Norm Minimization Method Research Articles

Efficient enhancement of low-rank tensor completion via thin QR decomposition.

Low-rank tensor completion (LRTC), which aims to complete missing entries from tensors with partially observed terms by utilizing the low-rank structure of tensors, has been widely used in various real-world issues. The core tensor nuclear norm minimization (CTNM) method based on Tucker decomposition is one of common LRTC methods. However, the CTNM methods based on Tucker decomposition often have a large computing cost due to the fact that the general factor matrix solving technique involves multiple singular value decompositions (SVDs) in each loop. To address this problem, this article enhances the method and proposes an effective CTNM method based on thin QR decomposition (CTNM-QR) with lower computing complexity. The proposed method extends the CTNM by introducing tensor versions of the auxiliary variables instead of matrices, while using the thin QR decomposition to solve the factor matrix rather than the SVD, which can save the computational complexity and improve the tensor completion accuracy. In addition, the CTNM-QR method's convergence and complexity are analyzed further. Numerous experiments in synthetic data, real color images, and brain MRI data at different missing rates demonstrate that the proposed method not only outperforms in terms of completion accuracy and visualization, but also conducts more efficiently than most state-of-the-art LRTC methods.

Sparse optimization over polyhedral sets based on weighted ℓ-norm minimization

Finding the sparsest solutions over a polyhedral set is a challenging problem that has received a great deal of attention in the research community. This problem is typically NP-hard owing to the non-convexity and non-continuity of the ℓ 0 -norm. In this study, it is demonstrated that any solution of ℓ 0 -minimization over a polyhedral set must be one of the ‘extreme efficient solutions’ (EESs) of a special multiobjective optimization problem. Moreover, a new weighted ℓ 1 -norm minimization method using the Goldman–Tucker Theorem for linear optimization problems is established. Some essential features of this algorithm are examined, and numerical tests are performed to illustrate the efficiency of the proposed strategy. The numerical performance of the proposed method is compared with conventional reweighted ℓ 1 -algorithms and several compressed sensing solvers. Additionally, the method is applied to address the compressed sensing problem in magnetic resonance imaging (MRI) reconstruction.

Morphing from the TV-Norm to the l 0 -Norm.

For piecewise constant objects, the images can be reconstructed with under-sampled measurements. The gradient image of a piecewise image is sparse. If a sparse solution is a desired solution, an -norm minimization method is effective to solve an under-determined system. However, the -norm is not differentiable, and it is not straightforward to minimize an -norm. This paper suggests a function that is like the -norm function, and we refer to this function as meta -norm. The subdifferential of the meta -norm has a simple explicit expression. Thus, it is straightforward to derive a gradient descent algorithm to enforce the sparseness in the solution. In fact, the proposed meta norm is a transition that varies between the TV-norm and the -norm. As an application, this paper uses the proposed meta -norm for few-view tomography. Computer simulation results indicate that the proposed meta -norm effectively guides the image reconstruction algorithm to a piecewise constant solution. It is not clear whether the TV-norm or the -norm is more effective in producing a sparse solution. Index Terms-Inverse problem, optimization, total-variation minimization, -norm minimization, few-view tomography, iterative algorithm, image reconstruction.

A fast matrix completion method based on truncated$ {\mathit{L}}_{2, 1} $ norm minimization

<abstract> <p>In recent years, a truncated nuclear norm regularization (TNNR) method has obtained much attention from researchers in machine learning and image processing areas, because it is much more accurate on matrices with missing data than other traditional methods based on nuclear norm. However, the TNNR method is reported to be very slow, due to its large number of singular value decomposition (SVD) iterations. In this paper, a truncated $ {\boldsymbol{L}}_\bf{2, 1} $ norm minimization method was presented for fast and accurate matrix completion, which is abbreviated as TLNM. In the proposed TLNM method, the truncated nuclear norm minimization model of TNNR was improved to a truncated $ {\boldsymbol{L}}_\bf{2, 1} $ norm minimization model that aimed to optimize the truncated $ {\boldsymbol{L}}_\bf{2, 1} $ Norm and a weighted noisy matrix simultaneously for improving the accuracy of TLNM. Using Qatar Riyal (QR) decomposition to calculate the orthogonal bases for reconstructing recovery results, the proposed TLNM method is much faster than the TNNR method. Adequate results for color images validate the effectiveness and efficiency of TLNM comparing with TNNR and other competing methods.</p> </abstract>

Failure diagnosis for time-modulated arrays based on compressed sensing

Time-modulated arrays (TMAs) have a high design degrees of freedom (DoFs) to improve radiation performance, while they are prone to failure due to their hardware characteristics. In this article, we propose a novel technique to diagnose impaired TMAs based on compressed sensing (CS). The TMA diagnosis problem is reformulated as a sparse signal recovery problem at the center frequency and sidebands. Then, a method based on the difference of convex sets theory and sequential convex programming (DCS-SCP) is developed to implement diagnosis for impaired TMAs. Using a small number of far-field measurements at the same position but different frequencies, the joint recovery of the equivalent excitations at the center frequency and sidebands is realized by a mixed l 0 / l 2 -norm minimization method. The numerical simulation and the successful comparison with the state-of-the-art algorithms demonstrate the superiority of the proposed methods in terms of noise robustness and diagnosis accuracy.

A mixed-field sources localization algorithm based on high-order cumulant matrix reconstruction for general symmetric array

Sparse arrays are able to generate more lags to extend the array aperture, which is a distinct advantage in mixed-field localization. To exploit these lags, existing algorithms in the known literature can be mainly divided into two types: the subspace-based algorithm and the sparsity-based algorithm. However, the former algorithm cannot fully utilize the time delay information provided by sparse array, and the second algorithm has basis mismatch problem. In this paper, an interpolation processing method based on atomic norm is proposed to solve the sparse array localization problem. The high-order cumulant matrix is reconstructed by the interpolation method to generate an augmented cumulant matrix without holes, which can make full use of all the lags. Then, the atomic norm minimization method is used to recover the sparse matrix after interpolation in a gridless way. The matrix after recovery enables gridless direction-of-arrival (DOA) estimation. After the interpolation reconstruction, more lags can be exploited, the degrees of freedom are further increased. The proposed algorithm can not only make full use of the array receiving information but also avoid the base mismatch problem, and the accuracy of DOA estimation is improved. Numerical simulations verify the superiority of the proposed algorithm compared with the existing algorithms.

A Fast Matrix Completion Method Based on Matrix Bifactorization and QR Decomposition

The problem of recovering the missing values in an incomplete matrix, i.e., matrix completion, has attracted a great deal of interests in the fields of machine learning and signal processing. A matrix bifactorization method, which is abbreviated as MBF, is a fast method of matrix completion that has a better speed than the traditional nuclear norm minimization methods. However, it may become inaccurate and slow when solving matrices of not low rank. In this paper, an improved fast and accurate MBF method based on Qatar Riyal (QR) decomposition is proposed, which can be called FMBF-QR. On one side, the optimization problem of MBF is improved to be an iteratively reweighted L 2 , 1 norm minimization problem to enhance the accuracy of MBF. On the other side, the minimization problem of FMBF-QR is optimized very efficiently by using QR decomposition for improving the speed of MBF. Sufficient experimental results verify that FMBF-QR can converge with a higher accuracy and a faster speed than the traditional matrix completion methods.

An Image Denoising Model via the Reconciliation of the Sparsity and Low-Rankness of the Dictionary Domain Coefficients

Sparse coding has achieved great success in various image restoration tasks. However, if the sparse representation coefficients of the structure (low-frequency information) and texture (high-frequency information) components of the image are under the same penalty constraint, the restoration effect may not be ideal. In this paper, an image denoising model combining mixed norm and weighted nuclear norm as regularization terms is proposed. The proposed model simultaneously exploits the group sparsity of the high frequency and low-rankness of the low frequency in dictionary-domain. The mixed norm is used to constrain the high frequency part and the weighted nuclear norm is used to constrain the low frequency part. In order to make the proposed model easy to solve under the framework of alternative direction multiplier method (ADMM), iterative shrinkage threshold method and weighted nuclear norm minimization method are used to solve the two sub-problems. The validity of the model is verified experimentally through comparison with some state-of-the-art methods.

Structure-guided computed tomography reconstruction from limited-angle projections.

Limited-angle computed tomography (CT) imaging is one of the common imaging problems. The reconstructed images often encounter obvious artifacts and structure degradation. In recent years, the recoverability prior of image structure has been widely explored in limited-angle CT reconstruction, and the image quality has been greatly improved. However, the artifacts and structure degradation still exist. In this study, we establish a new reconstruction model based on weighted relative structure (wRS) determined by image gradients, which serves as weights to guide image reconstruction in order to reduce artifacts and preserve structures. Then, we develop an efficient algorithm using a surrogate function to solve this model. Moreover, this method is compared with some of other popular reconstruction methods, such as anisotropic total variation method and image gradient L0 norm minimization method and so on. Experiments on digital phantoms, real carved cheese and walnut projection are reported to demonstrate its superiority. Several quantitative indices including RMSE, PSNR, and SSIM of the reconstruction images from 90°-data of FORBILD head phantom are 0.0120, 43.52, and 0.9961. The experimental results indicate that the image obtained by our method is the closest to reference image. By comparing reconstruction images or their residual images, images reconstructed from real CT data, the experimental results of the residual images and the respective quantitative data analysis also demonstrate that the images reconstructed using our new method suffer from less artifacts and structure degradation.

A Novel Gradient Descent Least-Squares (GDLSs) Algorithm for Efficient Gridless Line Spectrum Estimation With Applications in Tomographic SAR Imaging

This paper presents a novel efficient method for gridless line spectrum estimation problem with single snapshot and sparse signals, namely the gradient descent least squares (GDLS) method. Conventional single snapshot (a.k.a. single measure vector or SMV) line spectrum estimation methods either rely on smoothing techniques that sacrificing the range and/or azimuth resolution, or adopt the sparsity constraint and utilize compressed sensing (CS) method by defining prior grids and resulting in the off-grid problem. Recently emerged atomic norm minimization (ANM) methods achieved gridless SMV line spectrum estimation, but its computational complexity is extremely high; thus it is practically infeasible in real applications with large problem scales. Our proposed GDLS method reformulates the line spectrum estimations problem into a least squares (LS) estimation problem and solves the corresponding objective function via gradient descent algorithm in an iterative fashion with efficiency. The convergence guarantee, computational complexity, as well as performance analysis for evenly distributed antenna array case are discussed in this paper. Numerical simulations show that the proposed GDLS algorithm outperforms the state-of-the-art methods e.g., CS and ANM, in terms of estimation performances. It can completely avoid the off-grid problem, and its computational complexity is significantly lower than ANM. Our method has been tested in tomographic SAR (TomoSAR) imaging applications via simulated and real experiment data. Results show great potential of the proposed method in terms of better cloud point performance and eliminating the gridding effect.

DOD and DOA Estimation From Incomplete Data Based on PARAFAC and Atomic Norm Minimization Method

In this article, we propose an efficient direction of departure (DOD) and direction of arrival (DOA) estimation method for bistatic multiple-input multiple-output (MIMO) radar with faulty arrays. A third-order tensor model is built, and the measurement 3-D structure can be better utilized than the traditional matrix model. Subsequently, the atomic norm minimization (ANM) technique is used to further improve the angle estimation performance. Furthermore, we found in the research process that when the faulty arrays still maintain the symmetry property, the measurement tensor can be converted to the real-valued domain by the forward–backward averaging technique and the unitary transform technique. The new algorithm we proposed exploits the multidimensional structure of the signal without estimating the signal subspace. Comparing with traditional matrix completion (MC) methods, it has a better performance in terms of robustness and resolving correlated targets. Also, the algorithm proposed in this article does not require angle pairing. Simulation results verify the effectiveness of the proposed algorithm.

Robust low tubal rank tensor completion via factor tensor norm minimization

Recent research has demonstrated that low tubal rank recovery based on tensor has received extensive attention. In this correspondence, we define tensor double nuclear norm and tensor Frobenius/nuclear hybrid norm to induce a surrogate for tensor tubal rank, and prove that they are equivalent to tensor Schatten-p norm for p=1/2 and p=2/3. Based on the definition, we propose two novel tractable tensor completion models called Double Nuclear norm regularized Tensor Completion (DNTC) and Frobenius/Nuclear hybrid norm regularized Tensor Completion (FNTC) by integrating these two norm minimization and factorization methods into a joint learning framework. Furthermore, we adopt invertible linear transforms to obtain low tubal rank tensors, which makes the model more flexible and effective. Two efficient algorithms are designed to solve the proposed tensor completion models by incorporating the convexity of the factor norms. Comprehensive experiments are conducted on synthetic and real datasets to achieve better results in comparison with some state-of-the-art approaches.

MMV-Based Sequential AoA and AoD Estimation for Millimeter Wave MIMO Channels

The fact that the millimeter-wave (mmWave) multiple-input multiple-output (MIMO) channel has sparse support in the spatial domain has motivated recent compressed sensing (CS)-based mmWave channel estimation methods, where the angles of arrivals (AoAs) and angles of departures (AoDs) are quantized using angle dictionary matrices. However, the existing CS-based methods usually obtain the estimation result through one-stage channel sounding that have two limitations: (i) the requirement of large-dimensional dictionary and (ii) unresolvable quantization error. These two drawbacks are irreconcilable; improvement of the one implies deterioration of the other. To address these challenges, we propose, in this paper, a two-stage method to estimate the AoAs and AoDs of mmWave channels. In the proposed method, the channel estimation task is divided into two stages, Stage I and Stage II. Specifically, in Stage I, the AoAs are estimated by solving a multiple measurement vectors (MMV) problem. In Stage II, based on the estimated AoAs, the receive sounders are designed to estimate AoDs. The dimension of the angle dictionary in each stage can be reduced, which in turn reduces the computational complexity substantially. We then analyze the successful recovery probability (SRP) of the proposed method, revealing the superiority of the proposed framework over the existing one-stage CS-based methods. We further enhance the reconstruction performance by performing resource allocation between the two stages. We also overcome the unresolvable quantization error issue present in the prior techniques by applying the atomic norm minimization method to each stage of the proposed two-stage approach. The simulation results illustrate the substantially improved performance with low complexity of the proposed two-stage method.

Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization

Abstract This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p {\ell^{p}} -quasi-norm minimization. Compared to ℓ 1 {\ell^{1}} -norm minimization, the choice of 0 < p < 1 {0<p<1} provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the ℓ 1 {\ell^{1}} -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both ℓ 1 {\ell^{1}} -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.

Efficient DOA Estimation Method for Reconfigurable Intelligent Surfaces Aided UAV Swarm

The conventional direction of arrival (DOA) estimation methods are performed with multiple receiving channels. In this paper, a changeling DOA estimation problem is addressed in a different scenario with only one full-functional receiving channel. A new unmanned aerial vehicle (UAV) swarm system using multiple lifted reconfigurable intelligent surface (RIS) is proposed for the DOA estimation. The UAV movement degrades the DOA estimation performance significantly, and the existing atomic norm minimization (ANM) methods cannot be used in the scenario with array perturbation. Specifically, considering the position perturbation of UAVs, a new atomic norm-based DOA estimation method is proposed, where an atomic norm is defined with the parameter of the position perturbation. Then, a customized semi-definite programming (SDP) method is derived to solve the atomic norm-based method, where different from the traditional SDP method, an additional transforming matrix is formulated. Moreover, a gradient descent method is applied to refine the estimated DOA and the position perturbation further. Simulation results show that the proposed method achieves much better DOA estimation performance in the RIS-aided UAV swarm system with only one receiving channel than various benchmark schemes.

Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery.

Recently, transform-based tensor nuclear norm (TNN) minimization methods have received increasing attention for recovering third-order tensors in multi-dimensional imaging problems. The main idea of these methods is to perform the linear transform along the third mode of third-order tensors and then minimize the nuclear norm of frontal slices of the transformed tensor. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform by solely using the observed tensor in a self-supervised manner. The proposed network makes use of the low-rank representation of the transformed tensor and data-fitting between the observed tensor and the reconstructed tensor to learn the nonlinear transform. Extensive experimental results on different data and different tasks including tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging demonstrate the superior performance of the proposed method over state-of-the-art methods.

Nonlinear analysis and minimum L2-norm control in memcapacitor-based hyperchaotic system via online particle swarm optimization

Memristor and memcapacitor are two novel memristive devices. Memristive nonlinear elements behave like synapses in the nervous system. In this study, an original physical model of HP memristor is presented based on the movement of the boundary between the doped and undoped regions by causing the charged dopants to drift. Furthermore, a charge-controlled memcapacitor is used to design a novel hyperchaotic oscillator. It is found that the hyperchaotic oscillator, which is based on memristor and memcapacitor, can realize high-security data encryption. Then, the problem of controlling chaos is addressed in the proposed memcapacitor-based hyperchaotic memristor oscillator using a simple feedback control. Moreover, in this study, a novel approach is used to stabilize chaos using the L2–norm minimization method. The feedback control is applied to minimize the L2–norm of state variables as the cost function. An online particle swarm optimization (PSO) technique is developed to design the feedback control using L2–norm minimization. The sensitivity to initial condition is examined for different initial conditions. The proposed technique can be used for chaos stabilization in a complex dynamic system.

Channel Estimation for Switch-Based Millimeter-Wave Communications via Atomic Norm

Channel estimation is a challenging issue in millimeter-wave massive multiple-input-multiple-output (MIMO) communication systems due to the large number of antennas in the transceiver. Existing methods are usually based on phase shifters which may not be a simple circuit at mmWave band. In this paper, we construct a switch-based architecture for analog processors from the coarray point of view and then propose an atomicℓ0-norm minimization problem. We then propose an efficient algorithm to solve this problem based on Wirtinger projection. Since the proposed method requires no angle discretization, it does not suffer from grid mismatch effect that greatly deteriorates the estimation performance of grid-based channel estimation methods. Compared to the atomic norm minimization (ANM) method, our method does not involve vectorization of the channel matrix and hence the dimensionality of the problem is much less than that of ANM. We show that our method is able to provide comparable estimation performance to ANM but with much less computational time. Extensive simulations are carried out to verify the effectiveness of our proposed method.

Millimeter-wave massive MIMO channel estimation based on majorization–minimization approach

Millimeter-wave massive MIMO systems acquire a high dimensional distorted channel matrix to perform channel estimation. The captured channel matrix is sparse and rank deficient due to low scattering at millimeter-wave bands. The reduced number of RF chains enable the indirect access of signal at each antenna. This makes it difficult to estimate the channel accurately. To address this issue, researchers have focussed on compressive sensing and convex regularization methods. The first one performs well in noiseless scenario and while the latter suffers from over shrinking of all eigenvalues equally and is not suitable for large dimensional matrices. In this paper, a majorization minimization based fast non-convex (MM-FNC) algorithm is proposed to solve non-convex regularization based channel estimation. The proposed algorithm estimates the low-rank channel with fast convergence properties for large dimensional arrays. Simulation results show that the proposed approach exhibits superior performance over the CS-based orthogonal matching pursuit (CS-OMP) method, nuclear norm minimization methods (NNM) and matrix factorization (MF) methods in terms of normalized mean square error and achievable spectral efficiency with respect to different signal-to-noise ratio level.

Low-rank matrix recovery via regularized nuclear norm minimization

In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix X (not necessary to be exactly low-rank) from its few noisy measurements b = A ( X ) + n with a bounded constraint ‖ n ‖ 2 ≤ ϵ , provided that the tk -order restricted isometry constant (RIC) of A satisfies a certain constraint related to t > 0 . Specifically, the obtained recovery condition in the case of t > 4 / 3 is found to be same with the sharp condition established previously by Cai and Zhang [10] to guarantee the exact recovery of any rank- k matrix via the constrained nuclear norm minimization method. More importantly, to the best of our knowledge, we are the first to establish the tk -order RIC based coefficient estimate of the robust null space property in the case of 0 < t ≤ 1 .