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The tensor t-product is a powerful tool for the analysis of and computation with third-order tensors. This paper discusses properties and the computation of eigentubes and eigenslices of third-order tensors under the t-product; the eigentubes and eigenslices are analogues of eigenvalues and eigenvectors for matrices. The computational methods considered and analysed include the tensor power method, tensor subspace iteration, and the tensor QR algorithm. Computed examples illustrate the performance of these methods. [ABSTRACT FROM AUTHOR]

This paper proposes a general inertial smoothing proximal gradient algorithm for solving the Capped- ℓ 1 exact continuous relaxation regularization model proposed by Yu and Zhang (2022) [29]. The proposed algorithm incorporates different extrapolations into the gradient and proximal steps. It is proved that, under some general parameter constraints, the singular values of any accumulation point of the sequence generated by the proposed algorithm have the common support set, and the zero singular values can be achieved in a finite number of iterations. Furthermore, any accumulation point is a lifted stationary point of the relaxation model. Numerical experiments illustrate the efficiency of the proposed algorithm on synthetic and real data, respectively. [ABSTRACT FROM AUTHOR]

Time-variant Yang-Baxter-like matrix equation (YBLME) with the disturbances of noises is the hotspot in various scientific disciplines. The existing research, either can not handle the noise, or rely on the build-in numerical algorithm provided by MATLAB. Therefore, it is necessary to further study the digital-computer discrete solution of this problem. In this paper, the authors propose a noise-handling ten-instant discrete solution for the time-variant YBLME. By defining the matrix-form error function, Zhang neural network (ZNN) is exploited to design the continuous-time noise-handling ZNN model. For potential digital hardware (i.e., digital computer) realization, a discrete-time ZNN-based noise-handling ten-instant (DTZNH10i) algorithm, which is capable of handling three types of noises, is proposed on the basis of a Zhang time discretization (ZTD) formula. The convergence and precision of the proposed DTZNH10i algorithm are investigated and discussed. Theoretical analyses show the correctness and effectiveness of the proposed algorithm. The paper offers three examples of time-variant YBLME with disturbances of constant, linear, and bounded random noises to illustrate the efficacy and convergence of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n × n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem. • The eigen-problems of reduced biquaternion matrices are further studied based on the complex representation form. • Propose new algebraic algorithms for finding the eigenvalues and the eigenvectors of a reduced biquaternion matrix. • An n × n reduced biquaternion matrix has infinite eigenvalues. • There are multiple eigenvalues corresponding to the same eigenvector of a reduced biquaternion matrix. • The proposed method is more comprehensive and can find more eigenvalues of a reduced biquaternion matrix. [ABSTRACT FROM AUTHOR]

In multi-view multi-label learning, instances can be described by a variety of view features, and they are also associated with a set of labels. Most of the existing multi-view multi-label learning methods extract common and private features from views. However, these methods ignore the fact that the extracted features have different importance for multi-label prediction. To address this issue, this paper proposes a multi-view multi-label learning framework with view feature attention allocation. First, the common and private features between different views are obtained. Then, the original features are compared with the common features to obtain the similarity. Next, the attention weight matrix can be obtained by multiplying the similarity with the synergistic feature matrix. Finally, the acquired attention is used to reconstruct the synergistic feature matrix that indicates the semantic information of the view for multi-label prediction. To verify the effectiveness of the proposed algorithm, experiments are conducted on seven multi-view multi-label datasets, and five advanced algorithms are taken for performance comparison. The extensive experimental results demonstrate the superiority of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper we present a linear algorithm that estimates some physical parameters of a continuous-time system, described by an analytical mathematical model, when not all the state variables can be measured. The algorithm starts from the well-known subspace methods and applies some linear transformations to recover, at least partially, the estimated model in physical coordinates. Some analytical investigations and numerical experiments are shown for this method, which has general application within linear time-invariant (LTI) dynamical systems. [ABSTRACT FROM AUTHOR]

• We construct 2D bézier curves with monotone curvature using class a matrices. • A new condition for class a matrices based on its singular values is proved. • An algorithm is provided utilizing the condition for easier class a curve creation. • Several aesthetic curves are given to demonstrate the effectiveness of our approach. In this paper, we construct 2D Bézier curves with monotone curvature using Class A matrices. A new sufficient condition for Class A matrices based on its singular values, is provided and proved, generalizing the 2D typical curves proposed by Mineur et al. (1998). An algorithm is provided utilizing the condition for easier Class A curve creation. Several 2D aesthetic curve examples are constructed to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Box-constrained optimization problems in the real m × n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In recent years, multi-view clustering has been widely concerned and studied. Being its data generally has high dimension and noise, how to effectively deal with redundant features in data from various views and improve the clustering effect is an important issue in multi-view clustering. In this paper, a robust multi-view clustering algorithm Multi-view K -means clustering algorithm based on redundant and sparse feature learning (RSFMVKM) considering redundancy and sparse feature learning is proposed, which can effectively reduce the dimensions of data from different views. Firstly, the K -means algorithm is extended to multi-view clustering, and the importance of each views is represented by learning the weight of each view. Secondly, the sparse feature representation is obtained by applying the norm constraint to the projection matrix in a single view, and the cosine similarity is used to depict the redundancy matrix, which can remove the redundant information and improve the clustering effect. Moreover, the proposed algorithm can effectively avoid the generation of empty classes, and the clustering index matrix is discrete, so the clustering results can be obtained directly. Finally, the proposed algorithm is compared with six state-of-the-art multi-view clustering algorithms on six public datasets, and the experimental results show that the algorithm is effective. • By feature learning, the dimensionality of data is effectively reduced and redundant features are removed. • Control the importance of each view by learning weights. • Using the coordinate descent method, the clustering results can be obtained directly and no empty class is generated. [ABSTRACT FROM AUTHOR]

In this paper, a new dissipation-preserving scheme is established for weakly dissipative perturbations of oscillatory Hamiltonian systems. The system exhibits a nonlinear oscillatory structure. The main oscillation is governed by a matrix M and the damping is governed by a matrix Γ. The new scheme preserves the oscillatory structure of the systems by incorporating the matrix M in the scheme based on the idea of ERKN methods. Meanwhile, the discrete gradient and splitting are used to construct the scheme such that the numerical solution possesses a nearly correct damping rate of the system. A main feature of the new scheme is that a relatively large stepsize can be chosen since the convergence of the implicit iterations in the scheme is shown to be independent of the matrices M and Γ. Three numerical experiments of perturbed Hamiltonian systems are conducted to show the effectiveness and the efficiency of the new scheme in comparison with the traditional discrete gradient methods. [ABSTRACT FROM AUTHOR]

For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that a prediction-correction matrix splitting iteration algorithm is constructed from the view of numerical optimization to solve the derived equation instead, which is inspired by the idea of adaptive parameter update. The novel algorithm adopts the prediction and correction two-step iteration, which uses information with delay to define the iterations. The global convergence results are established and the algorithm enjoys at least a Q -linear convergence rate under some suitable conditions. Further, a preconditioned version is also presented. Compared with some well-known algorithms, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to compute the minimal-norm SOF. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library. [ABSTRACT FROM AUTHOR]

• Algebra matrix method integrates the structural factor and the state information of social networks. • Monte Carlo simulation experiments verify the effectiveness of the algebra matrix evaluation. • The evaluation deviations of diffusion threshold are shown best performance comparing with three popular mean field methods. • The weighted average strategy is proposed as applications. Tacit knowledge is the knowledge existing in human brain which is not easy to be recorded or quantified, and often is learned in the face-to-face interactions. The tacit knowledge diffusion depends on the decision-making of tacit knowledge owners, and the expression of explicit knowledge carriers. However, the comprehensive influence of the tacit knowledge owners, explicit knowledge carriers and the relations of them were not attracted enough attention. In this paper, an algebra matrix method is used to integrate the multidimensional information of network structures and the nodes' states. By the algebra matrix method, the diffusion threshold of the tacit knowledge is calculated, which is called algebra matrix evaluation. This evaluation method is proven to be effective by comparing with Monte Carlo simulations on three types of artificial networks and five reals. With applications of the algebra matrix evaluation, we construct a co-author network according the data of the academic papers from 1980 to 2017 on Aminer platform, and define states of tacit knowledge owners and the explicit knowledge carriers by the scholar's career lengths and the paper's cited quantities respectively. It is found that the thresholds of tacit knowledge diffusion are decreasing with the expansions of the scale of the largest connected components, whether tacit knowledge diffuses in the co-author networks or in the largest connected components. And with the evolution of cumulative co-author network, the diffusion thresholds of tacit knowledge in the largest connected component decrease in ladder-like with unequal steps. Furthermore, it is find ignoring the state factor will lead to the deviation in the evaluation of tacit knowledge diffusion thresholds, which is 16.33% in the largest connected components and 45.07% in the whole network. [ABSTRACT FROM AUTHOR]

In multi-view ensemble clustering, the correctly-partitioned data objects should be assigned with a higher weight, thereby helping to decrease the influence of incorrectly-partitioned data objects. Therefore, different data objects should be treated separately instead of being set the same view weight as traditional solutions. In this paper, a multi-view ensemble clustering approach is proposed using joint affinity matrix, which is generated by sample-level weight. Firstly, a new concept of core data objects is defined according to the influence index and Gaussian Mixed Model, and basic partitions and sample-level weights can be yielded for every view. Secondly, a joint affinity matrix, which maintains pairwise similarities of all data objects, is generated using the sample-level weights. Consequently, data objects can be effectively assigned to the correct partition. Thirdly, a multi-view ensemble clustering algorithm is proposed using the joint affinity matrix. In the end, experimental results on benchmark datasets validate the efficacy of the algorithm with state-of-the-art baselines. • Core data objects are redefined using Gaussian Mixed Model and nearest neighbors. • A single view clustering method without the specified number of clusters is proposed. • A new fusion mechanism is proposed to generate joint affinity matrix. • A multi-view ensemble clustering algorithm is proposed using joint affinity matrix. [ABSTRACT FROM AUTHOR]

A structured eigen-problem A x + F (x) = λ x is studied in this paper, where in applications A ∈ R n × n is an irreducible Stieltjes matrix. Under certain restrictions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of A , the Newton-like algorithm for this eigen-problem is well defined and converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method. [ABSTRACT FROM AUTHOR]

• The calculation of graph similarity matrix in spectral clustering is computational complex for the large high-dimensional data sets. • The similarity matrix can be obtained using matrix completion method to improve the computational efficiency. • The Schatten capped p norm used in this paper integrates the rank norm, nuclear norm, and the Schatten p norm. • The split Bregman algorithm based on the Schatten capped p norm and randomized singular value decomposition is developed to accelerate the convergence of matrix completion. Spectral clustering (SC) is a widely used technique to perform group unsupervised classification of graph signals. However, SC is sometimes computationally intensive due to the need to calculate the graph similarity matrices on large high-dimensional data sets. This paper proposes an efficient SC method that rapidly calculates the similarity matrix using a matrix completion algorithm. First, a portion of the elements in the similarity matrix are selected by a blue noise sampling mask, and their similarity values are calculated directly from the original dataset. After that, a split Bregman algorithm based on the Schatten capped p norm is developed to rapidly retrieve the rest of the matrix elements. Finally, spectral clustering is performed based on the completed similarity matrix. A set of simulations based on different data sets are used to assess the performance of the proposed method. It is shown that for a sufficiently large sampling rate, the proposed method can accurately calculate the completed similarity matrix, and attain good clustering results while improving on computational efficiency. [ABSTRACT FROM AUTHOR]

• Frequency-dependent mass and stiffness matrices of bars and beams are derived. • The scope of the investigation is broadened by applying symbolic computation. • The equivalency of new theory with current dynamic stiffness theory is established. • The Wittrick-Williams algorithm is applied to the new theory. • The investigation paves the way to include damping in dynamic stiffness research. Starting from the solutions of the governing differential equations of motion in free vibration, the frequency dependent mass and stiffness matrices of bar and beam elements have been derived in this paper, but importantly, their equivalency with the corresponding dynamic stiffness matrix is established. In sharp contrast to series solutions, reported in the literature, explicit expressions for each term of the frequency dependent mass and stiffness matrices of bar and beam elements are generated in concise form through the application of symbolic computation and their relationship with the single dynamic stiffness matrix (which contains both the mass and stiffness properties) for each of the two element types is highlighted. The theory is demonstrated by numerical results. By splitting the dynamic stiffness matrix into frequency dependent mass and stiffness matrices and at the same time retaining the exactness of results, the investigation paves the way for future research to overcome the difficulty to include damping in the dynamic stiffness research which has not been possible earlier. Furthermore, the frequency dependent mass and stiffness matrices derived in this paper permit the application of the Wittrick-Williams algorithm to compute with certainty the exact natural frequencies of structures comprising bar and beam elements. [ABSTRACT FROM AUTHOR]

• Structural proximity measure to cluster high-dimensional OD matrices. • Prior identification of subspaces to address multi-density OD database. • Two-level DBSCAN approach to identify optimal parameters. • Estimation of typical OD matrices from clusters of typical travel patterns. • Proposed approach performed better than k-medoids, spectral, and hierarchical. Limited studies exist in the literature on demand related travel patterns, the analysis of which requires a rich database of Origin Destination (OD) matrices with appropriate clustering algorithms. This paper develops a methodological framework to explore typical travel patterns from multi-density high dimensional matrices and estimate typical OD corresponding to those patterns. The contributions of the paper are multi-fold. First, to cluster high-dimensional OD matrices, we deploy geographical window-based structural similarity index (GSSI) as proximity measure in the DBSCAN algorithm that captures both OD structure and network related attributes. Second, to address the issue of multi-density data points, we propose clustering on individual subspaces. Third, we develop a simple two-level approach to identify optimum DBSCAN parameters. Finally, as proof-of-concept, the proposed framework is applied on proxy OD matrices from real Bluetooth data (B-OD) from Brisbane City Council region. The OD matrix clusters, typical travel patterns, and typical B-OD matrices are estimated for this study region. The analysis reveals nine typical travel patterns. The methodology was also found to perform better when GSSI was used instead of Euclidian distance as a proximity measure, and two-level DBSCAN instead of K-medoids, Spectral, and Hierarchical methods. The framework is generic and applicable for OD matrices developed from other data sources and any spatiotemporal context. DBSCAN is chosen for this study because it does not require a pre-determined number of clusters, and it identifies outliers as noise. [ABSTRACT FROM AUTHOR]

Fuzzy orders, especially T -preorders, can express the vague priority over a list of alternatives. The cardinal consistency of such relations is achieved through the T -transitivity condition, while the ordinal consistency of the final decision at a given threshold α ∈ (0 , 1 ] is not guaranteed by the T -consistency. This paper investigates the problem of ordinal consistency, as the minimum requirement for a reliable judgment, of a partial preference induced by a fuzzy relation at a given level α ∈ (0 , 1 ] , so-called α -preference relation. We first define new concepts of T − L -cyclic and T -consistency at a given level α for fuzzy relations. Based on digraph theory, a new methodology is designed for finding the locations of all consistent and inconsistent L -cycles of an α -preference. An algorithm is proposed to eliminate the ordinal inconsistency through the T M -transitivity. Meanwhile, a new T -consistency index is introduced to measure the acceptable consistency level of fuzzy relations. Furthermore, some results concerning the ordinal consistency of T -preorders are applied to design another algorithm for creating a consistent collective judgment based on initial fuzzy assessments of alternatives in a group ranking problem. The proposed method constructs the consistent fuzzy order matrix for each case (expert), then calculates the group consensus by aggregating these matrices. The collective fuzzy order is then checked for consistency level. Finally, a numerical example with a real dataset is given to illustrate the application of the proposed method in a ranking problem. • New definitions of (in)consistency for fuzzy relations are proposed. • We consider the link between these concepts and L-cycles in the associated digraph. • Design of new algorithms to find and eliminate all inconsistent cycles of judgment. • A new consistency index, called T-consistency, is defined. • Construction of a consistent judgment based on a total fuzzy T-preorder. [ABSTRACT FROM AUTHOR]

Mean Centering (MC) is an important data preprocessing technique, which has a wide range of applications in data mining, machine learning, and multivariate statistical analysis. When the data set is large, this process will be time-consuming. In this paper, we propose an efficient quantum MC algorithm based on the block-encoding technique, which enables the existing quantum algorithms can get rid of the assumption that the original data set has been classically mean-centered. Specifically, we first adopt the strategy that MC can be achieved by multiplying by the centering matrix C , i.e., removing the row means, column means and row-column means of the original data matrix X can be expressed as X C , C X and C X C , respectively. This allows many classical problems involving MC, such as Principal Component Analysis (PCA), to directly solve the matrix algebra problems related to X C , C X or C X C. Next, we can employ the block-encoding technique to realize MC. To achieve it, we first show how to construct the block-encoding of the centering matrix C , and then further obtain the block-encodings of X C , C X and C X C. Finally, we describe one by one how to apply our MC algorithm to PCA and other algorithms. [ABSTRACT FROM AUTHOR]

In order to obtain better denoising results, this paper proposes the Robust Low-Rank Analysis with Adaptive Weighted Tensor (AWTD) method for image denoising tasks. On one hand, it uses the latest adaptive weight tensor, which obtains the low-rank approximation of the tensor by adding adaptive weights to the unfolding matrix of the tensor. The adaptive weight tensor can effectively retain useful singular values and better preserve the low-rank properties of the unfolding matrix. On the other hand, the proposed algorithm considers the spatial information and spectral information at the same time: for the RGB images, it retains the structural information inside the image patch and the connection between different channels (the spatial information of the image); for the hyperspectral images, it also retains the spectral information of the hyperspectral images. The experimental results show that the proposed method is superior to other test methods. • The adaptive weight tensor can effectively retain useful singular values and better preserve the low-rank properties of the unfolding matrix. • The proposed algorithm considers the spatial information and spectral information at the same time. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approximation algorithm for solving generalized Lyapunov matrix equations. We also present a convergence analysis for this algorithm. In each step of this algorithm two standard Lyapunov matrix equations with real coefficient matrices should be solved. Then we determine the optimal parameter to minimize the corresponding spectral radius of iteration matrix to obtain fastest speed of convergence. Finally some numerical examples are given to prove the capability of the present algorithm and a comparison is made with the existing results. [ABSTRACT FROM AUTHOR]

Randomized block coordinate descent type methods have been demonstrated to be efficient for solving large-scale optimization problems. Linear convergence to the unique solution is established if the objective function is strongly convex. In this paper we propose a randomized block coordinate descent algorithm for solving the matrix least squares problem min X ∈ R m × n ‖ C − AXB ‖ F with A ∈ R p × m , B ∈ R n × q , and C ∈ R p × q . We prove that the proposed algorithm converges linearly to the unique minimum norm least squares solution (i.e., A † C B † ) without the strong convexity assumption. Instead, we only need that B has full row rank. Numerical experiments are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Link prediction refers to predicting the connection probability between two nodes in terms of existing observable network information, such as network structural topology and node properties. Although traditional similarity-based methods are simple and efficient, their generalization performance varies widely in different networks. In this paper, we propose a novel link prediction approach ICP based on inductive matrix completion, which recoveries node connection probability matrix by applying node features to a low-rank matrix. The approach first explores a comprehensive node feature representation by combining different structural topology information with node importance properties via feature construction and selection. The selected node features are then used as the input of a supervised learning task for solving the low-rank matrix. The node connection probability matrix is finally recovered by a bi-linear function, which predicts the connection probability between two nodes with their features and the low-rank matrix. In order to demonstrate the ICP superiority, we took eleven related efforts including two recent methods proposed in 2020 as baseline methods, and it is shown that ICP has stable performance and good universality in twelve different real networks. Compared with the baseline methods, the improvements of ICP in terms of the average AUC results are ranging from 3.81% ∼ 12.77% and its AUC performance is improved by 0.08% ∼ 3.54% compared with the best baseline method. The limitation of ICP lies in its high computational complexity due to the feature construction, but the complexity can be reduced by replacing complex features with node semantic attributes if there are additional data available. Moreover, it provides a potential link prediction solution for large-scale networks, since inductive matrix completion is a supervised learning task, in which the underlying low-rank matrix can be solved by representative nodes instead of all their nodes. • A novel link prediction approach based on inductive matrix completion is proposed. • A comprehensive node feature representation is employed. • Link prediction is based on the node features and the factorized matrices. • Comparisons have been done with both classical and state-of-the-art methods. • The approach provides a potential link prediction solution for large-scale networks. [ABSTRACT FROM AUTHOR]

Many classical synchronization problems such as the assembly line crew scheduling problem (ALCS), some data association problems or multisensor tracking problems can be formulated as finding intra-column rearrangements for a single matrix representing costs, distances, similarities or time requirements. In this paper, we consider an extension of these problems to the case of multiple matrices, reflecting various possible instances (scenarios). To approximate optimal rearrangements, we introduce the Block Swapping Algorithm (BSA) and a further customization of it that we call the customized Block Swapping Algorithm (Cust BSA). A numerical study shows that the two algorithms we propose – in particular Cust BSA – yield high-quality solutions and also deal efficiently with high-dimensional set-ups. [ABSTRACT FROM AUTHOR]

• In this paper, we proposed an algorithm to solve mixed solutions of the matrix Equation ∑ i = 1 t A i X i B i = E with sub-matrix constraints. We also prove that the iterative solution sequence generated by the algorithm is convergent. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case, for both small-scale and large-scale matrices. The work belongs to the field of numerical algebra, and has been widely concerned. We put forward and analyze in details an iterative method to find the mixed solutions of a matrix equation with sub-matrix constraints. The convergence of the approximated solution sequence generated by the iterative method is investigated, showing that if the constrained matrix equation is consistent, the mixed solution group can be obtained after a finite number of iterations. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case for both small-scale and large-scale matrices. [ABSTRACT FROM AUTHOR]

• A new latent variables model which is a combination of linear and nonlinear models is introduced for data. • A novel deep neural network structure is designed to solve the matrix completion problem. • A new regularization technique is proposed to mitigate over-fitting. • Reconstruction accuracy and regularizer quality is evaluated. Matrix completion has received a vast amount of attention and research due to its wide applications in various study fields. Existing methods of matrix completion consider only nonlinear (or linear) relations among entries in a data matrix and ignore linear (or nonlinear) relationships latent. This paper introduces a new latent variables model for the data matrix which is a combination of linear and nonlinear models and designs a novel deep-neural-network-based matrix completion algorithm to address both linear and nonlinear relations among the entries of the data matrix. The proposed method consists of two branches. The first branch learns the latent representations of columns and reconstructs the columns of the partially observed matrix through a series of hidden neural network layers. The second branch does the same for the rows. In addition, based on multi-task learning principles, we enforce these two branches work together and introduce a new regularization technique to reduce over-fitting. More specifically, the missing entries of data are recovered as a main task and manifold learning is performed as an auxiliary task. The auxiliary task constrains the weights of the network; therefore, it can be considered as a regularizer, improving the main task and reducing over-fitting. Experimental results obtained on synthetic data and several real-world data verify the effectiveness of the proposed method compared with state-of-the-art matrix completion methods. [ABSTRACT FROM AUTHOR]

• Derivation of rank-Greville algorithm, maintaining a general rank factorization. • Exploiting rank-deficiency to reach an asymptotic computational complexity of O(mr) for updating a least-squares solution when adding an observation. • Publically available implementation in Python3, using Numpy. Updating a linear least-squares solution can be critical for near real-time signal-processing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A ∈ R n × m with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least-squares algorithm exploiting the rank-deficiency of A , achieving the update of the minimum-norm least-squares solution in O (m r) operations and, therefore, solving the linear least-squares problem from scratch in O (n m r) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity. [ABSTRACT FROM AUTHOR]