Minimum Norm Research Articles

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.

Extreme Few-View Tomography without Training Data.

There are fewer than 10 projection views in extreme few-view tomography. The state-of-the-art methods to reconstruct images with few-view data are compressed sensing based. Compressed sensing relies on a sparsification transformation and total variation (TV) norm minimization. However, for the extreme few-view tomography, the compressed sensing methods are not powerful enough. This paper seeks additional information as extra constraints so that extreme few-view tomography becomes possible. In transmission tomography, we roughly know the linear attenuation coefficients of the objects to be imaged. We can use these values as extra constraints. Computer simulations show that these extra constraints are helpful and improve the reconstruction quality.

Efficient 2D DOA Estimation via Decoupled Projected Atomic Norm Minimization

This paper presents an efficient two-dimensional (2D) direction of arrival (DOA) estimation method, termed as decoupled projected atomic norm minimization (D-PANM), to solve the angle-ambiguity problem. It first introduces a novel atomic metric via projecting the original atom set onto a smoothing space, based on which we formulate an equivalent semi-definite programming (SDP) problem. Then, two relatively low-complexity decoupled Toeplitz matrices can be obtained to estimate the DOAs. We further exploit the structural information hidden in the newly constructed data to avoid pair matching for the azimuth and elevation angles when the number of sensors is odd, and then propose a fast and feasible decoupled alternating projections (D-AP) algorithm, reducing computational complexity to a great extent. Numerical simulations are performed to demonstrate that the proposed algorithm is no longer restricted by angle ambiguity scenarios, but instead provides a more stable estimation performance, even when multiple signals share the same angles in both azimuth and elevation dimensions. Additionally, it greatly improves the resolution, with control of the computation load compared with the existing atomic norm minimization (ANM) algorithm.

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.

Does size matter? Evidence from municipal splits

AbstractWe contribute to the limited knowledge of the consequences of municipal splits by estimating how break‐ups of seven Swedish municipalities affected per capita expenditures. To predict what would have happened had the break‐ups not taken place, we apply the matrix completion method with nuclear norm minimization. We find that smaller municipalities not necessarily imply higher per capita expenditures. Instead, expenditures increase in some cases, are unaffected in others, and in others, decrease. The results point to the complex nature of territorial reforms and underscore the perils of policy recommendations that take uniform outcomes of either amalgamations or break‐ups for granted.

Gridless compressed covariance sensing for parameter identification of nonuniform and undersampled blade tip timing data

Due to the harsh working environments, rotating blades are susceptible to failures, which endanger the operational safety of turbomachinery; thus, it is important to monitor rotating blades to promptly detect incipient failure. Blade tip timing (BTT) is a promising approach for monitoring the health of rotating blades owing to its noncontact nature, high measurement efficiency, and long service life. However, the large gap between the high vibration frequency and low sampling frequency results in severe undersampling of the BTT signal. Although advanced methods have been proposed to reconstruct the spectrum or identify parameters from BTT signal, almost all of them face basis mismatch issue caused by frequency discretization and suffers from inevitable reconstruction errors. To overcome these limitations, we developed a gridless compressed covariance sensing (GCCS) approach for the sparse spectral estimation and parameter identification of BTT data. Unlike existing methods relying on frequency discretization, GCCS directly performs a continuous range of spectrum reconstruction and parameter estimation. We derived that GCCS is equivalent to a weighted atomic norm minimization (ANM) with the weight function linked to the Capon spectrum, which implies the advantages of GCCS over methods based on conventional ANM. GCCS involves two core steps, first, covariance (matrix) recovery from compressive samples is formulated as a semidefinite programming; second, the extended Prony’s method is employed as a postprocessing technique to retrieve vibration parameters from the covariance estimations, thereby achieving continuous parameter estimation. The quantitative comparison in the simulations and experiments shows the effectiveness of GCCS in overcoming basis mismatch and improving the accuracy and consistency of the parameter estimation.

A lagrange programming neural network approach for nuclear norm optimization.

This article proposes a continuous-time optimization approch instead of tranditional optimiztion methods to address the nuclear norm minimization (NNM) problem. Refomulating the NNM into a matrix form, we propose a Lagrangian programming neural network (LPNN) to solve the NNM. Moreover, the convergence condtions of LPNN are presented by the Lyapunov method. Convergence experiments are presented to demonstrate the convergence of LPNN. Compared with tranditional algorithms of NNM, the proposed algorithm outperforms in terms of image recovery.

Lp quasi-norm minimization: algorithm and applications

Sparsity finds applications in diverse areas such as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts and need less storage. This paper proposes a heuristic method for retrieving sparse approximate solutions of optimization problems via minimizing the ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{p}$$\\end{document} quasi-norm, where 0<p<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<p<1$$\\end{document}. An iterative two-block algorithm for minimizing the ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{p}$$\\end{document} quasi-norm subject to convex constraints is proposed. The proposed algorithm requires solving for the roots of a scalar degree polynomial as opposed to applying a soft thresholding operator in the case of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{1}$$\\end{document} norm minimization. The algorithm’s merit relies on its ability to solve the ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{p}$$\\end{document} quasi-norm minimization subject to any convex constraints set. For the specific case of constraints defined by differentiable functions with Lipschitz continuous gradient, a second, faster algorithm is proposed. Using a proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm’s speed while proving its convergence. We present various applications where the proposed algorithm excels, namely, sparse signal reconstruction, system identification, and matrix completion. The results demonstrate the significant gains obtained by the proposed algorithm compared to other ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _{p}$$\\end{document} quasi-norm based methods presented in previous literature.

Synthesis of ship systems optimal regulators based on matrix inequalities

The output and state regulators synthesis for ship control systems based on linear matrix inequalities using the CVX software package for solving optimization problems by the method of semi-definite programming is considered. Estimates parameters for regulators providing asymptotically stable systems regimes on Lyapunov inequalities solutions sets are obtained. An algorithm for controlling the ship course using a robust regulator in state feedback, followed by the use of CVX technologies to implement the minimum energy consumption control algorithm, is presented. The control vector estimation is reduced to the minimization of the L2 norm and it is performed in two stages, namely, the best mode is first determined, and then the mode under constraints is defined. The use of Lyapunov inequalities with the calculations implementation in CVX format on proprietary solvers has allowed us to obtain compact program texts and expand the set of restrictions introduced. In the algorithm for solving the boundary value problem of controlling a discrete model of an object, in contrast to existing optimization methods, a mode of transfer from the initial state to the final one with the right boundary, not necessarily equal to zero, is implemented. For a class of observable and controllable time–invariant systems, using the Krylov’s function contained in the MATLAB gallery of functions, L2‑norms to be minimized are formed. Examples of regulator calculations with a given degree of stability and optimal systems, confirming proposed solutions correctness, are given.

Minimum norm partial eigenstructure assignment approach for vibration system with time delay

AbstractIn this paper, the minimum norm partial eigenstructure assignment problems (PESAPs) in vibration system with time delay is studied via acceleration‐velocity‐displacement active control. A new explicit solution is given for solving PESAPs using orthogonality relations of symmetric vibration system. A new constructive technique is presented to solve the PESAPs in vibration system with time delay by singular value decomposition method. An optimization algorithm is proposed to solve the minimum norm PESAPs by a quasi‐Newton optimization method. The proposed algorithm works without using the knowledge of receptance matrices and without solving the Sylvester equation. The effectiveness of the proposed algorithm is verified by the results of two numerical examples.

Non-Synchronous Measurements Acoustic Imaging Method Based on Capped Nuclear Norm Minimization

In this work, the difference convex function based on capped nuclear norm minimization (CNNM-DC) method, a cross-spectral matrix (CSM) completion iterative algorithm with excellent noise immunity, is proposed to realize acoustic imaging of non-synchronous measurements (NSM) under the condition of low signal-to-noise ratio (SNR). Compared with the CSM completion algorithm based on the nuclear norm minimization (NNM) model, the truncated nuclear norm regularization (TNNR) model, and the weighted nuclear norm minimization (WNNM) model, the proposed method can obtain the acoustic image with narrow main lobes and no sidelobe under the condition of low SNR. The simulation results show that, under conditions of low SNR, the proposed method effectively reduces the width of the main lobe, suppresses the side lobes, and achieves more accurate imaging results than the previous method. Finally, experiments were conducted to verify the feasibility of CNNM-DC. The experimental results show that the proposed method is an accurate acoustic imaging algorithm for NSM under low SNR, which lays a foundation for acoustic imaging in industrial occasions with strong background noise interference.

Gridless Evolutionary Approach for Line Spectral Estimation With Unknown Model Order.

Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic l0 norm (i.e., the continuous analog of l0 norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of the atomic l0 norm, such as the nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However, the relaxations give rise to a resolution limit, subsequently leading to biased model order and convergence error. To overcome the above shortcomings of relaxation, we propose a novel idea of simultaneously estimating the frequencies and model order using the atomic l0 norm. To accomplish this idea, we build a multiobjective optimization model. The measurement error and the atomic l0 norm are taken as the two optimization objectives. The proposed model directly exploits the model order via the atomic l0 norm, thus breaking the resolution limit. We further design a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations. One is a variable-length coding and search strategy. It flexibly codes and interactively searches diverse solutions with different model orders. These solutions act as steppingstones that helpfully exploring the variable and open-ended frequency search space and provide extensive potentials toward the optima. Another innovation is a model-order pruning mechanism, which heuristically prunes less contributive frequencies within the solutions, thus significantly enhancing convergence and diversity. Simulation results confirm the superiority of our approach in both frequency estimation and model-order selection.

Asymmetric Supervised Fusion-Oriented Hashing for Cross-Modal Retrieval.

Hashing technologies have been widely applied for large-scale multimodal retrieval tasks owing to their excellent performance in search and storage tasks. Although some effective hashing methods have been proposed, it is still difficult to handle the intrinsic linkages that exist among different heterogeneous modalities. Moreover, optimizing the discrete constraint problem through a relaxation-based strategy results in a large quantization error and leads to a suboptimal solution. In this article, we present a novel asymmetric supervised fusion-oriented hashing method, named (ASFOH), which investigates three novel schemes to remedy the above issues. Specifically, we first explicitly formulate the problem as matrix decomposition into a common latent representation and a transformation matrix, combined with an adaptive weight scheme and nuclear norm minimization to ensure the information completeness of multimodal data. Then, we associate the common latent representation with the semantic label matrix, thereby increasing the discriminative capability of the model by constructing an asymmetric hash learning framework, thus, making the generated hash codes more compact. Finally, an efficient discrete optimization iterative algorithm based on nuclear norm minimization is proposed to decompose the nonconvex multivariate optimization problem into several subproblems with analytical solutions. Comprehensive experiments on the MIRFlirck, NUS-WIDE, and IARP-TC12 datasets testify that ASFOH outperforms the compared state-of-the-art approaches.

An efficient algorithm for the minimal least squares solution of linear systems with indefinite symmetric matrices

In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form LDLt using Jacobi rotations in order to increase the pivot´s absolute value. Furthermore, Rook´s pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form.

Hypersurfaces of the Space form for Which the Covariant Derivative of Their Ricci Tensors has Vanishing Minimal Norm Tensor

Hypersurfaces of the Space form for Which the Covariant Derivative of Their Ricci Tensors has Vanishing Minimal Norm Tensor

Quaternion weighted Schatten [formula omitted]-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM’s empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.

Partial quadratic eigenvalue assignment for vibration systems using receptances and system matrices

In this paper, we consider the partial quadratic eigenvalue assignment problem (PQEAP) for vibration system by active feedback control, such that some “troublesome” eigenpairs are replaced by some suitable chosen or newly measured ones, while the remaining unknown eigenpairs are kept unchanged. With the assumptions that the mass and stiffness matrices are nonsingular, we give a new proof of the spectral decomposition of quadratic symmetric pencil. And then we provide some necessary and sufficient conditions of the feedback matrices so that the closed-loop can preserve the no spill-over property. With these conditions, we characterize a set of parametric solutions to the PQEAP using receptances, system matrices and a few open-loop eigenpairs. Finally, we propose a gradient-based optimization method for solving the minimum norm solution to the PQEAP. Numerical experiments illustrate the effectiveness and robustness of the proposed algorithm.

Multitask control of aerial manipulator robots with dynamic compensation based on numerical methods

This paper presents a control scheme for aerial manipulators which allows to solve different motion problems: end-effector position control, end-effector trajectory tracking control and path-following control. The scheme has two cascaded controllers: i) the first controller is a minimum norm controller based on numerical methods, it solves the three motion control problems just by modifying the controller references. Also, since the aerial manipulator robot is a redundant system, i.e., it has extra degrees of freedom to accomplish the task, it is possible to set other control objectives in a hierarchical order. As a secondary objective of the control it is proposed to maintain a desired configuration for the robotic arm during the task. ii) The second cascade controller is designed to compensate the dynamics of the system which main objective is to drive the velocity errors to zero. The coupled dynamic model of the robotic system (hexarotor and robotic arm) is presented. This model is usually developed as a function of the forces and torques. However, in this work, it is written as a function of reference velocities which are usual references for these vehicles. The proposed control algorithms are given with the corresponding stability and robustness analysis. Finally, to validate the control scheme, experimental tests are performed in a partially structured environment with an aerial manipulator conformed by an aerial platform and a 3DOF robotic arm.

NMGMDA: a computational model for predicting potential microbe–drug associations based on minimize matrix nuclear norm and graph attention network

The prediction of potential microbe–drug associations is of great value for drug research and development, especially, methods, based on deep learning, have been achieved significant improvement in bio-medicine. In this manuscript, we proposed a novel computational model named NMGMDA based on the nuclear norm minimization and graph attention network to infer latent microbe–drug associations. Firstly, we created a heterogeneous microbe–drug network in NMGMDA by fusing the drug and microbe similarities with the established drug–microbe associations. After this, by using GAT and NNM to calculate the predict scores. Lastly, we created a fivefold cross validation framework to assess the new model NMGMDA's progressiveness. According to the simulation results, NMGMDA outperforms some of the most advanced methods, with a reliable AUC of 0.9946 on both MDAD and aBioflm databases. Furthermore, case studies on Ciprofloxacin, Moxifoxacin, HIV-1 and Mycobacterium tuberculosis were carried out in order to assess the effectiveness of NMGMDA even more. The experimental results demonstrated that, following the removal of known correlations from the database, 16 and 14 medications as well as 19 and 17 microbes in the top 20 predictions were validated by pertinent literature. This demonstrates the potential of our new model, NMGMDA, to reach acceptable prediction performance.

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

&lt;abstract&gt; &lt;p&gt;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.&lt;/p&gt; &lt;/abstract&gt;