Restricted Isometric Property Research Articles

An efficient optimization of measurement matrix for compressive sensing

Compressive sensing (CS) is a new paradigm for signal acquisition and reconstruction, which can reconstruct the signal at less than the Nyquist sampling rate. The sampling of the signal occurs through a measurement matrix (MM); thus, MM generation is significant in the context of the CS framework. In this paper, an optimization algorithm is introduced for the generation of the MM of CS based on Restricted Isometric Property (RIP) mandates that eigenvalues of the sensing matrix fall within an interval also minimizes the mutual coherence of the sensing matrix (i.e. the product of the MM and sparsifying matrix). A novel gradient-based iterative optimization method is used to reduce the eigenvalues of the sensing matrix by SVD decomposition. Meanwhile, the proposed algorithm can also reduce the operational complexity. Experimental results and analysis prove that the optimized MM reduces the maximum mutual and average mutual coherence between the MM and the sparsifying basis, which shows the effectiveness of the proposed algorithm over some state-of-art works.

The null space property of the weighted ℓr − ℓ1 minimization

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This paper studies NSP of the weighted [Formula: see text] ([Formula: see text]) minimization. Several versions of NSP of the weighted [Formula: see text] minimization including the weighted [Formula: see text] NSP, the weighted [Formula: see text] stable NSP, the weighted [Formula: see text] robust NSP and the [Formula: see text] weighted [Formula: see text] robust NSP for [Formula: see text], are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted [Formula: see text] minimization are established. Furthermore, we show that to some extent, the weighted [Formula: see text] stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022).

Deterministic Construction of Compressed Sensing Measurement Matrix with Arbitrary Sizes via QC-LDPC and Arithmetic Sequence Sets

It is of great significance to construct deterministic measurement matrices with good practical characteristics in Compressed Sensing (CS), including good reconstruction performance, low memory cost and low computing resources. Low-density-parity check (LDPC) codes and CS codes can be closely related. This paper presents a method of constructing compressed sensing measurement matrices based on quasi-cyclic (QC) LDPC codes and arithmetic sequence sets. The cyclic shift factor in each submatrix of QC-LDPC is determined by arithmetic sequence sets. Compared with random matrices, the proposed method has great advantages because it is generated based on a cyclic shift matrix, which requires less storage memory and lower computing resources. Because the restricted isometric property (RIP) is difficult to verify, mutual coherence and girth are used as computationally tractable indicators to evaluate the measurement matrix reconstruction performance. Compared with several typical matrices, the proposed measurement matrix has the minimum mutual coherence and superior reconstruction capability of CS signal according to one-dimensional (1D) signals and two-dimensional (2D) image simulation results. When the sampling rate is 0.2, the maximum (minimum) gain of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) is up to 2.89 dB (0.33 dB) and 0.031 (0.016) while using 10 test images. Meanwhile, the reconstruction time is reduced by nearly half.

Sparse Kronecker product decomposition: a general framework of signal region detection in image regression

Abstract This paper aims to present the first Frequentist framework on signal region detection in high-resolution and high-order image regression problems. Image data and scalar-on-image regression are intensively studied in recent years. However, most existing studies on such topics focussed on outcome prediction, while the research on region detection is rather limited, even though the latter is often more important. In this paper, we develop a general framework named Sparse Kronecker Product Decomposition (SKPD) to tackle this issue. The SKPD framework is general in the sense that it works for both matrices and tensors represented image data. Our framework includes one-term, multi-term, and nonlinear SKPDs. We propose nonconvex optimization problems for one-term and multi-term SKPDs and develop path-following algorithms for the nonconvex optimization. Under a Restricted Isometric Property, the computed solutions of the path-following algorithm are guaranteed to converge to the truth with a particularly chosen initialization even though the optimization is nonconvex. Moreover, the region detection consistency could also be guaranteed. The nonlinear SKPD is highly connected to shallow convolutional neural networks (CNN), particularly to CNN with one convolutional layer and one fully-connected layer. Effectiveness of SKPD is validated by real brain imaging data in the UK Biobank database.

Coherence Based Sufficient Condition for Support Recovery Using Generalized Orthogonal Matching Pursuit

In an underdetermined system, compressive sensing can be used to recover the support vector. Greedy algorithms will recover the support vector indices in an iterative manner. Generalized Orthogonal Matching Pursuit (GOMP) is the generalized form of the Orthogonal Matching Pursuit (OMP) algorithm where a number of indices selected per iteration will be greater than or equal to 1. To recover the support vector of unknown signal ‘x’ from the compressed measurements, the restricted isometric property should be satisfied as a sufficient condition. Finding the restricted isometric constant is a non-deterministic polynomial-time hardness problem due to that the coherence of the sensing matrix can be used to derive the sufficient condition for support recovery. In this paper a sufficient condition based on the coherence parameter to recover the support vector indices of an unknown sparse signal ‘x’ using GOMP has been derived. The derived sufficient condition will recover support vectors of P-sparse signal within ‘P’ iterations. The recovery guarantee for GOMP is less restrictive, and applies to OMP when the number of selection elements equals one. Simulation shows the superior performance of the GOMP algorithm compared with other greedy algorithms.

Compressed data separation via unconstrained l1-split analysis

We investigate the problems of reconstruction signals’ distinct subcomponents, that are approximately sparse in terms of morphologically different dictionaries, from a few noisy linear measurements. We show that under common assumptions on a restricted isometric property adapted to the composed dictionary and a mutual coherence condition between the two different dictionaries, the two distinct subcomponents can be stably and approximately recovered via the unconstrained [Formula: see text]-split analysis, with an error bound that is in general unimprovable for the Gaussian noises.

Adaptive Compressed Spectrum Sensing Using Neural Networks in Cognitive Radio Networks

In today's era of Internet of Things (IoT) applications, Cognitive Radio (CR) is recognized as the most promising technology to access the unused spectrum. Such CR networks can be easily built using Wide-band spectrum sensing. However, the processing of wide-band signals involves a high sampling rate. In such scenario, Compressed Spectrum Sensing (CSS) overcomes the challenges of real-time signal recovery and sampling. CSS uses Sparse signals which are widely used in many applications as they aid in processing large data. Beyond sparsity, all the real-world signals will have special structures (like Restricted Isometric property, Null space property). Moreover, it is very hard to determine the domain in which the signal is sparse. To find this we need a random measurement matrix which plays an important role in extracting the sparse coefficients of the signal. In this paper, a customized neural network is employed to identify the peculiar structures of sparse signals for efficient recovery in real time. The neural network learns and trains an adaptive measurement matrix from the sparse signals to reduce the sensing overhead. The transmitter and receiver systems are configured using Universal Software Radio Peripheral (USRP) boards with LabVIEW® and MATLAB® extensions for peer to peer communications. The implementation results depict the superior performance of neural network based recovery in assimilating the additional structures of real-time signals.

ANN-Assisted CoSaMP Algorithm for Linear Electromagnetic Imaging of Spatially Sparse Domains

Greedy pursuit algorithms (GPAs) are widely used to reconstruct sparse signals. Even though many electromagnetic (EM) inverse scattering problems are solved on sparse investigation domains, GPAs have rarely been used for this purpose. This is because (i) they require a priori knowledge of the sparsity level in the investigation domain, which is often not available in EM imaging applications, and (ii) the EM scattering matrix does not satisfy the restricted isometric property. In this work, these challenges are respectively addressed by (i) using an artificial neural network (ANN) to estimate the sparsity level, and (ii) adding a Tikhonov regularization term to the diagonal elements of the scattering matrix. These enhancements permit the compressive sampling matching pursuit (CoSaMP) algorithm to be efficiently used to solve the two-dimensional EM inverse scattering problem, which is linearized using the Born approximation, on spatially sparse investigation domains. Numerical results, which demonstrate the efficiency and applicability of the proposed ANN-enhanced CoSaMP algorithm, are provided.

Enhanced Expressive Power and Fast Training of Neural Networks by Random Projections

Random projections are able to perform dimension reduction efficiently for datasets with nonlinear low-dimensional structures. One well-known example is that random matrices embed sparse vectors into a low-dimensional subspace nearly isometrically, known as the restricted isometric property in compressed sensing. In this paper, we explore some applications of random projections in deep neural networks. We provide the expressive power of fully connected neural networks when the input data are sparse vectors or form a low-dimensional smooth manifold. We prove that the number of neurons required for approximating a Lipschitz function with a prescribed precision depends on the sparsity or the dimension of the manifold and weakly on the dimension of the input vector. The key in our proof is that random projections embed stably the set of sparse vectors or a low-dimensional smooth manifold into a low-dimensional subspace. Based on this fact, we also propose some new neural network models, where at each layer the input is first projected onto a low-dimensional subspace by a random projection and then the standard linear connection and non-linear activation are applied. In this way, the number of parameters in neural networks is significantly reduced, and therefore the training of neural networks can be accelerated without too much performance loss.

A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence.

In this paper, the problem of constructing the measurement matrix in compressed sensing is addressed. In compressed sensing, constructing a measurement matrix of good performance and easy hardware implementation is of interest. It has been recently shown that the measurement matrices constructed by Logistic or Tent chaotic sequences satisfy the restricted isometric property (RIP) with a certain probability and are easy to be implemented in the physical electric circuit. However, a large sample distance that means large resources consumption is required to obtain uncorrelated samples from these sequences in the construction. To solve this problem, we propose a method of constructing the measurement matrix by the Chebyshev chaotic sequence. The method effectively reduces the sample distance and the proposed measurement matrix is proved to satisfy the RIP with high probability on the assumption that the sampled elements are statistically independent. Simulation results show that the proposed measurement matrix has comparable reconstruction performance to that of the existing chaotic matrices for compressed sensing.

Deterministic construction of array QC CS measurement matrices based on Singer perfect difference sets

Low-density parity-check (LDPC) codes and compressed sensing (CS) share many common environments. In this study, a novel approach for constructing a new class of deterministic sparse sensing matrices based on array quasi-cyclic (QC) LDPC codes via Singer perfect difference sets is proposed. In contrast to random and the other deterministic matrices, the proposed framework would be highly desirable as it is generated based on circulant permutation matrices, which requires less memory for storage and lower computational cost for sensing. Since the restricted isometric property is very difficult to verify, then the mutual coherence and the girth are two computationally tractable criteria that the authors used to assess the CS recovery capabilities of sensing matrices. In addition, inspired by LDPC codes, they extract a necessary condition for the proposed measurement matrix to have effective values for girth as large as g ≥ 6 and 8 . Comprehensive one-dimensional (1D) and 2D simulations verify that their proposed sensing matrix has minimum coherence and superior CS recovery abilities in comparison with the corresponding random Gaussian, Bernoulli, and the other deterministically generated matrices. Furthermore, the required physical storage space and the complexity of the hardware implementation are greatly reduced due to being sparse and QC in structure.

Scalable Structured Compressive Video Sampling With Hierarchical Subspace Learning

Conventional scalable compressive video sampling cannot sufficiently exploit the structured sparsity within video sequences. This paper proposes a novel quality scalable structured compressive video sampling (SS-CVS) framework with hierarchical subspace learning to support video transmission over heterogeneous network conditions. The proposed framework incorporates the union of data-driven subspaces (UoDS) model to introduce structured sparsity into sensing matrix for progressive acquisition of measurements. Hierarchical subspace learning is developed to generate the subspaces with their bases based on adaptive subspace clustering in a progressive fashion. Consequently, two hierarchical structures are derived to enable progressive mapping of subspaces and bases on the structured sensing matrices to support quality scalability. To guarantee the convergence of hierarchical subspace learning, the training set is updated with the adaptive group set from the reconstructed reference frames. It is demonstrated that SS-CVS can guarantee stable recovery for each quality layer under the constraints of block restricted isometric property (RIP). Furthermore, the proposed SS-CVS model is generalized to tensor subspaces for scalable compressive sampling of high-dimensional signals. Experimental results demonstrate that the proposed algorithm can achieve quality scalabilities with an improved reconstructed performance in comparison to the state-of-the-art scalable compressive video sampling methods.

Effective Prediction for Rock Burst Dataset Using Classification Algorithms with Particle Swarm

Rock burst and Slope Stability is one of the common failures in hard rock mining and civil construction. This study focuses on the prediction of rock burst and Slope Stability classification with case instances using cloud models and attribution weight. First, cloud models are introduced briefly related to the rock burst and Slope Stability classification problem. Then, the attribution weight method is presented to quantify the contribution of each rock burst and Slope Stability indicator for classification. In addition, analysis and prediction of slope stability is of great importance in geotechnical engineering. With the development of economics, the number of slopes is increasing and landslides caused by slope instability have become one of the three major geological disasters in the world along with earthquakes and volcanoes. To reduce or prevent landslide damage, slope stability analysis and stabilization are required. However, accurately predicting slope stability is challenging because of the complexity of slope structures and the difficulty to determine the precise input data associated with key geotechnical parameters the proposed methodology PSO feature extraction preserves important distance relationships, such as : The Random forest, Naive Bayes of each object of the original dataset. This leads to preservation of any mining operation that depends on the ordering of distances between objects, such as Random forest, Naive Bays -search, SVM, J.48 and MLP classification, as well as many visualization techniques. In particular, it establishes a restricted isometric property, i.e., tight bounds on the contraction/expansion of the original distances.

A General Framework for Understanding Compressed Subspace Clustering Algorithms

Subspace clustering (SC) refers to the problem of clustering unlabeled high-dimensional data into a union of low-dimensional linear subspaces. In many practical scenarios, one may have access to only the compressed data due to constraints of measurement or computation. In this paper, based on the recently proposed restricted isometric property of Gaussian random projection for low-dimensional subspaces, we propose a general framework for analyzing the performance of various subspace clustering algorithms when applied to the compressed data. Our framework captures the connection between the problems of compressed subspace clustering (CSC) and noisy subspace clustering. With the existing study on noisy SC, our framework makes it possible to easily extend the results in noisy SC to CSC. In this paper, we select the most commonly used subspace clustering algorithms, i.e. , sparse SC (SSC), SSC-orthogonal matching pursuit (SSC-OMP), and thresholding based SC (TSC), as representatives and analyze their performance using the proposed framework. Our results are consistent with related work obtained by previous researchers, where our methodology is much more direct and has more universal implications. Finally, the practicability and efficiency of CSC are verified by numerical experiments.

Iteratively reweighted algorithm for signals recovery with coherent tight frame

We consider the problem of compressed sensing with a coherent tight frame and design an iteratively reweighted least squares algorithm to solve it. To analyze the problem, we propose a sufficient null space property under a tight frame (sufficient D‐NSP). We show that, if a measurement matrix A satisfies the sufficient D‐NSP of order s, then an s‐sparse signal under the tight frame can be exactly recovered. Furthermore, if A satisfies the restricted isometric property with tight frame D of order 2bs, then it also satisfies the sufficient D‐NSP of order as with a < b and b sufficiently large. We prove the convergence of the algorithm based on the sufficient D‐NSP and give the upper error bounds. In numerical experiments, we use the discrete cosine transform, discrete Fourier transform, and Haar wavelets to verify the effectiveness of this algorithm. With increasing measurement number, the signal‐to‐noise ratio increases monotonically.

A Data-Gathering Scheme with Joint Routing and Compressive Sensing Based on Modified Diffusion Wavelets in Wireless Sensor Networks.

Compressive sensing (CS)-based data gathering is a promising method to reduce energy consumption in wireless sensor networks (WSNs). Traditional CS-based data-gathering approaches require a large number of sensor nodes to participate in each CS measurement task, resulting in high energy consumption, and do not guarantee load balance. In this paper, we propose a sparser analysis that depends on modified diffusion wavelets, which exploit sensor readings’ spatial correlation in WSNs. In particular, a novel data-gathering scheme with joint routing and CS is presented. A modified ant colony algorithm is adopted, where next hop node selection takes a node’s residual energy and path length into consideration simultaneously. Moreover, in order to speed up the coverage rate and avoid the local optimal of the algorithm, an improved pheromone impact factor is put forward. More importantly, theoretical proof is given that the equivalent sensing matrix generated can satisfy the restricted isometric property (RIP). The simulation results demonstrate that the modified diffusion wavelets’ sparsity affects the sensor signal and has better reconstruction performance than DFT. Furthermore, our data gathering with joint routing and CS can dramatically reduce the energy consumption of WSNs, balance the load, and prolong the network lifetime in comparison to state-of-the-art CS-based methods.

Mobility-Aware Reconstruction Algorithm for Correlated Nonzero Neighborhood Structured Downlink Channel in Massive MIMO

This paper investigates downlink (DL) channel estimation of frequency division duplex (FDD) massive multiple-input-multiple-output (MIMO) with a finite number of base station (BS) antennas. Based on the performance analysis of DL channel in FDD, it is demonstrated that the interference effect cannot be ignored in the finite region. Also, excessive training overhead is required in conventional channel estimation methods when the number of BS antennas goes large. Therefore, it is necessary to design an algorithm for alleviating the interference in addition to the reduction of pilot overhead. In this paper, a general model is presented to exploit certain sparsity structure of massive MIMO channel in beam domain, which is called nonzero-neighborhood (NZN) structure. Considering the smooth variation of channel paths during two consecutive estimation intervals, a compressed sensing-based algorithm is proposed to efficiently estimate and track NZN-structured DL channel. The convergence behavior and computational complexity of the proposed mobility-aware subspace pursuit (MA-SP) algorithm are analyzed based on restricted isometric property. It is demonstrated that the application of proposed MA-SP algorithm efficiently reduces the required pilot length in channel estimation. Interference analysis shows that inter-cell interference is considerably alleviated in the proposed channel estimation phase without any prior information about interference. Simulation results demonstrate the effectiveness of the proposed MA-SP technique compared with the previous methods.

A Sparse and Low-Rank Near-Isometric Linear Embedding Method for Feature Extraction in Hyperspectral Imagery Classification

A sparse and low-rank near-isometric linear embedding (SLRNILE) method has been proposed to make dimensionality reduction and extract proper features for hyperspectral imagery (HSI) classification. The SLRNILE stands on the theory of the John-Lindenstrauss lemma, and tries to estimate a sparse and low-rank projection matrix that satisfies the restricted isometric property (RIP) condition on all secants of the HSI data. The RIP condition guarantees that the desired linear mapping near-isometrically preserves nearest neighbor points of all HSI pixels. Seeking the desired mapping is then modeled into minimizing a Lagrange multipliers formulation. The alternating direction method of multipliers framework is utilized to solve the above convex program, and column generation techniques are adopted to alleviate the computation memory burden during the optimization procedure. Five experiments on three widely used HSI data sets are designed to completely test the performance of SLRNILE, and experimental results are compared against those of six state-of-the-art feature extraction methods, including principal component analysis, Laplacian eigenmaps, locality preserving projections, neighborhood preserving embedding, sparse nonnegative matrix underapproximation, and random projections. The results show that SLRNILE performs best among all the seven methods, and its computational time is longest of all but still bearable for regular users. Therefore, the SLRNILE can be a good choice for feature extraction in HSI classification.

One-bit sensing, discrepancy and Stolarsky's principle

A sign-linear one-bit map from the -dimensional sphere to the -dimensional Hamming cube is given by where . For , we estimate , the smallest integer so that there is a sign-linear map which has the -restricted isometric property, where we impose the normalized geodesic distance on and the Hamming metric on . Up to a polylogarithmic factor, , which has a dimensional correction in the power of . This is a question that arises from the one-bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the -average of the embedding error is equivalent to minimizing the discrete energy , where is the normalized geodesic distance. Bibliography: 39 titles.

Channel Estimation for Finite Scatterers Massive Multi-User MIMO System

In finite scattering propagation environment, when the number of scatterers is very small compared to the number of base station antennas and users in the cell and if the same scatterers are shared by all users, then the correlation among the users increases. Hence, the high-dimensional multi-user MIMO system is likely to have low-rank channel. To estimate the channel matrix, weighted nuclear norm minimization method is proposed in this paper. Iterative weighted singular value thresholding algorithm is used to solve the optimization problem. To recover the low-rank channel, a partial random Fourier matrix (PRFM) satisfying the restricted isometric property is adapted as the training matrix. The PRFM forces the iterative algorithm to converge in one iteration which reduces the huge computational complexity of the proposed channel estimation method. The mean square error and achievable sum rate are the criteria used to measure the performance of the proposed method. The results show that the proposed method outperforms the least square estimation method and the nuclear norm minimization method for various finite scatterers in different SNR levels.