Search Results (7)

Data acquisition and high-dimensional signal processing often require the recovery of sparse representations of signals to minimize the resources needed for data collection. ${\ell }_p$ quasi-norm minimization excels in exactly reconstructing sparse signals from fewer measurements, but it is NP-hard and challenging to solve. In this paper, we propose two distinct Iteratively Re-weighted ${\ell }_1$ Minimization (IR${\ell }_1$) formulations for solving this non-convex sparse recovery problem by introducing two novel reweighting strategies. These strategies ensure that the $ϵ$-regularizations adjust dynamically based on the magnitudes of the solution components, leading to more effective approximations of the non-convex sparsity penalty. The resulting IR${\ell }_1$ formulations provide first-order approximations of tighter surrogates for the original ${\ell }_p$ quasi-norm objective. We prove that both algorithms converge to the true sparse solution under appropriate conditions on the sensing matrix. Our numerical experiments demonstrate that the proposed IR${\ell }_1$ algorithms outperform the conventional approach in enhancing recovery success rate and computational efficiency, especially in cases with small values of p. Full article

A multiple-actuator fault isolation approach for overactuated electric vehicles (EVs) is designed with a minimal ${\ell }_1$-norm solution. As the numbers of driving motors and steering actuators increase beyond the number of controlled variables, an EV becomes an overactuated system, which exhibits actuator redundancy and enables the possibility of fault-tolerant control (FTC). On the other hand, an increase in the number of actuators also increases the possibility of simultaneously occurring multiple faults. To ensure EV reliability while driving, exact and fast fault isolation is required; however, the existing fault isolation methods demand high computational power or complicated procedures because the overactuated systems have many actuators, and the number of simultaneous fault occurrences is increased. The method proposed in this paper exploits the concept of sparsity. The underdetermined linear system is defined from the parity equation, and fault isolation is achieved by obtaining the sparsest nonzero component of the residuals from the minimal ${\ell }_1$-norm solution. Therefore, the locations of the faults can be obtained in a sequence, and only a consistently low computational load is required regardless of the isolated number of faults. The experimental results obtained with a scaled-down overactuated EV support the effectiveness of the proposed method, and a quantitative index of the sparsity condition for the target EV is discussed with a CarSim-connected MATLAB/Simulink simulation. Full article

The low-rank and sparse decomposition model has been favored by the majority of hyperspectral image anomaly detection personnel, especially the robust principal component analysis(RPCA) model, over recent years. However, in the RPCA model, ${\ell }_0$ operator minimization is an NP-hard problem, which is applicable in both low-rank and sparse items. A general approach is to relax the ${\ell }_0$ operator to ${\ell }_1$-norm in the traditional RPCA model, so as to approximately transform it to the convex optimization field. However, the solution obtained by convex optimization approximation often brings the problem of excessive punishment and inaccuracy. On this basis, we propose a non-convex regularized approximation model based on low-rank and sparse matrix decomposition (LRSNCR), which is closer to the original problem than RPCA. The WNNM and Capped ${\ell }_{2,1}$-norm are used to replace the low-rank item and sparse item of the matrix, respectively. Based on the proposed model, an effective optimization algorithm is then given. Finally, the experimental results on four real hyperspectral image datasets show that the proposed LRSNCR has better detection performance. Full article

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with ${\ell }_0$-quasi norm based on RKHS to depict this degraded problem. The underlying function in RKHS can be represented by the linear combination of reproducing kernels and their coefficients. Thus, we turn to estimate the related coefficients in the nonconvex minimization problem. An efficient algorithm is designed to solve the given nonconvex problem by the mathematical program with equilibrium constraints (MPEC) and proximal-based strategy. We theoretically prove that the sequences generated by the designed algorithm converge to the nonconvex problem’s local optimal solutions. Numerical experiment also demonstrates the effectiveness of the proposed method. Full article

Since the cloud radio access network (C-RAN) transmits information signals by jointly transmission, the multiple copies of information signals might be eavesdropped on. Therefore, this paper studies the resource allocation algorithm for secure energy optimization in a downlink C-RAN, via jointly designing base station (BS) mode, beamforming and artificial noise (AN) given imperfect channel state information (CSI) of information receivers (IRs) and eavesdrop receivers (ERs). The considered resource allocation design problem is formulated as a nonlinear programming problem of power minimization under the quality of service (QoS) for each IR, the power constraint for each BS, and the physical layer security (PLS) constraints for each ER. To solve this non-trivial problem, we first adopt smooth ${\ell }_0$ -norm approximation and propose a general iterative difference of convex (IDC) algorithm with provable convergence for a difference of convex programming problem. Then, a three-stage algorithm is proposed to solve the original problem, which firstly apply the iterative difference of convex programming with semi-definite relaxation (SDR) technique to provide a roughly (approximately) sparse solution, and then improve the sparsity of the solutions using a deflation based post processing method. The effectiveness of the proposed algorithm is validated with extensive simulations for power minimization in secure downlink C-RANs. Full article

Compressive sensing theory has attracted widespread attention in recent years and sparse signal reconstruction has been widely used in signal processing and communication. This paper addresses the problem of sparse signal recovery especially with non-Gaussian noise. The main contribution of this paper is the proposal of an algorithm where the negentropy and reweighted schemes represent the core of an approach to the solution of the problem. The signal reconstruction problem is formalized as a constrained minimization problem, where the objective function is the sum of a measurement of error statistical characteristic term, the negentropy, and a sparse regularization term, ℓ_p-norm, for 0 < p < 1. The ℓ_p-norm, however, leads to a non-convex optimization problem which is difficult to solve efficiently. Herein we treat the ℓ_p -norm as a serious of weighted ℓ₁-norms so that the sub-problems become convex. We propose an optimized algorithm that combines forward-backward splitting. The algorithm is fast and succeeds in exactly recovering sparse signals with Gaussian and non-Gaussian noise. Several numerical experiments and comparisons demonstrate the superiority of the proposed algorithm. Full article

The goal of sparse linear hyperspectral unmixing is to determine a scanty subset of spectral signatures of materials contained in each mixed pixel and to estimate their fractional abundances. This turns into an ℓ₀ -norm minimization, which is an NP-hard problem. In this paper, we propose a new iterative method, which starts as an ℓ₁ -norm optimization that is convex, has a unique solution, converges quickly and iteratively tends to be an ℓ₀ -norm problem. More specifically, we employ the arctan function with the parameter σ ≥ 0 in our optimization. This function is Lipschitz continuous and approximates ℓ₁ -norm and ℓ₀ -norm for small and large values of σ, respectively. We prove that the set of local optima of our problem is continuous versus σ. Thus, by a gradual increase of σ in each iteration, we may avoid being trapped in a suboptimal solution. We propose to use the alternating direction method of multipliers (ADMM) for our minimization problem iteratively while increasing σ exponentially. Our evaluations reveal the superiorities and shortcomings of the proposed method compared to several state-of-the-art methods. We consider such evaluations in different experiments over both synthetic and real hyperspectral data, and the results of our proposed methods reveal the sparsest estimated abundances compared to other competitive algorithms for the subimage of AVIRIS cuprite data. Full article