Approximated Function Based Spectral Gradient Algorithm for Sparse Signal Recovery

HKUST Electronic Theses Second-order algorithms for sparse signal recovery from intensity-only measurements by Ruixue Wen thesis 2022 1 online resource (xiv, 113 pages) : illustrations (some color) In this thesis, we study the sparse phase retrieval problem, which is to…Read more ›

The algorithms based on the technique of optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding (OT) were recently proposed for signal recovery, and they are very different from the traditional family of hard thresholding methods. However, the computational cost for OT-based algorithms remains high at the current stage of their development. This stimulates the development of the so-called natural thresholding (NT) algorithm and its variants in this paper. The family of NT algorithms is developed through the first-order approximation of the so-called regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model, and thus the computational cost for this family of algorithms is significantly lower than that of the OT-based algorithms. The guaranteed performance of NT-type algorithms for signal recovery from noisy measurements is shown under the restricted isometry property and concavity of the objective function of regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model. Empirical results indicate that the NT-type algorithms are robust and very comparable to several mainstream algorithms for sparse signal recovery.

The problem of sparse signal recovery from lower number of observations is formalized as a constrained minimization problem. Due to the dissimilar requirements of signal processing, the constraints and objective functions differ from each other. This paper presents an algorithm scheme base on alternating iteration to deal with the nonconvex objective function. The proposed method employs the log-penalty and proximity operator to seek the sparse solution. In each iteration, the introduction of nonconvex penalty decreases the penalization of large coefficients which contributes to a faster decrease of the function value. By making use of proximity operator, it reduces the computational complexity and achieves less number of iterations. In addition, the experimental results illustrate the effectiveness of the proposed method as well as the drawbacks of the algorithm.

This study concerns the issue of jointly enhancing noise robustness and promoting signal sparsity in Sparse Bayesian Learning (SBL), which aims at addressing the performance deficiency of sparse signal recovery due to uninformative data with low signal-to-noise ratios. In particular, the authors propose a hierarchical prior noise model with a signal-dependent parametrisation and incorporate it into developing the robust SBL algorithms for sparse signal recovery. The main contribution of the proposed approach is twofold. The first is the new consideration of noise-robustness enhancement in building SBL algorithms, which devotes to noise awareness in counteracting outliers in measurements. Specifically, the idea of signal-sparsity enforcing is extended to build a Least Absolute Deviation like loss criterion with the proposed hierarchical prior model of measurement noise. The second is the novelty of using the signal-dependent parametrisation in the proposed noise model. Indeed, the signal-dependent mechanism plays an indispensable role in producing the reliable noise parameter estimation jointly with updating signal model parameters under the fast SBL framework. In addition to numerical simulation studies, the real-life application of radio tomographic imaging is presented to validate the proposed approach.

Sparse optimization problems have been successfully applied to a wide range of research areas, and useful insights and elegant methods for proving the stability and convergence of neurodynamic algorithms have been yielded in previous work. This article develops several neurodynamic algorithms for sparse signal recovery by solving the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> regularization problem. First, in the framework of the locally competitive algorithm (LCA), modified LCA (MLCA) with finite-time convergence and MLCA with fixed-time convergence are designed. Then, the sliding-mode control (SMC) technique is introduced and modified, i.e., modified SMC (MSMC), which is combined with LCA to design MSMC-LCA with finite-time convergence and MSMC-LCA with fixed-time convergence. It is shown that the solutions of the proposed neurodynamic algorithms exist and are unique under the observation matrix satisfying restricted isometry property (RIP) condition, while finite-time or fixed-time convergence to the optimal points is shown via Lyapunov-based analysis. In addition, combining the notions of finite-time stability (FTS) and fixed-time stability (FxTS), upper bounds on the convergence time of the proposed neurodynamic algorithms are given, and the convergence results obtained for the MLCA and MSMC-LCA with fixed-time convergence are shown to be independent of the initial conditions. Finally, simulation experiments of signal recovery and image recovery are carried out to demonstrate the superior performance of the proposed neurodynamic algorithms.

This paper discusses compressive sensing with sparse measurement matrices. Sparse matrices have several attractive properties, like low computational complexity in both encoding and recovery, easy incremental updates to signals, and low storage requirement, etc. Typical examples of existing algorithms for sparse signal recovery with sparse measurement matrices include convex relaxation, matching pursuit, and Bayesian framework based approaches. In this paper, we propose an alternative technique to this problem. The proposed technique has a linear recovery complexity and relatively good empirical behavior. In this technique, we employ a permutation-based multi-dimensional measurement matrix, which is composed of several sub-matrices, each consisting of a block-diagonal matrix and a random permutation matrix. The measurement symbols generated from the same permutation matrix are referred to as a dimension. Such a measurement matrix brings some useful features to the measurement symbols. Fully exploiting these features enables us to design a simple and effective recovery algorithm. The proposed recovery algorithm employs an iterative process. In each iteration, the algorithm looks for measurement symbols with certain features, and uses such features to recover the source symbols related to these measurements. An iterative process is then applied, along with an interference cancellation operation, to reconstruct all source symbols gradually. The complexity of the proposed algorithm is relatively low, which grows linearly with the source signal length. Numerical results show that the proposed technique empirically offers a much lower sketch length than ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -minimization-based convex relaxation and Bayesian framework based algorithms. It achieves the empirical lower bound of sketch length and linear recovery complexity at the same time.

Numerous algorithms have been proposed recently for sparse signal recovery in Compressed Sensing (CS). In practice, the number of measurements can be very limited due to the nature of the problem and/or the underlying statistical distribution of the non-zero elements of the sparse signal may not be known a priori. It has been observed that the performance of any sparse signal recovery algorithm depends on these factors, which makes the selection of a suitable sparse recovery algorithm difficult. To take advantage in such situations, we propose to use a fusion framework using which we employ multiple sparse signal recovery algorithms and fuse their estimates to get a better estimate. Theoretical results justifying the performance improvement are shown. The efficacy of the proposed scheme is demonstrated by Monte Carlo simulations using synthetic sparse signals and ECG signals selected from MIT-BIH database.

In this paper, we consider the problem of finding the sparsest solution to underdetermined linear systems. Unlike the literatures which use the ℓ1 regularization to approximate the original problem, we consider the ℓ2/3 regularization which leads to a better approximation but a nonconvex, nonsmooth, and non-Lipschitz optimization problem. Through developing a fixed point representation theory associated with the two thirds thresholding operator for ℓ2/3 regularization solutions, we propose a fixed point iterative thresholding algorithm based on two thirds norm for solving the k-sparsity problems. Relying on the restricted isometry property, we provide subsequentional convergence guarantee for this fixed point iterative thresholding algorithm on recovering a sparse signal. By discussing the preferred regularization parameters and studying the phase diagram, we get an adequate and efficient algorithm for the high-dimensional sparse signal recovery. Finally, comparing with the existing algorithms, such as the standard ℓ1 minimization, the iterative reweighted ℓ2 minimization, the iterative reweighted ℓ1 minimization, and iterative Half thresholding algorithm, we display the results of the experiment which indicate that the two thirds norm fixed point iterative thresholding algorithm applied to sparse signal recovery and large scale imageries from noisy measurements can be accepted as an effective solver for ℓ2/3 regularization.

Ground Penetrating Radar (GPR) is essential for subsurface exploration. Conventional GPR 3D imaging demands dense spatial sampling along regular grids, which is both time-consuming and impractical in complex environments. In this work, we propose a novel method that combines sparse recovery techniques with a placement matrix to merge arbitrarily and sparsely sampled measurements into a regular grid framework. By exploiting the inherent sparsity of subsurface targets and using the Dantzig Selector with cross-validation, our method reconstructs the target reflectivity vector from random spatial sampling. The recovered data is then processed via the Back-Projection Algorithm (BPA) to generate high-resolution 3D images. Simulations demonstrate that our approach not only improves imaging quality under reduced sampling conditions but also efficiently handles arbitrary scanning paths by mapping irregular measurements onto the desired grid.

In this paper, we propose a novel sparse signal recovery algorithm called Trainable ISTA (TISTA). The proposed algorithm consists of two estimation units such as a linear estimation unit and a minimum mean squared error (MMSE) estimator-based shrinkage unit. The estimated error variance required in the MMSE shrinkage unit is precisely estimated from a tentative estimate of the original signal. The remarkable feature of the proposed scheme is that TISTA includes adjustable variables controlling a step size and the error variance for the MMSE shrinkage. The variables are adjusted by standard deep learning techniques. The number of trainable variables of TISTA is equal to the number of iteration rounds and it is much smaller than that of known learnable sparse signal recovery algorithms. This feature leads to highly stable and fast training processes of TISTA. Computer experiments show that TISTA is applicable to various classes of sensing matrices such as Gaussian matrices, binary matrices and matrices with large condition numbers. Numerical results also demonstrate that TISTA shows significantly faster convergence than those of AMP and LISTA in many cases.

In this paper, we propose a novel sparse signal recovery algorithm, DU-PSISTA (Deep Unfolded-Periodic Sketched Iterative Shrinkage-Thresholding Algorithm), which aims to balance computational efficiency and accuracy for recovering high-dimensional sparse signals. The proposed method periodically alternates between standard ISTA and a sketched variant that leverages random projections to reduce the dimensionality of gradient computations. This hybrid structure enables flexible control over the trade-off between accuracy and computational complexity through a pre-configurable period parameter. Furthermore, we incorporate deep unfolding into the algorithm that embeds learnable parameters such as step sizes and thresholding factors into each iteration. These parameters are optimized through data-driven training, enabling the model to adaptively improve convergence and performance. Numerical experiments confirm that our method achieves comparable recovery performance to conventional deep unfolded ISTA while reducing computation, especially when the period parameter and sketch size are properly selected.

A modified multiple OLS (m2OLS) algorithm for signal recovery in compressive sensing

In this paper, we introduce a new sparse signal recovery algorithm referred to as the matching pursuit with greedy tree search (GTMP). The tree search in our proposed method is implemented to minimize the cost function to improve the recovery performance of sparse signals. In addition, a pruning strategy is employed to each node of the tree for efficient implementation. In our performance guarantee analysis, we provide the condition that ensures the exact identification of the nonzero locations. Through empirical simulations, we show that GTMP is effective for sparse signal reconstruction and outperforms conventional sparse recovery algorithms.

Wearable ECG monitoring devices, often limited by power constraints, require efficient methods to reduce energy consumption and extend battery life for continuous health tracking. This work proposes a digital compressed sensing approach for ECG (DCS-ECG), featuring a suitable sensing matrix, effective sparse signal recovery algorithms, and an optimized number of bits for quantization (BM). The framework is designed to achieve high compression ratios (CR) with minimal signal distortion and parameter extraction without the reconstruction of the original signal. The following five sensing matrices are examined: deterministic block binary diagonal (DBBD) matrix, random binary (RBin) matrix, random Bernoulli (RBen) matrix, random Gaussian (RG) matrix, and random sparse fixed binary (RSB) matrix. These are paired with a sparsifying matrix (ψ), a hybrid of identity matrix (I), discrete sine transform matrix (S), and discrete cosine transform matrix (C) (i.e. ψ = [I S C]). The following five sparse reconstruction algorithms are evaluated: iterative hard thresholding (IHT) algorithm, orthogonal matching pursuit (OMP) algorithm, smoothed ℓ0 (SL0) algorithm, ℓ1 least square (L1LS) algorithm, and subspace matching pursuit (SMP) algorithm. The proposed DCS-ECG system was tested on ECG signals from the MIT-BIH arrhythmia database. The configuration using the DBBD sensing matrix and OMP recovery algorithm with the sparsifying matrix achieved a low average percentage of root mean square difference (PRD) of 3.75 ± 1.43% at CR = 12.5%, which also offers fast reconstruction. For CR = 12.5% and BM = 8, the CS measurement generation on the Arduino Due consumes 2.02 mJ, while reconstruction on the Raspberry Pi requires 1.13 J. The wireless transmission energy is 1.2973 J. Overall, the proposed CS-based system achieves an 86.24% total energy saving compared to the without CS approach, demonstrating its suitability for low-power IoT-enabled wearable edge ECG health monitoring applications.

Seeking sparse solutions of underdetermined linear systems is required in many areas of engineering and science such as signal and image processing. The efficient sparse representation becomes central in various big or high-dimensional data processing, yielding fruitful theoretical and realistic results in these fields. The mathematical optimization plays a fundamentally important role in the development of these results and acts as the mainstream numerical algorithms for the sparsity-seeking problems arising from big-data processing, compressed sensing, statistical learning, computer vision, and so on. This has attracted the interest of many researchers at the interface of engineering, mathematics and computer science. Sparse Optimization Theory and Methods presents the state of the art in theory and algorithms for signal recovery under the sparsity assumption. The up-to-date uniqueness conditions for the sparsest solution of underdertemined linear systems are described. The results for sparse signal recovery under the matrix property called range space property (RSP) are introduced, which is a deep and mild condition for the sparse signal to be recovered by convex optimization methods. This framework is generalized to 1-bit compressed sensing, leading to a novel sign recovery theory in this area. Two efficient sparsity-seeking algorithms, reweighted l1-minimization in primal space and the algorithm based on complementary slackness property, are presented. The theoretical efficiency of these algorithms is rigorously analysed in this book. Under the RSP assumption, the author also provides a novel and unified stability analysis for several popular optimization methods for sparse signal recovery, including l1-mininization, Dantzig selector and LASSO. This book incorporates recent development and the author's latest research in the field that have not appeared in other books.