Sparsity Order Research Articles

Sparsity‐aware adaptive block‐based compressive sensing

Conventional methods for block-based compressive sensing consider an equal number of samples for all blocks. However, the sparsity order of blocks in natural images could be different and, therefore, a various number of samples could be required for their reconstruction. In this study, the authors propose an adaptive block-based compressive sensing scheme, which collects a different number of samples from each block. The authors show that by adapting the sampling rate, in addition to reducing the whole required number of measurements, the reconstruction performance would be improved, simultaneously. Simulation results verify the effectiveness of the proposed scheme, especially for multi-level pixel value images like Mondrian test image.

Graphs with sparsity order at most two: the complex case

The sparsity order of a (simple undirected) graph is the highest possible rank (over or ) of the extremal elements in the matrix cone that consists of positive semidefinite matrices with prescribed zeros on the positions that correspond to non-edges of the graph (excluding the diagonal entries). The graphs of sparsity order 1 (for both and ) correspond to chordal graphs, those graphs that do not contain a cycle of length greater than three, as an induced subgraph, or equivalently, is a clique-sum of cliques. There exist analogues, though more complicated, characterizations of the case where the sparsity order is at most 2, which are different for and . The existing proof for the complex case, is based on the result for the real case. In this paper we provide a more elementary proof of the characterization of the graphs whose complex sparsity order is at most two. Part of our proof relies on a characterization of the -free graphs, with the path of length 3 and the stable set of cardinality 3, and of the class of clique-sums of such graphs.

Adaptive and Blind Wideband Spectrum Sensing Scheme Using Singular Value Decomposition

The Modulated Wideband Converter (MWC) can provide a sub-Nyquist sampling for continuous analog signal and reconstruct the spectral support. However, the existing reconstruction algorithms need a priori information of sparsity order, are not self-adaptive for SNR, and are not fault tolerant enough. These problems affect the reconstruction performance in practical sensing scenarios. In this paper, an Adaptive and Blind Reduced MMV (Multiple Measurement Vectors) Boost (ABRMB) scheme based on singular value decomposition (SVD) for wideband spectrum sensing is proposed. Firstly, the characteristics of singular values of signals are used to estimate the noise intensity and sparsity order, and an adaptive decision threshold can be determined. Secondly, optimal neighborhood selection strategy is employed to improve the fault tolerance in the solver of ABRMB. The experimental results demonstrate that, compared with ReMBo (Reduce MMV and Boost) and RPMB (Randomly Projecting MMV and Boost), ABRMB can significantly improve the success rate of reconstruction without the need to know noise intensity and sparsity order and can achieve high probability of reconstruction with fewer sampling channels, lower minimum sampling rate, and lower approximation error of the potential of spectral support.

Adaptive joint sparse recovery algorithm based on Tabu Search

This paper introduces a novel metaheuristic methodology to address the Multiple Measurement Vectors problem using a well-known greedy search strategy. A modified version of Tabu Search algorithm is utilized to determine a precise estimation of row support and then joint sparse samples are reconstructed using MMSE criterion. The proposed approach is more robust to the sparsity order variations and noise uncertainty, in comparison with the conventional MMV problem solvers. Furthermore, to avoid wastage of the sampling resources and reduce the implementation costs, a two-step joint sparse recovery framework is developed which the first step predicts and adjusts the optimum sampling rate and the second one reconstructs signal vectors applying obtained sampling rate. Numerical simulations demonstrate the superiority of proposed technique for both improving the performance and reducing the computational complexity and sampling costs.

Trace-Based Sparsity Order Estimation With Sparsely Sampled Random Matrices

In most of compressed sensing (CS) contributions, the sparsity of interesting signals is usually assumed to be prior. It may incur performance degradation of CS in such environments, where the sparsity order of channel occupancy changes dynamically. Therefore, sparsity order estimation (SOE) is vital, so that the sampling rate can be adjusted adaptively to exploit fully CS techniques. This paper studies the SOE problem for wideband cognitive radio. To investigate this issue, at first, the theoretical expression of sparsity order of the considered signals is derived via random matrix theory. The theoretical analysis shows that the sparsity order can be expressed with the ratio, which consists of the trace of measured signals' covariance matrix, compression sampling ratio, signal-to-noise ratio, and so on. And then, according to the theoretical expressions, a novel SOE approach is presented. In the proposed method, the trace of measured signals' covariance matrix is first calculated, and second, the sparsity order is estimated based on the trace. Compared with the previous work, the proposed algorithm, besides the lower computational complexity, has some superior performance, such as better robustness, smaller SOE error, and so on. Theoretical analysis and simulation results demonstrate the performance of the proposed approach.

JOINT COMPRESSIVE SENSING FRAMEWORK FOR SPARSE DATA/CHANNEL ESTIMATION IN NON-ORTHOGONAL MULTICARRIER SCHEME

Many wireless channel behavior exhibits approximate sparse modeling in time domain, therefore compressive sensing (CS) approaches are applied for more accurate wireless channel estimation than traditional least squares approach. However, the CS approach is not applied for multicarrier data information recovery because the transmitted symbol can be sparse neither in time domain nor in frequency domain. In this paper, a new Sparse Frequency Division Multiplexing (SFDM) approach is suggested to generate sparse multicarrier mapping in frequency domain based on the huge combinatorial domain. The subcarriers will be mapped in sparse manner according to data stream for taking advantages of multicarrier modulation with lower number of subcarriers. The number of activated subcarriers is designed to achieve the same as Orthogonal Frequency–Division Multiplexing data rate under lower signal-to-noise ratio. The proposed approach exploits the double sparsity of data symbol in the frequency domain, and channel sparsity in the time domain. The same CS approach for both data recovery and adaptive channel estimation in a unified sparsely manner is used. The suggested framework can be used with any non-orthogonal waveform shaping and can work efficiently without any prior information about neither the channel sparsity order nor searching for the optimum pilot patterns.

Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors

For a graph G with p vertices the closed convex cone S⪰0(G) consists of all real positive semidefinite p×p matrices whose sparsity pattern is given by G, that is, those matrices with zeros in the off-diagonal entries corresponding to nonedges of G. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. While the maximum rank of an extremal ray in S⪰0(G), known as the sparsity order of G, has been characterized for different classes of graphs, we here study all possible extremal ranks of S⪰0(G). We investigate when the geometry of the (±1)-cut polytope of G yields a polyhedral characterization of the set of extremal ranks of S⪰0(G). For a graph G without K5 minors, we show that appropriately chosen normal vectors to the facets of the (±1)-cut polytope of G specify the off-diagonal entries of extremal matrices in S⪰0(G). We also prove that for appropriately chosen scalars the constant term of the linear equation of each facet-supporting hyperplane is the rank of its corresponding extremal matrix in S⪰0(G). Furthermore, we show that if G is series-parallel then this gives a complete characterization of all possible extremal ranks of S⪰0(G). Consequently, the sparsity order problem for series-parallel graphs can be solved in terms of polyhedral geometry.

C–LASSO and its dual for sparse signal estimation from array data

We treat the estimation of a sparse set of sources emitting plane waves observed by a sensor array as a complex-valued LASSO (c–LASSO) problem where the usual ℓ1-norm constraint is replaced by the ℓ1-norm of a matrix D times the solution vector. When the sparsity order is given, algorithmically selecting a suitable value for the c–LASSO regularization parameter remains a challenging task. The corresponding dual problem is formulated and it is shown that the dual solution is useful for selecting the regularization parameter of the c-LASSO. The solution path of the c-LASSO is analyzed and this motivates an order-recursive algorithm for the selection of the regularization parameter and a faster iterative algorithm that is based on a further approximation. This greatly facilitates computation of the c-LASSO-path as we can predict the changes in the active indices as the regularization parameter is reduced. Using this regularization parameter, the directions of arrival for all sources are estimated.

Energy efficiency of compressed spectrum sensing in wideband cognitive radio networks

In cognitive radio networks, wideband spectrum sensing (WSS) has been advocated as an effective approach to increase the spectrum access opportunity of the secondary users. On the other hand, because of the increase of sampling rate and sensing time duration cost by the analog-to-digital converter (ADC), energy saving has been a significant problem for WSS. In this paper, taking advantage of the frequency-domain sparsity of the wideband spectrum, a WSS scheme combining compressed sensing and multi-band joint detection technique is proposed to reduce the energy consumption. Based on extensive analysis and simulation, we identify the sparsity order of the wideband spectrum, the received signal-to-noise ratio (SNR) of the primary signal, and the compression rate employed in sampling as three key factors that affect the sensing performance. In particular, we derive a closed-form analytical model of the scheme. Based on these observations, the energy efficiency, defined as the ratio of the spectrum access opportunity to the energy consumption, is maximized through the optimization process of the compression rate under the sensing performance constraints. We also indicate the uniqueness of the optimal compression rate for maximizing the energy efficiency. Numerical and simulation results show that our proposed scheme is more energy efficient if the wideband spectrum is sparser in the frequency domain.

Averaged Properties of the Residual Error in Sparse Signal Reconstruction

Greedy algorithms, like orthogonal matching pursuit (OMP), compressive sampling matching pursuit, and their modifications, are the popular methods of compressive sensing in sparse signal reconstruction. They are iterative in manner and often use a scaled value of the system noise energy as a reference threshold to decide when to stop the searching procedure. This letter presents an analytical analysis of the residual error and also of the difference of the successive residual errors in a general case of the greedy searching procedure, provided that the measurement error is an independent and identically distributed Gaussian stationary process. The obtained results show that the mean energy of the mentioned errors, just at the moment of perfect recovery of the signal support, depends only on the sparsity order and/or on the number of measurement samples. The results contributed to designing a practical stopping rule, which is a combination of those two errors. Numerical experiments with the new rule implemented in the OMP algorithm show that the number of executed iterations can be less spread around the true value of the signal sparsity in the mean sense. A quality of the sparse signal estimation is also preserved.

인지 무선 네트워크에서 Sub-Nyquist 샘플링을 활용한 협력 스펙트럼 센싱 기법

We propose cooperative spectrum sensing schemes based on sub-Nyquist sampling. As compressed sensing has recently attracted great attention, sparsity order estimation techniques also has been widely investigated. Thus, assuming that the sparsity order of channel occupancy can be obtained, we mathematically analyze the detection performance of sub-Nyquist sampling schemes according to various sampling rates and cooperative spectrum sensing schemes. Simulation results verify the performance of the proposed schemes.

Compressive Sparsity Order Estimation for Wideband Cognitive Radio Receiver

Compressive sensing (CS) has been widely investigated in the cognitive radio (CR) literature in order to reduce the hardware cost of sensing wideband signals assuming prior knowledge of the sparsity pattern. However, the sparsity order of the channel occupancy is time-varying and the sampling rate of the CS receiver needs to be adjusted based on its value in order to fully exploit the potential of CS-based techniques. In this context, investigating blind sparsity order estimation (SOE) techniques is an open research issue. To address this, we study an eigenvalue-based compressive SOE technique using asymptotic random matrix theory. We carry out detailed theoretical analysis for the signal plus noise case to derive the asymptotic eigenvalue probability distribution function (aepdf) of the measured signal's covariance matrix for sparse signals. Subsequently, based on the derived aepdf expressions, we propose a technique to estimate the sparsity order of the wideband spectrum with compressive measurements using the maximum eigenvalue of the measured signal's covariance matrix. The performance of the proposed technique is evaluated in terms of normalized SOE Error (SOEE). It is shown that the sparsity order of the wideband spectrum can be reliably estimated using the proposed technique for a variety of scenarios.

Deterministic Construction of Sparse Sensing Matrices via Finite Geometry

Compressed sensing is a novel sampling technique that provides a fundamentally new approach to data acquisition. Comparing with the traditional method, compressed sensing asserts that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. As a long-standing subject in combinatorial design theory, the packing design gives rise to deterministic sparse matrices with low coherence. With this framework in mind, we investigate a series of packing designs originated from finite geometry. The connection between the sensing matrix and finite geometry is revealed, using the packing design as a bridge. More specifically, we construct a series of $m \times n$ binary sensing matrices with sparsity order $k=\Theta (m^{1/3})$ or $k=\Theta (m^{1/2})$ . Moreover, we use an embedding operation to merge our binary matrices with matrices having low coherence. Comparing with the original binary matrices, this embedding operation generates modified matrices with better recovery performance. Numerical simulations show that our binary and modified matrices outperform several typical sensing matrices. The sparse property of our matrices helps to accelerate the recovery process, which is very preferable in practice.

Stochastic Gradient Pursuit for Adaptive Equalization of Sparse Multipath Channels

In this paper, a new heuristic algorithm for the sparse adaptive equalization problem, termed as stochastic gradient pursuit, is proposed. A decision-feedback equalization structure is used in order to effectively mitigate the effect of long multipath channels. Diverging from the commonly used approach of sparse channel identification, we exploit the sparsity of the inverse problem under the compressive sensing perspective. Also, an extension to the case where the sparsity order parameter is unknown, is developed. Simulation results verify that the proposed schemes exhibit faster convergence and improved tracking capabilities compared to conventional and other sparse aware equalization schemes, offering at the same time a reduced computational complexity.

Sparsity Order Estimation and its Application in Compressive Spectrum Sensing for Cognitive Radios

Compressive sampling techniques can effectively reduce the acquisition costs of high-dimensional signals by utilizing the fact that typical signals of interest are often sparse in a certain domain. For compressive samplers, the number of samples M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> needed to reconstruct a sparse signal is determined by the actual sparsity order S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> of the signal, which can be much smaller than the signal dimension N. However, S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> is often unknown or dynamically varying in practice, and the practical sampling rate has to be chosen conservatively according to an upper bound S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> of the actual sparsity order in lieu of S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> , which can be unnecessarily high. To circumvent such wastage of the sampling resources, this paper introduces the concept of sparsity order estimation, which aims to accurately acquire S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> prior to sparse signal recovery, by using a very small number of samples M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> less than M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> . A statistical learning methodology is used to quantify the gap between M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> in a closed form via data fitting, which offers useful design guideline for compressive samplers. It is shown that M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> ≥ 1.2S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> log(N/S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nz</sub> + 2) + 3 for a broad range of sampling matrices. Capitalizing on this gap, this paper also develops a two-step compressive spectrum sensing algorithm for wideband cognitive radios as an illustrative application. The first step quickly estimates the actual sparsity order of the wide spectrum of interest using a small number of samples, and the second step adjusts the total number of collected samples according to the estimated signal sparsity order. By doing so, the overall sampling cost can be minimized adaptively, without degrading the sensing performance.

A short note on compressed sensing with partially known signal support

This short note studies a variation of the compressed sensing paradigm introduced recently by Vaswani et al., i.e., the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. In this framework, we propose a reconstruction method based on a convex minimization program coined innovative Basis Pursuit DeNoise (or iBPDN). Under the common ℓ 2 -fidelity constraint made on the available measurements, this optimization promotes the ( ℓ 1 ) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results of Vaswani et al. to the cases of compressible signals and noisy measurements by showing that iBPDN is ℓ 2 − ℓ 1 instance optimal. The corresponding proof relies on a small adaption of the results of Candès in 2008 for characterizing the stability of the Basis Pursuit DeNoise (BPDN) program. We also emphasize an interesting link between our method and the recent work of Davenport et al. on the δ -stable embeddings and the cancel-then-recover strategy applied to our problem. For both approaches, reconstructions are indeed stabilized when the sensing matrix respects the Restricted Isometry Property for the same sparsity order.

On the Sparsity Order of a Graph and Its Deficiency in Chordality

Given a graph G on n nodes, let denote the cone consisting of the positive semidefinite matrices (with real or complex entries) having a zero entry at every off-diagonal position corresponding to a non edge of G. Then, the sparsity order of G is defined as the maximum rank of a matrix lying on an extreme ray of the cone . It is known that the graphs with sparsity order 1 are the chordal graphs and a characterization of the graphs with sparsity order 2 is conjectured in [1] in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with sparsity order 2 in the complex case and we give a decomposition result for the graphs with sparsity order in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also indicate how an inequality from [17] relating the sparsity order of a graph and its minimum fill-in can be derived from a result concerning the dimension of the faces of the cone .