Recovery Performance Of Sparse Signals Research Articles

Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames

Frames are recognized for their importance in many fields of communications, signal processing, quantum physics, and so on. In this paper, we design an incoherent tight frame by selecting some rows of a matrix that is the Kronecker product of Fourier and unitary matrices. The Kronecker-product-based frame allows its elements to have a small number of phases, regardless of the frame length, which is suitable for low-cost implementation. To obtain the Kronecker-product-based frame with low mutual coherence, we first derive an objective function by transforming the Gram matrix expression to compute the coherence. If the Hadamard matrix is employed as a unitary matrix, the objective function can be computed efficiently with low complexity. Then, we find a subsampling index set for the Kronecker-product-based frame by minimizing the objective function. In simulations, we show that the Kronecker-product-based frames can achieve similar mutual coherence to optimized harmonic frames of a large number of phases. We apply the frames to compressed sensing (CS) as the measurement matrices, where the Kronecker-product-based frames demonstrate reliable performance of sparse signal recovery.

Sensing matrix design for wideband channel estimation in millimeter‐wave hybrid multiple‐input multiple‐output system

AbstractHybrid analog/digital multiple‐input multiple‐output (MIMO) system is proposed to support low complexity multi‐stream communication in millimeter‐wave (mmWave) frequencies. The main idea of channel estimation methods for mmWave hybrid MIMO systems is based on leveraging the sparsity nature of the channel in the angle and delay domain. Sparse signal recovery algorithms have been used to reconstruct the channel coefficients. Due to the lack of information about the angle of arrival and angle of departure before channel estimation, random values are common choices for precoders and combiners. Moreover, the training pilot symbols are considered to be random variables. In this article, we propose sequential time domain sensing matrix design and sequential frequency domain sensing matrix design algorithms to design these variables in order to enhance sparse signal recovery performance and consequently the channel estimation accuracy. Our methods are based on reducing the correlation between the sensing matrix columns, which is defined in the sparse formulation of the channel estimation problem. We design the training radio frequency precoders/combiners phase angle values and pilot symbol vectors with the aid of a sequential procedure that has a reasonable complexity. Simulation results show that utilizing the designed values instead of random variables increased the performance of the channel estimation, especially when we deal with a higher number of variables.

Fast Variational Bayesian Inference for Temporally Correlated Sparse Signal Recovery

The performance of sparse signal recovery (SSR) can be enhanced by exploiting rich temporal correlation in the multiple snapshots of signal of interest. However, existing methods need to transform the temporally correlated multiple measurements SSR problem into its vectorization form, imposing huge computational cost for algorithmic realization. To overcome this drawback, we propose a novel formulation to model the temporal correlation so that variational Bayesian inference (VBI) can be applied to simplify the inference and a novel uncoupling trick is also proposed to reduce the computation. Theoretical and simulation results indicate that our method can bring a considerable computational complexity reduction and achieve a performance improvement for the temporally correlated SSR problem compared to the state of arts time-varying sparse Bayesian learning (TSBL) method.

Bayesian Compressive Sensing of Sparse Signals with Unknown Clustering Patterns.

We consider the sparse recovery problem of signals with an unknown clustering pattern in the context of multiple measurement vectors (MMVs) using the compressive sensing (CS) technique. For many MMVs in practice, the solution matrix exhibits some sort of clustered sparsity pattern, or clumpy behavior, along each column, as well as joint sparsity across the columns. In this paper, we propose a new sparse Bayesian learning (SBL) method that incorporates a total variation-like prior as a measure of the overall clustering pattern in the solution. We further incorporate a parameter in this prior to account for the emphasis on the amount of clumpiness in the supports of the solution to improve the recovery performance of sparse signals with an unknown clustering pattern. This parameter does not exist in the other existing algorithms and is learned via our hierarchical SBL algorithm. While the proposed algorithm is constructed for the MMVs, it can also be applied to the single measurement vector (SMV) problems. Simulation results show the effectiveness of our algorithm compared to other algorithms for both SMV and MMVs.

Signal recovery under cumulative coherence

This paper considers signal recovery in the framework of cumulative coherence. First, we show that the Lasso estimator and the Dantzig selector exhibit similar behavior under the cumulative coherence. Then we estimate the approximation equivalence between the Lasso and the Dantzig selector by calculating prediction loss difference under the condition of cumulative coherence. And we also prove that the cumulative coherence implies the restricted eigenvalue condition. Last, we illustrate the advantages of cumulative coherence condition for three class matrices, in terms of the recovery performance of sparse signals via extensive numerical experiments.

Recovery of Sparse Signals via Generalized Orthogonal Matching Pursuit: A New Analysis

As an extension of orthogonal matching pursuit (OMP) for improving the recovery performance of sparse signals, generalized OMP (gOMP) has recently been studied in the literature. In this paper, we present a new analysis of the gOMP algorithm using the restricted isometry property (RIP). We show that if a measurement matrix Φ ∈ ℜ m×n satisfies the RIP with isometry constant δ max{9,S+1}K ≤ 1/8, then gOMP performs stable reconstruction of all K-sparse signals x ∈ ℜ n from the noisy measurements y=Φx+v, within max{K,⌊8K/S⌋} iterations, where v is the noise vector and S is the number of indices chosen in each iteration of the gOMP algorithm. For Gaussian random measurements, our result indicates that the number of required measurements is essentially m=O(Klog n /K), which is a significant improvement over the existing result m=O(K 2 log n /K), especially for large K.

트리검색 기법을 이용한 희소신호 복원기법

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.

Deterministic Construction of Real-Valued Ternary Sensing Matrices Using Optical Orthogonal Codes

In this letter, a new class of real-valued matrices is presented for deterministic compressed sensing. A base matrix is constructed by cyclic shifts of binary sequences in an optical orthogonal code (OOC). Then, a Hadamard matrix is used for its extension, which ultimately produces a real-valued matrix that takes the entries of 0, -1 and +1 before normalization. The new sensing matrix forms a tight frame with small coherence, which theoretically guarantees the average recovery performance of sparse signals with uniformly distributed supports. Several example sensing matrices are presented by employing a special type of OOCs obtained from modular Golomb rulers.

Progressive fusion of reconstruction algorithms for low latency applications in compressed sensing

Recently, it has been shown that fusion of the estimates of a set of sparse recovery algorithms result in an estimate better than the best estimate in the set, especially when the number of measurements is very limited. Though these schemes provide better sparse signal recovery performance, the higher computational requirement makes it less attractive for low latency applications. To alleviate this drawback, in this paper, we develop a progressive fusion based scheme for low latency applications in compressed sensing. In progressive fusion, the estimates of the participating algorithms are fused progressively according to the availability of estimates. The availability of estimates depends on computational complexity of the participating algorithms, in turn on their latency requirement. Unlike the other fusion algorithms, the proposed progressive fusion algorithm provides quick interim results and successive refinements during the fusion process, which is highly desirable in low latency applications. We analyse the developed scheme by providing sufficient conditions for improvement of CS reconstruction quality and show the practical efficacy by numerical experiments using synthetic and real-world data.

Deterministic construction of Fourier-based compressed sensing matrices using an almost difference set

In this paper, a new class of Fourier-based matrices is studied for deterministic compressed sensing. Initially, a basic partial Fourier matrix is introduced by choosing the rows deterministically from the inverse discrete Fourier transform (DFT) matrix. By row/column rearrangement, the matrix is represented as a concatenation of DFT-based submatrices. Then, a full or a part of columns of the concatenated matrix is selected to build a new M × N deterministic compressed sensing matrix, where M = p r and N = L(M + 1) for prime p, and positive integers r and L ≤ M - 1. Theoretically, the sensing matrix forms a tight frame with small coherence. Moreover, the matrix theoretically guarantees unique recovery of sparse signals with uniformly distributed supports. From the structure of the sensing matrix, the fast Fourier transform (FFT) technique can be applied for efficient signal measurement and reconstruction. Experimental results demonstrate that the new deterministic sensing matrix shows empirically reliable recovery performance of sparse signals by the CoSaMP algorithm.

Correction to “Generalized Orthogonal Matching Pursuit” [Dec 12 6202-6216

As an extension of orthogonal matching pursuit (OMP) improving the recovery performance of sparse signals, generalized OMP (gOMP) has recently been studied in the literature. In this paper, we present a new analysis of the gOMP algorithm using restricted isometry property (RIP). We show that if the measurement matrix $\mathbf{\Phi} \in \mathcal{R}^{m \times n}$ satisfies the RIP with $$\delta_{\max \left\{9, S + 1 \right\}K} \leq \frac{1}{8},$$ then gOMP performs stable reconstruction of all $K$-sparse signals $\mathbf{x} \in \mathcal{R}^n$ from the noisy measurements $\mathbf{y} = \mathbf{\Phi x} + \mathbf{v}$ within $\max \left\{K, \left\lfloor \frac{8K}{S} \right\rfloor \right\}$ iterations where $\mathbf{v}$ is the noise vector and $S$ is the number of indices chosen in each iteration of the gOMP algorithm. For Gaussian random measurements, our results indicate that the number of required measurements is essentially $m = \mathcal{O}(K \log \frac{n}{K})$, which is a significant improvement over the existing result $m = \mathcal{O}(K^2 \log \frac{n}{K})$, especially for large $K$.