Sparse Recovery Performance Research Articles

Meshless Search SR-STAP for Airborne Radar Based on Meta-Heuristic Algorithms

The sparse recovery (SR) space-time adaptive processing (STAP) method has excellent clutter suppression performance under the condition of limited observation samples. However, when the cluttering is nonlinear in a spatial-Doppler profile, it will cause an off-grid effect and reduce the sparse recovery performance. A meshless search using a meta-heuristic algorithm (MH) can completely eliminate the off-grid effect in theory. Therefore, genetic algorithm (GA), differential evolution (DE), particle swarm optimization (PSO), and grey wolf optimization (GWO) methods are applied to SR-STAP for selecting exact clutter atoms in this paper. The simulation results show that MH-STAP can estimate the clutter subspace more accurately than the traditional algorithm; PSO-STAP and GWO-STAP showed better clutter suppression performance in four MH-STAP methods. To search for more accurate clutter atoms, PSO and GWO are combined to improve the method’s capacity for global optimization. Meanwhile, the fitness function is improved by using prior knowledge of the clutter distribution. The simulation results show that the improved PSO-GWO-STAP algorithm provides excellent clutter suppression performance, which solves the off-grid problem better than does single MH-STAP.

Improved analytical learning proximal operator method for sparse recovery

Sparse recovery (SR) has received considerable attention in the last decades. Various iterative algorithms have been proposed to solve the SR problem, but most of them are unsatisfactory in terms of convergence accuracy and speed. Recently, deep unfolding methods achieve a dramatic improvement in convergence by unfolding iterative SR algorithms into deep neural networks and learning their parameters from training data, e.g., Learning Proximal Operator Method (LePOM) and Analytical LePOM (ALePOM). However, according to theoretical analysis, the recovery error bound of ALePOM should be further reduced as its network parameters are learned to fit all training data. To solve this problem, we design a new network named Improved ALePOM (I-ALePOM) by introducing a Long Short-Term Memory (LSTM) cell into each layer of ALePOM, which helps to adaptively compute the thresholds and step-sizes for each training data. By doing so, the recovery error bound becomes tighter and higher sparse recovery performance can be achieved. Furthermore, we extend the proposed I-ALePOM from the 1-D vector form to the 2-D matrix form, reducing the memory and computation burdens for practical sparse matrix recovery applications. Various numerical simulations verify that the proposed method has a higher performance than the state-of-the-art methods in the literature.

A Unified Framework for Constructing Nonconvex Regularizations

Over the past decades, many individual nonconvex methods have been proposed to achieve better sparse recovery performance in various scenarios. However, how to construct a valid nonconvex regularization function remains open in practice. In this paper, we fill in this gap by presenting a unified framework for constructing the nonconvex regularization based on the probability density function. Meanwhile, a new nonconvex sparse recovery method constructed via the Weibull distribution is studied.

A sensing matrix optimization method for sparse random stepped-frequency signal based on autocorrelation function

Based on the theory of sensing matrix optimization and the characteristics of step frequency waveform, this paper proposes a sensing matrix optimization method for sparse random step frequency signal based on autocorrelation function. Firstly, on the basis of constructing sparse reconstruction model, the internal relationship between the waveform parameters of sparse random step frequency signal and the construction mode of sensing matrix is given. Secondly, the relationship between the mutual coherence matrix and the autocorrelation function matrix of signal is analyzed, and the conclusion that under specific conditions, these two matrices are the same. Therefore, the optimal design of sensing matrix is transformed into the optimal design of sparse random stepped frequency signal based on autocorrelation function. Finally, a sparse waveform design method based on the joint constraint of the maximum and average sidelobes of the autocorrelation function is proposed. Simulation experiments verify the relationship between the mutual coherence matrix and the autocorrelation function matrix of signal, and through waveform design, the sensing matrix optimization and the improving performance of signal sparse recovery are realized.

Clustering structure based multiple measurement vectors model and its algorithm

Most multi-measurement vector models are based on the ideal assumption of shared sparse structure. However, due to time varying and multiple focuses of complex data, it is often difficult to meet the assumption in reality. Therefore, people have been working hard to utilise various sparse structures to make up for the problem. In this paper, we take the clustering structure of signals into account and propose the Clustering Structure based Multiple Measurement Vectors (CS-MMV) model, which not only utilises clustering characteristic between coefficients but also considers clustering structure within coefficients. Furthermore, we extend a classic algorithm to implement the new model. Experiments on simulation data and two face benchmarks show that the new model is more suitable for complex data with clustered structure, and the extended algorithm can effectively improve the performance of sparse recovery.

An Off-Grid STAP Algorithm Based On Local Mesh Splitting With Bistatic Radar System

The purpose of this letter is to solve the space-time off-grid problem caused by the extremely inhomogeneous clutter spectrum under the space/airborne bistatic system. For the estimation of inhomogeneous clutter spectrum, the method of sparse recovery (SR) is usually adopted. However, performance of sparse recovery may degrade since the real clutter point often does not fall in the center of the space-time grid. To deal with this issue, a local mesh splitting algorithm is proposed in this letter. First, select the space-time vector atoms which are most relevant to the residuals by subspace projection. Then, iterative local mesh splitting is performed around the selected atoms to match the real clutter points. Finally, the clutter subspace is reconstructed from all matched atoms selected by global and local iterations. The simulation results show that the method can obtain better accuracy with lower computational effort.

Nonconvex Robust Low-Rank Tensor Reconstruction via an Empirical Bayes Method

Low-rank tensor reconstruction has attracted a great deal of research interest in signal processing, image processing and machine learning. To deal with outliers that are now ubiquitous in many modern applications, a representative approach is to decompose and reconstruct the data into a low-rank component and a sparse outlier component. As surrogate functions of the matrix rank and the sparsity that are non-convex and discontinuous, the nuclear norm and the $\ell _1$ norm are often exploited for recovering higher order tensors and removing outliers, respectively. However, the nuclear norm and the $\ell _1$ norm are loose approximations of the matrix rank and the sparsity, respectively. Furthermore, the tensor nuclear norm is not guaranteed to be the tightest convex envelope of a multilinear rank. In this paper, we develop a new method for robust tensor reconstruction. By capitalizing on the empirical Bayes method, whose performance has been shown to act as state-of-the-art to the performance of standard sparse and low-rank matrix recovery, we formulate the objective function of the empirical Bayes method for robust tensor reconstruction, and then adjust the objective function to facilitate algorithm derivation, while keeping desirable attributes including concavity and symmetry of the objective function of the empirical Bayes method. We demonstrate the superior performance of the proposed algorithm in comparison with state-of-the-art alternatives by conducting experiments on both synthetic data and real data.

Efficient gradient‐based dictionary optimisation method for time‐varying array incorporating with structure characteristic

To guarantee good sparsity reconstruction quality, a suitable dictionary should be as orthogonal as possible. Also, to reduce the coherence of a dictionary, a sensing matrix optimisation model should be formulated. In this model, a high-dimension O (N) problem has to be considered because the sparse representation base is overcompleted, thus leading to a heavy computational load. This study proposes an efficient gradient-based method to address the dictionary optimisation problem of the time-varying arrays, whose elements relatively move in an arbitrary but known way. The dictionary optimisation models associated with different array geometries are formulated with distinct structure characteristic, and the Toeplitz and circulant properties are incorporated into the corresponding models to reduce the complexity by dimension reduction from O (N) to O (2) and O (1), respectively. An alternating minimisation approach for sensing matrix design (SMD) is derived by the gradient descent method, and an adaptive stepsize selection method is derived to further reduce complexity. Numerous simulations are conducted, and simulation results demonstrate that the presented methods have lower computational complexity but similar sparse recovery performance as the normalisation-constrained SMD method. Also, the effectiveness of the proposed structured optimisation framework to design a sensing matrix of a time-varying array is verified.

Statistical performance of convex low-rank and sparse tensor recovery

Low-rank or sparse tensor recovery finds many applications in computer vision and machine learning. The recently proposed regularized multilinear regression and selection (Remurs) model assumes the true tensor to be simultaneously low-Tucker-rank and sparse, and has been successfully applied in fMRI analysis. However, the statistical performance of Remurs-like models is still lacking. To address this problem, a minimization problem based on a newly defined tensor nuclear-l1-norm is proposed, to recover a simultaneously low-Tucker-rank and sparse tensor from its degraded observations. Then, an M-ADMM-based algorithm is developed to efficiently solve the problem. Further, the statistical performance is analyzed by establishing a deterministic upper bound on the estimation error for general noise. Also, under Gaussian noise, non-asymptotic upper bounds for two specific settings, i.e., noisy tensor decomposition and random Gaussian design, are given. Experiments on synthetic datasets demonstrate that the proposed theorems can precisely predict the scaling behavior of the estimation error.

Angular Positions Estimation of Spatially Extended Targets for MIMO Radar Using Complex Spatiotemporal Sparse Bayesian Learning

Aiming at the problem of scattering centers resolving and angular positions estimation of spatially extended targets, a high-resolution and high-accuracy angle estimation method based on multi-task group sparse model and collocated MIMO radar is proposed, which is helpful to obtain the structure information of targets and improve the success rate of target recognition. Characterized by sparse and clustered distribution in space, angular positions estimation of multiple closely-spaced and correlated point scattering targets belonging to a spatially extended target can be modeled as a multi-task group sparse problem and can be solved by multi-task group sparse recovery. To overcome the sparse recovery performance degradation caused by the high correlation in group sparse solution matrix and to improve the accuracy and robustness of angle estimation, a complex spatiotemporal sparse Bayesian learning (CST-SBL) algorithm which exploits spatiotemporal correlation structures of the solution matrix is proposed to reconstruct angular positions. Compared with previous work, the proposed approach achieves high-resolution and high-accuracy estimation performance, especially in cases of low SNR and few snapshots. The theoretical analysis and simulation results validate the effectiveness of the proposed technique.

Joint Mutual Coherence and Total Coherence Pilot Design for OFDM Channel Estimation

Existing works design the pilot based on the mutual coherence criterion or total coherence criterion alone. However, using the mutual coherence criterion or total coherence criterion alone to design the pilot will yield the sensing matrix which corresponds to a certain pilot with a large total coherence or mutual coherence. However, the sensing matrix with a large mutual coherence or total coherence will deteriorate the sparse recovery performance and is not optimal from the perspective of the sparse recovery. Therefore, in this paper, we jointly design the pilot based on both the mutual coherence criterion and the total coherence criterion for sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems. We first prove that the optimal pilot pattern with respect to the total coherence criterion is the cyclic difference set and given a pilot pattern, the pilot power allocation can be cast as a second order cone programming problem with respect to the total coherence criterion. A joint mutual coherence and total coherence pilot design algorithm is proposed to design the pilot jointly while cyclic difference set does not exist for practical OFDM systems. The proposed algorithm first optimizes the pilot pattern with the mutual coherence criterion assuming that the pilot power is equal. After obtaining the pilot pattern, the pilot power is allocated with the total coherence criterion. Simulation results have shown that the proposed pilot design algorithm can achieve the best performance in terms of normalized mean square error and bit error rate in comparison with existing pilot design methods.

Robust Affine Subspace Clustering via Smoothed $$\ell _{0}$$ ℓ 0 -Norm

In the past few years, sparse representation based method has been used in many fields with breathtaking speed due to its superior sparse recovery performance. Sparse subspace clustering (SSC), as one of its application hot-spots, has attracted considerable attention. Traditional sparse subspace clustering methods employ the $$\ell _{1}$$ -norm to induce sparse representation of data points. Typically, the use of $$\ell _{1}$$ -regularization instead of the $$\ell _{0}$$ one can make the objective function convex while it also causes large errors on large coefficients in some cases. In this work, we propose using the non-convex smoothed $$\ell _{0}$$ -norm to replace the $$\ell _{0}$$ one for affine subspace clustering. This leads to a non-convex minimization problem. We then propose an effective method to solve the problem which minimizes the objective function by using the gradient method and proximal projection. In addition, the proposed algorithm is robust to noise and can provide a fast solution. Extensive experiments on real datasets demonstrate the effectiveness of our proposed method.

Deep clipping noise mitigation using ISTA with the specified observations for LED‐based DCO‐OFDM system

Deep clipping is beneficial for the optical orthogonal frequency division multiplexing (O-OFDM) system, since it can lower the peak-to-average power ratio, reduce the direct current requirement in light emitting diodes (LEDs), and relax the bit-resolution requirement in digital-to-analogue converters (DACs). However, it is accompanied by more signal distortions. In this study, a deep clipping noise mitigation scheme using iterative shrinkage/thresholding algorithm (ISTA) with three steps is proposed to improve bit error rate (BER) performance of the LED-based DCO-OFDM systems. In the first step, the estimated observation interference is eliminated from the received symbols to minimise the negative effect of channel noise. In the second step, two strategies are presented to generate the specified observations thus reduce the component of measurement noise in the whole observation vector. In the last step, combining the generalised cross validation and the estimation of observation interference, the appropriate regularisation parameter are calculated for ISTA to improve the robustness of the sparse recovery performance. They use simulations to show that the proposed scheme can correct the deep clipping noise with favourable reconstruction quality, which significantly improves the BER performance and therefore assist the LED non-linearity mitigation.

Sparse Recovery Conditions and Performance Bounds for $\ell _p$-Minimization

In sparse recovery, a sparse signal ${\mathbf x}\in \mathbb {R}^N$ with $K$ nonzero entries is to be reconstructed from a compressed measurement $\mathbf y=Ax$ with ${\mathbf A}\in \mathbb {R}^{M\times N}$ ( $M ). The $\ell _p$ $(0\leq p pseudonorm has been found to be a sparsity inducing function superior to the $\ell _1$ norm, and the null space constant (NSC) and restricted isometry constant (RIC) have been used as key notions in the performance analyses of the corresponding $\ell _p$ -minimization. In this paper, we study sparse recovery conditions and performance bounds for the $\ell _p$ -minimization. We devise a new NSC upper bound that outperforms the state-of-the-art result. Based on the improved NSC upper bound, we provide a new RIC upper bound dependent on the sparsity level $K$ as a sufficient condition for precise recovery, and it is tighter than the existing bound for small $K$ . Then, we study the largest choice of $p$ for the $\ell _p$ -minimization problem to recover any $K$ -sparse signal, and the largest recoverable $K$ for a fixed $p$ . Numerical experiments demonstrate the improvement of the proposed bounds in the recovery conditions over the up-to-date counterparts.

Complex-Image-Based Sparse SAR Imaging and its Equivalence

Using sparse signal processing to replace matched filtering (MF) in synthetic aperture radar (SAR) imaging has shown significant potential to improve image quality. Due to the huge computational cost needed, it is difficult to apply conventional observation-matrix-based sparse SAR imaging method for large-scene reconstruction. The azimuth-range decouple method is able to minimize the computational complexity and achieve image performance similar to that obtained by the observation-matrix-based algorithm. However, there still exist two difficult problems in sparse SAR imaging, i.e., real-time processing and lack of raw data. To solve these problems, this paper presents a novel complex-image-based sparse SAR imaging method. It is found that if the input MF-recovered SAR complex image is obtained via fully sampled raw data, the proposed method can achieve an identical high-resolution image to that obtained by the azimuth-range decouple algorithm. The computational complexity is also decreased to the same order as that of MF, which makes the real-time sparse SAR imaging become possible. In addition, it should be noted that even though without raw data, the proposed method can still obtain impressive sparse recovery performance by using only the available complex image. Performance analysis and experimental results on real data validate the proposed method.

Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions

Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers of l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy and computational tradeoffs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.

Measurement matrix design for CS-MIMO radar using multi-objective optimization

In this paper, we design a measurement matrix for a compressive sensing-multiple-input multiple-output radar in the presence of clutter and interference. To optimize the measurement matrix, three main criteria are considered simultaneously to improve detection and sparse recovery performance while suppressing clutter and interference. To this end, we consider three well-known criteria including Bhattacharyya distance, mutual coherency of sensing matrix, and signal-to-clutter-plus-interference ratio. Due to the use of simultaneous multi-objective functions, a multi-objective optimization (MOO) framework is exploited. Some numerical examples are provided to illustrate the achieved improvement of our proposed method in target detection and sparse recovery performance. Simulation results show that the proposed MOO technique for measurement matrix design can achieve superior performance in target detection compared with Gaussian random measurement matrix technique.

Sparse signal recovery via exponential metric approximation

Sparse signal recovery problems are common in parameter estimation, image processing, pattern recognition, and so on. The problem of recovering a sparse signal representation from a signal dictionary might be classified as a linear constraint £0-quasinorm minimization problem, which is thought to be a Non-deterministic Polynomial-time (NP)-hard problem. Although several approximation methods have been developed to solve this problem via convex relaxation, researchers find the nonconvex methods to be more efficient in solving sparse recovery problems than convex methods. In this paper a nonconvex Exponential Metric Approximation (EMA) method is proposed to solve the sparse signal recovery problem. Our proposed EMA method aims to minimize a nonconvex negative exponential metric function to attain the sparse approximation and, with proper transformation, solve the problem via Difference Convex (DC) programming. Numerical simulations show that exponential metric function approximation yields better sparse recovery performance than other methods, and our proposed EMA-DC method is an efficient way to recover the sparse signals that are buried in noise.

Identification-While-Scanning of a Multi-Aircraft Formation Based on Sparse Recovery for Narrowband Radar.

It is known that the identification performance of a multi-aircraft formation (MAF) of narrowband radar mainly depends on the time on target (TOT). To realize the identification task in one rotated scan with limited TOT, the paper proposes a novel identification-while-scanning (IWS) method based on sparse recovery to maintain high rotating speed and super-resolution for MAF identification, simultaneously. First, a multiple chirp signal model is established for MAF in a single scan, where different aircraft may have different Doppler centers and Doppler rates. Second, based on the sparsity of MAF in the Doppler parameter space, a novel hierarchical basis pursuit (HBP) method is proposed to obtain satisfactory sparse recovery performance as well as high computational efficiency. Furthermore, the parameter estimation performance of the proposed IWS identification method is analyzed with respect to recovery condition, signal-to-noise ratio and TOT. It is shown that an MAF can be effectively identified via HBP with a TOT of only about one hundred microseconds for IWS applications. Finally, some numerical experiment results are provided to demonstrate the effectiveness of the proposed method based on both simulated and real measured data.

Square-Root Lasso With Nonconvex Regularization: An ADMM Approach

Square-root least absolute shrinkage and selection operator (Lasso), a variant of Lasso, has recently been proposed with a key advantage that the optimal regularization parameter is independent of the noise level in the measurements. In this letter, we introduce a class of nonconvex sparsity-inducing penalties to the square-root Lasso to achieve better sparse recovery performance over the convex counterpart. The resultant formulation is converted to a nonconvex but multiconvex optimization problem, i.e., it is convex in each block of variables. Alternating direction method of multipliers is applied as the solver, according to which two efficient algorithms are devised for row-orthonormal sensing matrix and general sensing matrix, respectively. Numerical experiments are conducted to evaluate the performance of the proposed methods.