Non-negativity Constraints Research Articles

Bounded Degree Nonnegative Counting CSP

Constraint satisfaction problems (CSP) encompass an enormous variety of computational problems. In particular, all partition functions from statistical physics, such as spin systems, are special cases of counting CSP (#CSP). We prove a complete complexity classification for every counting problem in #CSP with nonnegative valued constraint functions that is valid when every variable occurs a bounded number of times in all constraints. We show that, depending on the set of constraint functions \(\mathcal {F} \) , every problem in the complexity class # \(\text{CSP}(\mathcal {F}) \) defined by \(\mathcal {F} \) is either polynomial-time computable for all instances without the bounded occurrence restriction, or is #P-hard even when restricted to bounded degree input instances. The constant bound in the degree depends on \(\mathcal {F} \) . The dichotomy criterion on \(\mathcal {F} \) is decidable. As a second contribution, we prove a slightly modified but more streamlined decision procedure (from [14]) to test for the tractability of # \(\text{CSP}(\mathcal {F}) \) . This procedure on an input \(\mathcal {F} \) tells us which case holds in the dichotomy for # \(\text{CSP}(\mathcal {F}) \) . This more streamlined decision procedure enables us to fully classify a family of directed weighted graph homomorphism problems. This family contains both P-time tractable problems and #P-hard problems. To our best knowledge, this is the first family of such problems explicitly classified that are not acyclic , thereby the Lovász-goodness criterion of Dyer-Goldberg-Paterson [24] cannot be applied.

Moving force identification based on the nonnegative flexible conjugate gradient least square method and experimental verification

Moving force identification (MFI) is an approach aimed at acquiring the time history of unknown moving loads acting on a bridge, which is actually an inverse problem. The accuracy and efficiency of MFI are significantly influenced by the ill-posedness problem during the process of solving the vehicle-bridge interaction system, especially when the vehicle axles getting on or getting off the bridge. The objective of the study is thus to reduce the effect of ill-posedness problem on the accuracy and efficiency of MFI through numerical and experimental methods. A nonnegative flexible conjugate gradient least square (NN-FCGLS) method is first developed through enriching the conjugate gradient least square method with the flexible Krylov subspace technique and nonnegative constraint conditions. Then, the solution of the NN-FCGLS method is continuously updated with the Krylov subspace technique and the Karush-Kuhn-Tucker conditions at each iteration to obtain a stable solution for ill-posed dynamic force identification. Finally, laboratory experiments were carried out with a simply supported aluminum alloy beam. Numerical simulation results show that the NN-FCGLS method is insensitive to vehicle speed and vehicle wheelbase, and has better accuracy, stronger robustness and higher computational efficiency than the time domain method and the conjugate gradient least square method. The relative percent errors of all ten cases do not exceed 7.8 % under the interference of 20 % noise level. The experimental results shows that the correlation coefficients between measured responses and reconstructed responses are larger than 0.92 in all cases. Experimental studies demonstrate the accuracy and efficiency of the proposed NN-FCGLS method in the practice of MFI.

Semi-supervised multi-view concept decomposition

Concept Factorization (CF), as a novel paradigm of representation learning, has demonstrated superior performance in multi-view clustering tasks. It overcomes limitations such as the non-negativity constraint imposed by traditional matrix factorization methods and leverages kernel methods to learn latent representations that capture the underlying structure of the data, thereby improving data representation. However, existing multi-view concept factorization methods fail to consider the limited labeled information inherent in real-world multi-view data. This often leads to significant performance loss. To overcome these limitations, we propose a novel semi-supervised multi-view concept factorization model, named SMVCF. In the SMVCF model, we first extend the conventional single-view CF to a multi-view version, enabling more effective exploration of complementary information across multiple views. We then integrate multi-view CF, label propagation, and manifold learning into a unified framework to leverage and incorporate valuable information present in the data. Additionally, an adaptive weight vector is introduced to balance the importance of different views in the clustering process. We further develop targeted optimization methods specifically tailored for the SMVCF model. Finally, extensive experiments on four diverse datasets with varying label ratios demonstrated the effectiveness and superiority of the proposed SMVCF in multi-view clustering tasks.

Discretized tensor-based model of total focusing method: A sparse regularization approach for enhanced ultrasonic phased array imaging

The total focusing method (TFM) is considered as the standard in ultrasonic phased array imaging and plays a vital role in industrial non-destructive testing (NDT). By utilizing the full matrix capture (FMC) dataset, the TFM can focus on every point within the specified imaging region, and is more accurate than the traditional ultrasonic phased array imaging methods. However, the TFM is essentially a delay and sum technique that often operates linearly on the time-domain signals and takes no prior information into account, so its image quality remains inadequate when dealing with defects in close proximity or scattering materials. To address this problem, this paper formulates the imaging principle of the TFM as a Boolean matrix and establishes the discretized tensor-based model. Subsequently, the model is addressed by employing the sparse regularization strategy, taking into some characteristics of industrial NDT. Regarding the solution algorithm, due to the generation of negative values by the fast iterative shrinkage threshold algorithm (FISTA), this paper introduces the rectified linear unit (ReLU) function as a non-negative constraint and presents a dedicated solution algorithm (ReLU-FISTA) for acquiring detection results. Through verification of simulation and experiment, the proposed approach exhibits superior capabilities of defect characterization and noise suppression when compared to the TFM, leading to substantial enhancements in image quality.

Fast Robust Matrix Completion via Entry-Wise l0 -Norm Minimization.

Matrix completion (MC) aims at recovering missing entries, given an incomplete matrix. Existing algorithms for MC are mainly designed for noiseless or Gaussian noise scenarios and, thus, they are not robust to impulsive noise. For outlier resistance, entry-wise lp -norm with and M-estimation are two popular approaches. Yet the optimum selection of p for the entrywise lp -norm-based methods is still an open problem. Besides, M-estimation is limited by a breakdown point, that is, the largest proportion of outliers. In this article, we adopt entrywise l0 -norm, namely, the number of nonzero entries in a matrix, to separate anomalies from the observed matrix. Prior to separation, the Laplacian kernel is exploited for outlier detection, which provides a strategy to automatically update the entrywise l0 -norm penalty parameter. The resultant multivariable optimization problem is addressed by block coordinate descent (BCD), yielding l0 -BCD and l0 -BCD-F. The former detects and separates outliers, as well as its convergence is guaranteed. In contrast, the latter attempts to treat outlier-contaminated elements as missing entries, which leads to higher computational efficiency. Making use of majorization-minimization (MM), we further propose l0 -BCD-MM and l0 -BCD-MM-F for robust non-negative MC where the nonnegativity constraint is handled by a closed-form update. Experimental results of image inpainting and hyperspectral image recovery demonstrate that the suggested algorithms outperform several state-of-the-art methods in terms of recovery accuracy and computational efficiency.

Estimating ensemble weights for bagging regressors based on the mean–variance portfolio framework

This paper presents a novel ensemble learning framework inspired by modern portfolio optimization to address regression problems. This formulation in the ensemble learning field allows determining the ensemble’s weights considering the error predictions of the base learners, the variability in their predictions, and the covariance among their predictions’ errors, which is completely aligned with the bias–variance–covariance theory. Under the framework proposed, four potential instantiations have also been provided. The first two ensemble models impose the non-negativity constraints on the ensemble’s weights (along with the equality constraint that the ensemble’s weights sum up to one) and are solved with the active set algorithm. The second two ensemble models do not include the non-negativity constraints on the ensemble’s weights (which in the financial literature are called the non-shorting constraints), giving rise to a convex quadratic programming (QP) problem (as the matrix included in the quadratic term is symmetric and positive definite) that is solved by the Lagrangian procedure. Extensive experiments with regression datasets evaluate the proposed ensemble framework. Comparisons with other state-of-the-art ensemble methods confirm that the ensemble framework yields the best overall performance.

Secrecy-Capacity Bounds for Visible Light Communications With Signal-Dependent Noise

In physical-layer security, secrecy capacity is an important performance metric. This work aims to determine the secrecy capacity for an indoor visible light communication system consisting of a transmitter, a legitimate receiver and an eavesdropping receiver. In such a system, both signal-independent noise and signal-dependent noise are considered. Under nonnegativity and average optical intensity constraints, lower and upper bounds on secrecy capacity are derived by the variational method, the dual expression of the secrecy capacity, and the concept of “the optimal input distribution that escapes to infinity”. By an asymptotic analysis at large optical intensity, there is a small gap between the asymptotic upper and lower bounds. Then, by adding a peak optical intensity constraint, we further analyze the exact and asymptotic secrecy-capacity bounds. For practical considerations, the effects of imperfect channel state information, multi-photodiode eavesdropper, and artificial noise on secrecy performance are also discussed. Finally, the derived secrecy-capacity bounds are verified by numerical results.

Source characteristics of polycyclic aromatic hydrocarbons and polychlorinated biphenyls in surface soils of Shenyang, China: A comparison of two receptor models combined with Monte Carlo simulation

The surface soil concentrations of 16 PAHs and 15 PCBs were simultaneously determined by gas chromatography-tandem mass spectrometry in 21 locations of urban areas of Shenyang. The average concentrations of PAHs and PCBs were 26.40 ± 34.68 mg/kg and 48.03 ± 27.47 μg/kg, respectively. Factor analysis with nonnegative constraints (FA-NNC) and absolute principal component score with multiple linear regression (APCS-MLR) model were used to explore and evaluate the sources of PAHs and PCBs in the study area. The results of FA-NNC showed that PAHs in soils were mainly from traffic emissions (49.64%), coal combustion (46.88%) and petrogenic source (3.49%). The PCBs in soils were mainly from commercial and high temperature combustion mixed sources (20.3%), combustion and industry emission mixed sources (21.1%), electrical equipment sources (22.2%) and traffic emission sources (36.4%). The results of APCS-MLR were consistent with those of FA-NNC. The uncertainty of FA-NNC and APCS-MLR model was analyzed by Monte Carlo simulation method. The results revealed the reliability of the two receptor models on source apportionment. The estimated carcinogenic risks indicated that the risks of PAHs in soils exceed the acceptable range (10−6-10−4), while the risks of PCBs were below the acceptable risk level of 10−6.

NT-DPTC: A non-negative temporal dimension preserved tensor completion model for missing traffic data imputation

Missing traffic data imputation is an important step in the intelligent transportation systems. Low rank approximation is an important method for the missing traffic data imputation, especially the low rank matrix/tensor completion. However, the low-rank matrix completion has a huge time cost and the low-rank tensor completion cannot fully extract the information and maintain non-negativity. Therefore, we propose a novel non-negative temporal dimension preserved tensor completion (NT-DPTC) model with ideas: a) proposing a novel dimension preserved (DP) tensor decomposition method, which decomposes a tensor into three latent factor tensors to fully extract the intrinsic feature, b) using a Sigmoid mapper for releasing non-negative constraint from the training process and increasing flexibility, c) exploiting temporal constrains and AdamW schemes for obtaining a higher accuracy and faster convergence. Experiments on four industrial application scenarios highlight the superiority of our proposed model when compared with the existing state-of-the-art models.

Smooth deconvolution of low-field NMR signals

BackgroundLow resolution nuclear magnetic resonance (LR-NMR) is a common technique to identify the constituents of complex materials (such as food and biological samples). The output of LR-NMR experiments is a relaxation signal which can be modelled as a type of convolution of an unknown density of relaxation times with decaying exponential functions, plus random Gaussian noise. The challenge is to estimate that density, a severely ill-posed problem. A complication is that non-negativity constraints need to be imposed in order to obtain valid results. Significance and noveltyWe present a smooth deconvolution model for solution of the inverse estimation problem in LR-NMR relaxometry experiments. We model the logarithm of the relaxation time density as a smooth function using (adaptive) P-splines while matching the expected residual magnetisations with the observed ones. The roughness penalty removes the singularity of the deconvolution problem, and the estimated density is positive by design (since we model its logarithm). The model is non-linear, but it can be linearized easily. The penalty has to be tuned for each given sample. We describe an efficient EM-type algorithm to optimize the smoothing parameter(s). ResultsWe analyze a set of food samples (potato tubers). The relaxation spectra extracted using our method are similar to the ones described in the previous experiments but present sharper peaks. Using penalized signal regression we are able to accurately predict dry matter content of the samples using the estimated spectra as covariates.

Surface analysis insight note: Multivariate curve resolution of an X‐ray photoelectron spectroscopy image

This Insight Note follows a series of three previous insight notes on X‐ray photoelectron spectroscopy image analysis that focused on the importance of analyzing the raw data, the use of summary statistics, and principal component analysis (PCA). The same X‐ray photoelectron spectroscopy image data set was analyzed in all three notes. We now show an analysis of this same data set using multivariate curve resolution (MCR). MCR is a widely used exploratory data analysis method. Because of MCR's nonnegativity constraints, it has the important advantage of producing factors that look like real spectra. That is, both its scores and loadings are positive, so its results are often more interpretable than those from PCA. The requirements for preprocessing data are also, in general, lower for MCR compared with PCA. To help determine the number of factors that best describe the data set, a series of MCR models with different numbers of factors was created. Based on the chemical reasonableness of its factors, a two‐factor model was selected. Scores plots/images show the regions of the image that correspond to these two factors.

Direction-of-arrival and power spectral density estimation using a single directional microphone and group-sparse optimization

In this paper, two approaches are proposed for estimating the direction of arrival (DOA) and power spectral density (PSD) of stationary point sources by using a single, rotating, directional microphone. These approaches are based on a method previously presented by the authors, in which point source DOAs were estimated by using a broadband signal model and solving a group-sparse optimization problem, where the number of observations made by the rotating directional microphone can be lower than the number of candidate DOAs in an angular grid. The DOA estimation is followed by the estimation of the sources’ PSDs through the solution of an overdetermined least squares problem. The first approach proposed in this paper includes the use of an additional nonnegativity constraint on the residual noise term when solving the group-sparse optimization problem and is referred to as the Group Lasso Least Squares (GL-LS) approach. The second proposed approach, in addition to the new nonnegativity constraint, employs a narrowband signal model when building the linear system of equations used for formulating the group-sparse optimization problem, where the DOAs and PSDs can be jointly estimated by iterative, group-wise reweighting. This is referred to as the Group-Lasso with l_1-reweighting (GL-L1) approach. Both proposed approaches are implemented using the alternating direction method of multipliers (ADMM), and their performance is evaluated through simulations in which different setup conditions are considered, ranging from different types of model mismatch to variations in the acoustic scene and microphone directivity pattern. The results obtained show that in a scenario involving a microphone response mismatch between observed data and the signal model used, having the additional nonnegativity constraint on the residual noise can improve the DOA estimation for the case of GL-LS and the PSD estimation for the case of GL-L1. Moreover, the GL-L1 approach can present an advantage over GL-LS in terms of DOA estimation performance in scenarios with low SNR or where multiple sources are closely located to each other. Finally, it is shown that having the least squares PSD re-estimation step is beneficial in most scenarios, such that GL-LS outperformed GL-L1 in terms of PSD estimation errors.

Reduction to the pole near the equator with a Fourier-domain equivalent dipole layer

Reduction to the pole (RTP) transforms the phase of the observed total-field anomaly, making magnetic interpretation easier. However, RTP operations are notoriously unstable at low latitudes and produce singularities at the equator. The equivalent-source approach is an appropriate method to address the well-known difficulties of RTP at low latitudes. Although we routinely use equivalent-source RTP algorithms in the spatial domain, they consume massive computational resources. In contrast, the equivalent-source method in the Fourier domain is more efficient and also invalid for RTP because of the inevitable use of the RTP operator in the inversion. To overcome this problem, we introduce a simple approximation of the RTP operator instead and then develop a novel algorithm to calculate the magnetic moments of the equivalent dipole layer from the total-field anomaly near the equator. Because the calculations are mainly performed in the Fourier domain, this algorithm is more efficient than the conventional spatial-domain method. We also impose a nonnegative constraint on the dipole layer to improve the results and significantly reduce the undesired striation associated with the low-latitude RTP. We evaluate the effectiveness of our approach with synthetic data near the equator, and the results indicate that the algorithm can produce an RTP field that is an accurate representation of the real field at the pole, even when accounting for the existence of noise. The new approach is applied to field data collected near the geomagnetic equator in the South China Sea. In the central part of the anomaly, we obtain a robust spindle shape of the RTP field, which is identical to the shape of a basalt formation in that location. Recent 1:1,000,000 scale marine geologic mapping has confirmed this presence, proving that our solution is an applicable approach to the problem.

Multiview Deep Matrix Factorization for Shared Compact Representation

Multiview learning aims to learn beneficial patterns from heterogeneous data sources and has captured growing attention in recent years. Most of the previous research studies focused on searching for an effective feature embedding of downstream tasks using diverse optimization algorithms, however, very limited work has been conducted to explore the connection between multiview learning and deep neural networks of structure sharing hidden layers. In this article, we propose a multiview deep matrix factorization model to learn a shared compact representation from multiview data. First, the proposed model constructs a multiview auto-encoder architecture with one shared encoder and multiple decoders, where each view corresponds to a factorization and the shared encoder leads to a common hidden layer. Accordingly, matrix factorizations from multiview data share the last hidden layer for a high-level semantic representation. Second, the nonnegativity constraint of the learned representation is transformed to the projection operation, which can be easily achieved by activating weights of the shared encoder network. Third, this network is trained with a joint loss of the reconstruction error and the compactness loss. By employing the clustering layer, the proposed method serves as an end-to-end multiview clustering method. Finally, comprehensive experiments on nine real-world datasets demonstrate the superiority of the proposed method against state-of-the-art multiview clustering methods.

An Adaptive Divergence-Based Non-Negative Latent Factor Model

A High-dimensional and incomplete (HDI) matrix is regularly adopted to portray the inherent non-negativity of interactions among numerous nodes, which is involved in countless industrial applications driven by big data. An inherently non-negative latent factor (LF) model can take out the intrinsical features from such data conveniently and effectually due to its unimpeded training process. However, it constructs the learning objective relying on a standard Euclidean distance, thereby seriously restricting its representative ability to HDI data generated by different domains. To address this issue, this work proposes an adaptive divergence-based non-negative LF (ADNLF) model following: 1) constructing a generalized objective function based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <inline-formula> <tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\beta$</tex-math> </inline-formula> </i> -divergence to inflate its ability to represent various HDI data; 2) connecting the optimization variables with output LFs by a smooth and single LF-dependent bridging function to satisfy the non-negativity constraints constantly; and 3) facilitating adaptive divergence in the learning objective through particle swarm optimization for high scalability. Empirical studies on eight HDI matrices validate that an ADNLF model evidently outstrips state-of-the-art models in terms of estimation accuracy as well as computational efficiency for missing data of an HDI dataset.

Multi-view clustering with Laplacian rank constraint based on symmetric and nonnegative low-rank representation

Multi-view spectral clustering has attracted more and more attention due to its promising advantages in data clustering, and many related methods have been proposed, among which the method based on low-rank representation have become increasingly popular. However, the similarity matrices learned by the existing methods inevitably contain some negative entries and are generally nonsymmetric. In order to guarantee weight consistency for each pair of data points and highly correlated data points to be together, we propose a new multi-view spectral clustering model based on the symmetric and nonnegative low-rank representation. This model can learn the view-specific consistent similarity matrix and also capture the local structure and discriminative information among data by imposing symmetric and nonnegative constraints on the representation matrix. By employing the compatible and complementary information from multi-view data points, it can also learn the consensus similarity matrix for all views and perform the spectral clustering simultaneously. By the augmented Lagrangian multiplier method, an alternating iterative optimization algorithm is designed for solving this model and its computational complexity and convergence is analyzed. Comprehensive experiments on multiple multi-view data sets also verify the effectiveness and superiority of the proposed approach.

On Recovery of a Non-Negative Relaxation Spectrum Model from the Stress Relaxation Test Data.

The relaxation spectra, from which other material functions used to describe mechanical properties of materials can be uniquely determined, are important for modeling the rheological properties of polymers used in chemistry, food technology, medicine, cosmetics, and many other industries. The spectrum, being not directly accessible by measurement, is recovered from relaxation stress or oscillatory shear data. Only a few models and identification methods take into account the non-negativity of the real spectra. In this paper, the problem of recovery of non-negative definite relaxation spectra from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test is considered. A new hierarchical identification scheme is developed, being applicable both for relaxation time and frequency spectra. Finite-dimensional parametric classes of models are assumed for the relaxation spectra, described by a finite series of power-exponential and square-exponential basis functions. The related models of relaxation modulus are given by compact analytical formula, described by the products of power of time and the modified Bessel functions of the second kind for the time spectrum, and by recurrence formulas based on products of power of time and complementary error functions for frequency spectrum. The basis functions are non-negative. In result, the identification task was reduced to a finite-dimensional linear-quadratic problem with non-negative unknown model parameters. To stabilize the solution, an additional smoothing constraint is introduced. Dual approach was used to solve the stated optimal identification task resulting in the hierarchical two-stage identification scheme. In the first stage, dual problem is solved in two levels and the vector of non-negative model parameters is computed to provide the best fit of the relaxation modulus to experiment data. Next, in second stage, the optimal non-negative spectrum model is determined. A complete scheme of the hierarchical computations is outlined; it can be easily implemented in available computing environments. The model smoothness is analytically studied, and the applicability ranges are numerically examined. The numerical studies have proved that using developed models and algorithm, it is possible to determine non-negative definite unimodal and bimodal relaxation spectra for a wide class of polymers. However, the examples also demonstrated that if the basis functions are non-negative and the model is properly selected for a given type of the real spectrum (unimodal, multimodal), the optimal model determined without non-negativity constraint can be non-negative in the dominant range of its arguments, especially in the wide neighborhood of the spectrum peaks.

HL-nets: Physics-informed neural networks for hydrodynamic lubrication with cavitation

Recently, physics-informed neural networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs). In this study, we establish a deep learning computational framework, HL-nets, for computing the flow field of hydrodynamic lubrication involving cavitation effects. The Swift-Stieber (SS) and the Jakobsson-Floberg-Olsson (JFO) cavitation conditions are implemented in the PINNs to solve the Reynolds equation. For the non-negativity constraint of the SS cavitation condition, a penalizing scheme with a residual of the non-negativity and an imposing scheme with a continuous differentiable non-negative function are proposed. For the complementarity constraint of the JFO cavitation condition, the pressure and cavitation fraction are taken as the neural network outputs, and the residual of the Fischer-Burmeister (FB) equation constrains their complementary relationships. Multi-task learning (MTL) methods are applied to balance the loss terms of functions and constraints described above. To estimate the accuracy of HL-nets, we present a numerical solution of the Reynolds equation for oil-lubricated bearings involving cavitation. The results indicate that the proposed HL-nets can highly accurately simulate hydrodynamic lubrication involving cavitation phenomena. The imposing scheme can effectively improve the accuracy of the training results of PINNs, and it is expected to have great potential to be applied to different fields where the non-negativity constraint is needed.

A Multilevel Active-Set Preconditioner for Box-Constrained Pressure Poisson Solvers

Efficiently solving large-scale box-constrained convex quadratic programs (QPs) is an important computational challenge in physical simulation. We propose a new multilevel preconditioning scheme based on the active-set method and combine it with modified proportioning with reduced gradient projections (MPRGP) to efficiently solve such QPs arising from pressure Poisson equations with non-negative pressure constraints in fluid animation. Our method employs a purely algebraic multigrid method to ensure the solvability of the coarser level systems and to merge only algebraically-connected components, thereby avoiding performance degradation of the preconditioner. We present a filtering scheme to efficiently apply our multilevel preconditioning only to unconstrained subsystems of the pressure Poisson system while reusing the hierarchy constructed per simulation step. We demonstrate the effectiveness of our method over previous approaches in various examples.

Affine Subspace Robust Low-Rank Self-Representation: from Matrix to Tensor.

Low-rank self-representation based subspace learning has confirmed its great effectiveness in a broad range of applications. Nevertheless, existing studies mainly focus on exploring the global linear subspace structure, and cannot commendably handle the case where the samples approximately (i.e., the samples contain data errors) lie in several more general affine subspaces. To overcome this drawback, in this paper, we innovatively propose to introduce affine and nonnegative constraints into low-rank self-representation learning. While simple enough, we provide their underlying theoretical insight from a geometric perspective. The union of two constraints geometrically restricts each sample to be expressed as a convex combination of other samples in the same subspace. In this way, when exploring the global affine subspace structure, we can also consider the specific local distribution of data in each subspace. To comprehensively demonstrate the benefits of introducing two constraints, we instantiate three low-rank self-representation methods ranging from single-view low-rank matrix learning to multi-view low-rank tensor learning. We carefully design the solution algorithms to efficiently optimize the proposed three approaches. Extensive experiments are conducted on three typical tasks, including single-view subspace clustering, multi-view subspace clustering, and multi-view semi-supervised classification. The notably superior experimental results powerfully verify the effectiveness of our proposals.