Sparse Signal Recovery Research Articles

Application of compressive sensing techniques for advanced image processing and digital image transmission

The field of compressive sensing (CS) has emerged as a transformative approach in the acquisition and processing of high-dimensional data. This paper presents a comprehensive study on the application of compressive sensing techniques to advanced image processing and digital image transmission. By leveraging the inherent sparsity in natural images, CS allows for significant reductions in the amount of data required for accurate reconstruction, thereby overcoming the limitations imposed by the traditional Shannon-Nyquist sampling theorem. We explore the theoretical foundations of CS, including the principles of sparsity and incoherence, and provide a detailed overview of the Orthogonal Matching Pursuit (OMP) algorithm, a prominent greedy algorithm used for sparse signal recovery. Experimental results demonstrate the efficacy of CS in improving image reconstruction quality, as evidenced by enhancements in peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM). Additionally, we discuss the practical implementation of CS in single-pixel cameras and its potential impact on future imaging technologies. The findings suggest that CS offers a robust framework for efficient image acquisition and processing, making it a valuable tool for various applications in multimedia, medical imaging, and remote sensing.

Stochastic IHT with Stochastic Polyak Step-size for Sparse Signal Recovery

Stochastic IHT with Stochastic Polyak Step-size for Sparse Signal Recovery

Hard threshold acceleration algorithms for sparse signal recovery

Hard threshold acceleration algorithms for sparse signal recovery

Continuous-Time Sparse Signal Recovery

Continuous-Time Sparse Signal Recovery

An unrolled half-quadratic approach for sparse signal recovery in spectroscopy

This paper addresses the problem of sparse signal reconstruction frequently arising in analytical chemistry. Spectroscopic restoration is an important task that was tackled using two distinct approaches. First, model-based iterative methods were deployed. These methods rely on a strong theoretical background. However they face practical obstacles in terms of hyperparameter tuning. Conversely, approaches based on deep neural networks are fast and easy to implement. Nonetheless, they are often described as black-box tools, and their stability/robustness are ongoing research topics. In this paper, we propose a deep neural network based on unrolling a Majorization-Minimization (MM) Half-Quadratic (HQ) algorithm. This allows us to build interpretable layers mirroring iterations, making it possible to learn automatically data-driven hyperparameters such as regularization and stepsizes. Furthermore, we propose a dictionary of custom activation functions derived from potentials used in the original variational model. This interpretation of activations can be useful for analyzing the stability of neural networks. The efficiency of our method in comparison to iterative and learning-based methods is showcased through various experiments conducted on realistic mass spectrometry (MS) databases with various blur kernels and noise levels. All experiments show an improved reconstruction quality of chemical signals, while maintaining a low execution time at the test phase. All the results are reproducible, through the code shared on a repository available at https://github.com/GHARBIMouna/Unrolled-Half-Quadratic.

Approximating the zero-norm penalized sparse signal recovery using a hierarchical Bayesian framework

Sparse Bayesian learning (SBL) with its self-regulation and uncertainty estimation features, has become a popular topic in the field of sparse signal recovery. However, it is a challenging problem to employ zero-norm penalty on signals in the hierarchical framework of SBL. In this paper, an SBL approach that approximates the zero-norm penalized spare signal recovery is proposed based on the idea that signals can be modeled using generalized Gaussian distribution (GGDs). It should be noted that the mean and variance of GGDs are related to its shape parameters, making the direct use of GGDs in Bayesian inference impractical. To solve this problem, we derives a proposition and further build a new hierarchical Bayesian framework (HrBayFw). The proposed HrBayFw is equivalent to assign ℓp-norm penalty to signals, and the parameter p can be designed to approximate 0. Thus, the zero-norm penalized spare signal recovery can be realized using SBL, and the use of the proposition enables tractable marginalization over all parameters. The main advantage of the proposed approach is that it achieves a lower reconstruction error than current Bayesian methods. Numerical simulations evidence that the proposed SBL approach achieves better accuracy performance in terms of normalized-mean-square-error comparing with other contemporary algorithms.

Federated Consensus-Based Algorithm for Stable Recovery of Sparse Signals

Federated Consensus-Based Algorithm for Stable Recovery of Sparse Signals

Stable Recovery of Sparse Signals With Non-Convex Weighted $r$-Norm Minus 1-Norm

Stable Recovery of Sparse Signals With Non-Convex Weighted $r$-Norm Minus 1-Norm

On the Doppler Squint Effect in OTFS Systems Over Doubly-Dispersive Channels: Modeling and Evaluation

Extensive work has demonstrated the excellent performance of orthogonal time frequency space (OTFS) modulation in high-mobility scenarios. Time-variant wideband channel estimation serves as one of the key compositions of OTFS receivers since the data detection requires accurate channel state information (CSI). In practical wideband OTFS systems, the Doppler shift brought by the high mobility is frequency-dependent, which is referred to as the Doppler Squint Effect (DSE). Unfortunately, DSE was ignored in overall prior estimation schemes employed in OTFS systems, which leads to severe performance loss in channel estimation and the consequent data detection. In this paper, we investigate DSE of wideband time-variant channel in delay-Doppler domain and concentrate on the characterization of OTFS channel coefficients considering DSE. The formulation and evaluation of OTFS input-output relationship are provided for both ideal and rectangular waveforms considering DSE. The channel estimation is therefore formulated as a sparse signal recovery problem and an orthogonal matching pursuit (OMP)-based scheme is adopted to solve it. Simulation results confirm the significance of DSE and the performance superiority compared with traditional channel estimation approaches ignoring DSE.

An Improved GOMP Sparse Channel Estimation for Vehicle-to-Vehicle Communications

Reliable channel estimation is critical for wireless communication performance, especially in vehicle-to-vehicle (V2V) communication scenarios. Aiming at the major challenges of channel tracking and estimating as the highly dynamic nature of vehicle environments, an improved generalized orthogonal matching pursuit (iGOMP) is proposed for V2V channel estimation. The iGOMP algorithm transforms the channel estimation problem into a sparse signal recovery problem and replaces the classical inner product criterion with the Dice atom matching criterion. Additionally, the atomic weak progressive selection method is integrated to avoid the suboptimal selection of atoms from the redundant dictionary using the inner product criterion. The proposed iGOMP method can achieve optimal channel estimation by iterating feedback results. Simulation results demonstrate that the iGOMP method has superior estimation accuracy, mean square error (MSE), and bit error rate (BER) performance compared with traditional channel estimation methods in V2V communications.

Common structured sparsity based multi-user channel estimation for double-RIS assisted millimeter wave systems

Compared to a single reconfigurable intelligent surface (RIS) system, a double RIS system can achieve better passive beamforming gain, while requiring more accurate channel state information (CSI). However, due to the presence of double-reflection links, channel estimation with low pilot overhead is quite challenging. In this paper, we investigate channel estimation of double-RIS assisted multi-user millimeter wave (mmWave) communication systems. Firstly, the channel estimation problem for double-reflection links is formulated as a sparse signal recovery problem. Then, we analyze and exploit the common structured sparsity of the cascaded channel among multiple users, i.e., the common row and column sparsity in the angular channel matrix, and the corresponding common sparsity after being transformed into the channel vector. Inspired by this, we further propose a novel common structured sparsity based orthogonal matching pursuit (CSS-OMP) algorithm for multi-user joint channel estimation. Simulation results show that CSS-OMP algorithm can provide more accurate channel estimation with reduced pilot overhead compared to the conventional OMP algorithm.

Semi-tensor product-based one-bit compressed sensing

The area of one-bit compressed sensing (1-bit CS) focuses on the recovery of sparse signals from binary measurements. Over the past decade, this field has witnessed the emergence of well-developed theories. However, most of the existing literature is confined to fully random measurement matrices, like random Gaussian and random sub-Gaussian measurements. This limitation often results in high generation and storage costs. This paper aims to apply semi-tensor product-based measurements to 1-bit CS. By utilizing the semi-tensor product, this proposed method can compress high-dimensional signals using lower-dimensional measurement matrices, thereby reducing the cost of generating and storing fully random measurement matrices. We propose a regularized model for this problem that has a closed-form solution. Theoretically, we demonstrate that the solution provides an approximate estimate of the underlying signal with upper bounds on recovery error. Empirically, we conduct a series of experiments on both synthetic and real-world data to demonstrate the proposed method’s ability to utilize a lower-dimensional measurement matrix for signal compression and reconstruction with enhanced flexibility, resulting in improved recovery accuracy.

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 ReLU-based hard-thresholding algorithm for non-negative sparse signal recovery

In numerous applications, such as DNA microarrays, face recognition, and spectral unmixing, we need to acquire a non-negative K-sparse signal x from an underdetermined linear model y=Ax+v, where A is a sensing matrix and v is a noise vector. In this paper, we first propose a ReLU-based hard-thresholding algorithm (RHT) to recover x by taking advantage of its non-negative sparsity. Two sufficient conditions for stable recovery with RHT are then developed, which are respectively based on the restricted isometry property (RIP) and mutual coherence of the sensing matrix A. As far as we know, these two sufficient conditions are the best for hard-thresholding-type algorithms. Numerical experiments show that RHT has better overall recovery performance in the recovery non-negative sparse signals than the non-negative least squares (NNLS) algorithm, some hard-thresholding-type algorithms including the iterative hard-thresholding (IHT) algorithm, hard-thresholding pursuit (HTP), Newton-step-based iterative hard-thresholding algorithm (NSIHT), and Newton-step-based hard-thresholding pursuit (NSHTP), several non-negative sparse recovery approaches including non-negative orthogonal matching pursuit (NNOMP), fast NNOMP (FNNOMP), non-negative orthogonal least squares (NNOLS), and non-negative regularization (NNREG) method. In terms of efficiency, RHT has a similar performance to IHT, and is much more efficient than other tested algorithms.

Block sparse signal recovery via minimizing the block [formula omitted]-ratio sparsity

In this paper, we propose a method for block sparse signal recovery that minimizes the block q-ratio sparsity ‖z‖2,1/‖z‖2,qqq−1 with q∈[0,∞]. For the case of 1<q≤∞, we present the theoretical analyses and the computing algorithms for both cases of the ℓ2-bounded and ℓ2,∞-bounded noises. The corresponding unconstrained model is also investigated. Its superior performance in block sparse signal reconstruction is demonstrated by numerical experiments.

Generalized Penalized Constrained Regression: Sharp Guarantees in High Dimensions with Noisy Features

The generalized penalized constrained regression (G-PCR) is a penalized model for high-dimensional linear inverse problems with structured features. This paper presents a sharp error performance analysis of the G-PCR in the over-parameterized high-dimensional setting. The analysis is carried out under the assumption of a noisy or erroneous Gaussian features matrix. To assess the performance of the G-PCR problem, the study employs multiple metrics such as prediction risk, cosine similarity, and the probabilities of misdetection and false alarm. These metrics offer valuable insights into the accuracy and reliability of the G-PCR model under different circumstances. Furthermore, the derived results are specialized and applied to well-known instances of G-PCR, including l1-norm penalized regression for sparse signal recovery and l2-norm (ridge) penalization. These specific instances are widely utilized in regression analysis for purposes such as feature selection and model regularization. To validate the obtained results, the paper provides numerical simulations conducted on both real-world and synthetic datasets. Using extensive simulations, we show the universality and robustness of the results of this work to the assumed Gaussian distribution of the features matrix. We empirically investigate the so-called double descent phenomenon and show how optimal selection of the hyper-parameters of the G-PCR can help mitigate this phenomenon. The derived expressions and insights from this study can be utilized to optimally select the hyper-parameters of the G-PCR. By leveraging these findings, one can make well-informed decisions regarding the configuration and fine-tuning of the G-PCR model, taking into consideration the specific problem at hand as well as the presence of noisy features in the high-dimensional setting.

A truncated approximate difference algorithm for sparse signal recovery

In this paper, we study the regularization lp-norm minimization problem to recover the sparse signals. We first prove that every global optimal solution to the regularization lp-norm minimization problem also solves the l0-norm minimization problem if the certain conditions are satisfied, and then generate a truncated approximated difference algorithm to recover the sparse signals. At last, we provide some numerical simulations to test the performance of the truncated approximated difference algorithm, and the numerical results show that the proposed algorithm performs effectively in recovering the sparse signals compared with some state-of-art methods.

A Sparse Recovery Algorithm Using the Neurodynamic System With Predefined Time Convergence

Compressed sensing aims to compress sparse signals losslessly into significantly smaller samples and recover them if needed. One of the most critical approaches to recovering the sparse signal is minimizing l1-norm, the convex relaxation of l0-norm with the limit of linear measurement. However, sometimes such as the recovery of piecewise constant sparse signal, the need for controlling the convergence time has become an inevitable problem to solve. To solve this problem, the predefined time pseudo-inverse locally competitive algorithm (PPLCA) based on a nonlinear neurodynamic system is proposed in this paper, which is exceptionally effective if the signal is piecewise constant. Further, under the restricted isometry property (RIP), the predefined-time convergence rate of the proposed PPLCA is proven using Lyapunov’s stability theory. Finally, the numerical simulations show that the PPLCA could converge to the stationary point within the predefined time.

Sparse Signal Recovery through Long Short-Term Memory Networks for Compressive Sensing-Based Speech Enhancement

This paper presents a novel speech enhancement approach based on compressive sensing (CS) which uses long short-term memory (LSTM) networks for the simultaneous recovery and enhancement of the compressed speech signals. The advantage of this algorithm is that it does not require an iterative process to recover the compressed signals, which makes the recovery process fast and straight forward. Furthermore, the proposed approach does not require prior knowledge of signal and noise statistical properties for sensing matrix optimization because the used LSTM can directly extract and learn the required information from the training data. The proposed technique is evaluated against white, babble, and f-16 noises. To validate the effectiveness of the proposed approach, perceptual evaluation of speech quality (PESQ), short-time objective intelligibility (STOI), and signal-to-distortion ratio (SDR) were compared to other variants of OMP-based CS algorithms The experimental outcomes show that the proposed approach achieves the maximum improvements of 50.06%, 43.65%, and 374.16% for PESQ, STOI, and SDR respectively, over the different variants of OMP-based CS algorithms.

R-Lah distribution: Properties, limit theorems and an application to compressed sensing

We introduce and study the r-Lah distribution whose definition involves r-Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements.