalanced low-rank tensor completion method, which can flexibly estimate the low-rank incomplete tensor via decomposing it into a mixture of multiple latent tensor ring (TR) rank components. Specifically, each latent component is approximated using low-rank matrix factorization based on TR unfolding matrix. In addition, an effective proximal alternating minimization algorithm is developed and theoretically proved to maintain the global convergence property, that is, the whole sequence of iterates is convergent and converges to a critical point. Extensive experiments on both synthetic and real-world tensor data demonstrate that the proposed method achieves more favorable completion results with less computational cost when compared to the state-of-the-art tensor completion methods.

Time synchronization is a fundamental and paramount technology for a wireless network when a cooperative mission is performed. In this letter, the problem of how to achieve time synchronization and ranging for a mobile wireless network is addressed. The case of all nodes are in motion, equipped with a drifting clock, and capable of communicating with each other under unknown positions and velocities is handled. A network-wide model for synchronizing time-varying clocks and measuring relative distance is presented, from which the clock skews, clock offsets, and virtual polynomial coefficients related to the relative distances could be estimated. The Cramer-Rao Lower Bound (CRLB) is derived for the proposed model. The simulation results show that the proposed method is close to the best result of the existing available estimators and meets the CRLB. Furthermore, compared with the existing estimators, the proposed method is able to reduce the overall communication overheads and streamline the process.

Multi-view clustering is an unsupervised method which aims to enhance the clustering performance by combining the knowledge from multiple view data. Non-negative matrix factorization (NMF) is one of the most favourable multi-view clustering methods due to its strong representation ability of non-negative data. However, NMF only factorizes the data matrix into two non-negative factor matrices, which may limit its ability to learn higher level and more complex hierarchical information. To overcome this shortcoming, in this paper, we propose a multi-view clustering method based on deep graph regularized non-negative matrix factorization (MvDGNMF). MvDGNMF is able to extract more abstract representation by constructing a multilayer NMF model with graph Laplacian regularization and drive the last layer representation from each view to a common consensus representation. Meanwhile, an efficient algorithm using alternating multiplicative update rules is developed. Furthermore, in order to demonstrate the effectiveness of this proposed method, we employ several open datasets including image and text datasets to evaluate the clustering performance of MvDGNMF and the state-of-art methods.

Tensor decomposition methods have been widely applied to big data analysis as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the challenges posed by big data dilemma. This paper attempts to improve the scalability of tensor decompositions and makes two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms.

Dictionary learning plays a significant role in the field of machine learning. Existing works mainly focus on learning dictionary from a single domain. In this paper, we propose a novel projective double reconstructions (PDR) based dictionary learning algorithm for cross-domain recognition. Owing the distribution discrepancy between different domains, the label information is hard utilized for improving discriminability of dictionary fully. Thus, we propose a more flexible label consistent term and associate it with each dictionary item, which makes the reconstruction coefficients have more discriminability as much as possible. Due to the intrinsic correlation between cross-domain data, the data should be reconstructed with each other. Based on this consideration, we further propose a projective double reconstructions scheme to guarantee that the learned dictionary has the abilities of data itself reconstruction and data cross-reconstruction. This also guarantees that the data from different domains can be boosted mutually for obtaining a good data alignment, making the learned dictionary have more transferability. We integrate the double reconstructions, label consistency constraint and classifier learning into a unified objective and its solution can be obtained by proposed optimization algorithm that is more efficient than the conventional $\ell _{1}$ optimization based dictionary learning methods. The experiments show that the proposed PDR not only greatly reduces the time complexity for both training and testing, but also outperforms over the state-of-the-art methods.