Restricted Isometry Property Condition Research Articles

Greedy Block Coordinate Descent under Restricted Isometry Property

Exact support recovery of K-group sparse matrices X from the multiple measurement vectors (MMV) model Y = A X arises from many applications and has been extensively studied. In this paper, we investigate the restricted isometry property (RIP) based condition that guarantees exact support recovery of K-group sparse matrices X from the MMV model with greedy block coordinate descent (GBCD) algorithm in K iterations. We show that if A satisfies the RIP with \(\delta _{K+1}<1/\sqrt {K+1}\), then GBCD recovers the support of any K-group sparse matrix X in K iterations. This RIP condition is sharp since GBCD may fail to recover the support of a K-group sparse matrix in K iterations if \(\delta _{K+1}\geq 1/\sqrt {K+1}\). As far as we know, this is the first sharp condition for GBCD.

Orthogonal Matching Pursuit Under the Restricted Isometry Property

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary D in a Hilbert space H. Given an element f ∈ H, OMP generates a sequence of approximations fn, n = 1,2,..., each of which is a linear combination of n dictionary elements chosen by a greedy criterion. It is studied whether the approximations fn are in some sense comparable to best n term approximation from the dictionary. One important result related to this question is a theorem of Zhang [8] in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers n-sparse signal, whenever the dictionary D satisfies a Restricted Isometry Property (RIP) of order An for some constant A, and that the procedure is also stable in l 2 under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang’s theorem, formulated in the general context of n term approximation from a dictionary in arbitrary Hilbert spaces H. Namely, it is shown that OMP generates near best n term approximations under a similar RIP condition. AMS Subject Classification: 94A12, 94A15, 68P30, 41A46, 15A52

Single-pixel complementary compressive sampling spectrometer

A new type of compressive spectroscopy technique employing a complementary sampling strategy is reported. In a single sequence of spectral compressive sampling, positive and negative measurements are performed, in which sensing matrices with a complementary relationship are used. The restricted isometry property condition necessary for accurate recovery of compressive sampling theory is satisfied mathematically. Compared with the conventional single-pixel spectroscopy technique, the complementary compressive sampling strategy can achieve spectral recovery of considerably higher quality within a shorter sampling time. We also investigate the influence of the sampling ratio and integration time on the recovery quality.

Restricted Isometry Property on Banded Block Toeplitz Matrices with Application to Multi-Channel Convolutive Source Separation

In compressive sensing (CS), the restricted isometry property (RIP) is an important condition on measurement matrices which guarantees the recovery of sparse signals with undersampled measurements. It has been proved in the prior works that both random (e.g., i.i.d. Gaussian, Bernoulli, $\ldots$ ) and Toeplitz matrices satisfy the RIP with high probability. However, structured matrices, such as banded Toeplitz matrices have drawn more attention since their structures have the advantage of fast matrix multiplication which may decrease the computational complexity of recovery algorithms. In this paper, we show that banded block Toeplitz matrices satisfy the RIP condition with high probability. Banded block Toeplitz matrices can be used in the sparse multi-channel source separation. The banded block Toeplitz matrices decrease the computational complexity while they have fewer number of non-zero entries in comparison to the same dimensional banded Toeplitz matrices. Furthermore, our simulation results show that banded block Toeplitz matrices outperform banded Toeplitz matrices in signal estimation. The analytical RIP bound for banded block Toeplitz matrices is provided in this paper and the RIP bound of sparse Gaussian matrices is also obtained as an upper bound for banded block Toeplitz matrices. Our simulation and analytical results show that sparse Gaussian random matrices do satisfy the RIP condition with high probability. The probability of satisfying the RIP depends on the probability of zero entries.

Recovery of Sparse Signal and Nonconvex Minimization

In this paper, we present sufficient conditions in term of the restricted isometry property (RIP) to guarantee perfect recovery of sparse signal in the noiseless case and stable recovery in the noisy case via Lq-minimization, especially for nonconvex case 0&lt;q&lt;1. Using RIP condition, we present sufficient conditions to guarantee perfect recovery of sparse signal in the noiseless case and stable recovery in the noisy case via Lq-minimization.

On the perturbation of measurement matrix in non-convex compressed sensing

We study lp(0<p<1) minimization under both additive and multiplicative noise. Theorems are presented for completely perturbed lp(0<p<1) minimization. Theorems reveal that under suitable conditions the stability of lp minimization with certain values of 0<p<1 is limited by the noise level in the observation. The restricted isometry property condition and the worst case reconstruction error bound are given in terms of restricted isometry constant and relative perturbations. Simulation results are presented and compared to state-of-the-art methods.

Deterministic Sensing Matrices Based on Multidimensional Pseudo-Random Sequences

An approach is proposed for producing compressed sensing (CS) matrices via multidimensional pseudo-random sequences. The columns of these matrices are binary Gold code vectors where zeros are replaced by ?1. This technique is mainly applied to restore sub-Nyquist-sampled sparse signals, especially image reconstruction using block CS. First, for the specific requirements of message length and compression ratio, a set ? which includes all preferred pairs of m-sequences is obtained by a searching algorithm. Then a sensing matrix A M×N is produced by using structured hardware circuits. In order to better characterize the correlation between any two columns of A, the average coherence is defined and the restricted isometry property (RIP) condition is described accordingly. This RIP condition has strong adaptability to different sparse signals. The experimental results show that with constant values of N and M, the sparsity bound of A is higher than that of a random matrix. Also, the recovery probability may have a maximum increase of 20 % in a noisy environment.

High Resolution Direction of Arrival (DOA) Estimation Based on Improved Orthogonal Matching Pursuit (OMP) Algorithm by Iterative Local Searching

DOA (Direction of Arrival) estimation is a major problem in array signal processing applications. Recently, compressive sensing algorithms, including convex relaxation algorithms and greedy algorithms, have been recognized as a kind of novel DOA estimation algorithm. However, the success of these algorithms is limited by the RIP (Restricted Isometry Property) condition or the mutual coherence of measurement matrix. In the DOA estimation problem, the columns of measurement matrix are steering vectors corresponding to different DOAs. Thus, it violates the mutual coherence condition. The situation gets worse when there are two sources from two adjacent DOAs. In this paper, an algorithm based on OMP (Orthogonal Matching Pursuit), called ILS-OMP (Iterative Local Searching-Orthogonal Matching Pursuit), is proposed to improve DOA resolution by Iterative Local Searching. Firstly, the conventional OMP algorithm is used to obtain initial estimated DOAs. Then, in each iteration, a local searching process for every estimated DOA is utilized to find a new DOA in a given DOA set to further decrease the residual. Additionally, the estimated DOAs are updated by substituting the initial DOA with the new one. The simulation results demonstrate the advantages of the proposed algorithm.

Permutation Meets Parallel Compressed Sensing: How to Relax Restricted Isometry Property for 2D Sparse Signals

Traditional compressed sensing considers sampling a 1D signal. For a multidimensional signal, if reshaped into a vector, the required size of the sensing matrix becomes dramatically large, which increases the storage and computational complexity significantly. To solve this problem, the multidimensional signal is reshaped into a 2D signal, which is then sampled and reconstructed column by column using the same sensing matrix. This approach is referred to as parallel compressed sensing, and it has much lower storage and computational complexity. For a given reconstruction performance of parallel compressed sensing, if a so-called acceptable permutation is applied to the 2D signal, the corresponding sensing matrix is shown to have a smaller required order of restricted isometry property condition, and thus, lower storage and computation complexity at the decoder are required. A zigzag-scan-based permutation is shown to be particularly useful for signals satisfying the newly introduced layer model. As an application of the parallel compressed sensing with the zigzag-scan-based permutation, a video compression scheme is presented. It is shown that the zigzag-scan-based permutation increases the peak signal-to-noise ratio of reconstructed images and video frames.

A Sharp RIP Condition for Orthogonal Matching Pursuit

A restricted isometry property (RIP) conditionδK+KθK,1&lt;1is known to be sufficient for orthogonal matching pursuit (OMP) to exactly recover everyK-sparse signalxfrom measurementsy=Φx. This paper is devoted to demonstrate that this condition is sharp. We construct a specific matrix withδK+KθK,1=1such that OMP cannot exactly recover someK-sparse signals.

Sharp RIP bound for sparse signal and low-rank matrix recovery

This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix A satisfies the RIP condition δkA<1/3, then all k-sparse signals β can be recovered exactly via the constrained ℓ1 minimization based on y=Aβ. Similarly, if the linear map M satisfies the RIP condition δrM<1/3, then all matrices X of rank at most r can be recovered exactly via the constrained nuclear norm minimization based on b=M(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.

The Orthogonal Super Greedy Algorithm and Applications in Compressed Sensing

The general theory of greedy approximation is well developed. Much less is known about how specific features of a dictionary can be used to our advantage. In this paper, we discuss incoherent dictionaries. We build a new greedy algorithm which is called the orthogonal super greedy algorithm (OSGA). We show that the rates of convergence of OSGA and the orthogonal matching pursuit (OMP) with respect to incoherent dictionaries are the same. Based on the analysis of the number of orthogonal projections and the number of iterations, we observed that OSGA is times simpler (more efficient) than OMP. Greedy approximation is also a fundamental tool for sparse signal recovery. The performance of orthogonal multimatching pursuit, a counterpart of OSGA in the compressed sensing setting, is also analyzed under restricted isometry property conditions.

Subsampled circulant matrix based analogue compressed sensing

The modulated wideband converter (MWC) is an attractive analogue compressed sensing technique proposed recently. Unfortunately, the MWC has high hardware complexity owing to its parallel structure. To reduce the complexity, proposed is a novel subsampled circulant matrix based analogue compressed sensing (SCM-ACS) scheme. Using the cyclic shifts of the Zadoff-Chu sequence, the SCM-ACS scheme reduces the number of physical parallel channels from m to 1 with larger processing time, where m ranges from several dozen to several hundred. It is proved that when m=O(r log2M log3r) the measurement matrix of the SCM-ACS scheme satisfies the restricted isometry property condition with probability 1−M−O(1), where M is the length of the Zadoff-Chu sequence, and r is the sparsity of the input signal. Simulation results show that the SCM-ACS scheme outperforms the MWC on recovery performance.