Exact Recovery Of Sparse Signals Research Articles

Exact recovery of sparse signals with side information

Compressed sensing has captured considerable attention of researchers in the past decades. In this paper, with the aid of the powerful null space property, some deterministic recovery conditions are established for the previous ell _{1}–ell _{1} method and the ell _{1}–ell _{2} method to guarantee the exact sparse recovery when the side information of the desired signal is available. These obtained results provide a useful and necessary complement to the previous investigation of the ell _{1}–ell _{1} and ell _{1}–ell _{2} methods that are based on the statistical analysis. Moreover, one of our theoretical findings also shows that the sharp conditions previously established for the classical ell _{1} method remain suitable for the ell _{1}–ell _{1} method to guarantee the exact sparse recovery. Numerical experiments on both the synthetic signals and the real-world images are also carried out to further test the recovery performance of the above two methods.

A new result on recovery sparse signals using orthogonal matching pursuit

Orthogonal matching pursuit (OMP) algorithm is a classical greedy algorithm widely used in compressed sensing. In this paper, by exploiting the Wielandt inequality and some properties of orthogonal projection matrix, we obtained a new number of iterations required for the OMP algorithm to perform exact recovery of sparse signals, which improves significantly upon the latest results as we know.

Improved RIP‐based performance guarantee for sparse signal recovery via A*OMP

Combining the best-first tree search and orthogonal matching pursuit (OMP), A*OMP has been introduced that aims at solving the compressed sensing problem. The theoretical analysis of A*OMP is mainly presented and an improved condition for exact recovery of sparse signals via the A*OMP algorithm is proposed.

Exact Recovery of Sparse Signals Using Orthogonal Matching Pursuit: How Many Iterations Do We Need?

Orthogonal matching pursuit (OMP) is a greedy algorithm widely used for the recovery of sparse signals from compressed measurements. In this paper, we analyze the number of iterations required for the OMP algorithm to perform exact recovery of sparse signals. Our analysis shows that OMP can accurately recover all $K$ -sparse signals within $\lceil 2.8 \; K \rceil$ iterations when the measurement matrix satisfies a restricted isometry property (RIP). Our result improves upon the recent result of Zhang and also bridges the gap between Zhang's result and the fundamental limit of OMP at which exact recovery of $K$ -sparse signals cannot be uniformly guaranteed.

An Analysis of Block Sampling Strategies in Compressed Sensing

Compressed sensing is a theory which guarantees the exact recovery of sparse signals from a small number of linear projections. The sampling schemes suggested by current compressed sensing theories are often of little practical relevance, since they cannot be implemented on real acquisition systems. In this paper, we study a new random sampling approach that consists of projecting the signal over blocks of sensing vectors. A typical example is the case of blocks made of horizontal lines in the 2-D Fourier plane. We provide the theoretical results on the number of blocks that are sufficient for exact sparse signal reconstruction. This number depends on two properties named intra- and inter-support block coherence. We then show that our bounds coincide with the best so far results in a series of examples, including Gaussian measurements or isolated measurements. We also show that the result is sharp when used with specific blocks in time-frequency bases, in the sense that the minimum required amount of blocks to reconstruct sparse signals cannot be improved up to a multiplicative logarithmic factor. The proposed results provide a good insight on the possibilities and limits of block compressed sensing in imaging devices, such as magnetic resonance imaging, radio-interferometry, or ultra-sound imaging.

Improving A⋆ OMP: Theoretical and empirical analyses with a novel dynamic cost model

Best-first search has been recently utilized for compressed sensing (CS) by the A⋆ orthogonal matching pursuit (A⋆OMP) algorithm. In this work, we concentrate on theoretical and empirical analyses of A⋆OMP. We present a restricted isometry property (RIP) based general condition for exact recovery of sparse signals via A⋆OMP. In addition, we develop online guarantees which promise improved recovery performance with the residue-based termination instead of the sparsity-based one. We demonstrate the recovery capabilities of A⋆OMP with extensive recovery simulations using the adaptive-multiplicative (AMul) cost model, which effectively compensates for the path length differences in the search tree. The presented results, involving phase transitions for different nonzero element distributions as well as recovery rates and average error, reveal not only the superior recovery accuracy of A⋆OMP, but also the improvements with the residue-based termination and the AMul cost model. Comparison of the run times indicates the speed up by the AMul cost model. We also demonstrate a hybrid of OMP and A⋆OMP to accelerate the search further. Finally, we run A⋆OMP on sparse images to illustrate its recovery performance for more realistic coefficient distributions.

A User's Guide to Compressed Sensing for Communications Systems

This survey provides a brief introduction to compressed sensing as well as several major algorithms to solve it and its various applications to communications systems. We firstly review linear simultaneous equations as ill-posed inverse problems, since the idea of compressed sensing could be best understood in the context of the linear equations. Then, we consider the problem of compressed sensing as an underdetermined linear system with a prior information that the true solution is sparse, and explain the sparse signal recovery based on l1 optimization, which plays the central role in compressed sensing, with some intuitive explanations on the optimization problem. Moreover, we introduce some important properties of the sensing matrix in order to establish the guarantee of the exact recovery of sparse signals from the underdetermined system. After summarizing several major algorithms to obtain a sparse solution focusing on the l1 optimization and the greedy approaches, we introduce applications of compressed sensing to communications systems, such as wireless channel estimation, wireless sensor network, network tomography, cognitive radio, array signal processing, multiple access scheme, and networked control.

Interval-Passing Algorithm for Non-Negative Measurement Matrices: Performance and Reconstruction Analysis

We consider the Interval-Passing Algorithm (IPA), an iterative reconstruction algorithm for reconstruction of non-negative sparse real-valued signals from noise-free measurements. We first generalize the IPA by relaxing the original constraint that the measurement matrix must be binary. The new algorithm operates on any non-negative sparse measurement matrix. We give a performance comparison of the generalized IPA with the reconstruction algorithms based on 1) linear programming and 2) verification decoding. Then we identify signals not recoverable by the IPA on a given measurement matrix, and show that these signals are related to stopping sets responsible to failures of iterative decoding algorithms on the binary erasure channel (BEC). Contrary to the results of the iterative decoding on the BEC, the smallest stopping set of a measurement matrix is not the smallest configuration on which the IPA fails. We analyze the recovery of sparse signals on subsets of stopping sets via the IPA and provide sufficient conditions on the exact recovery of sparse signals. Reconstruction performance of the IPA using the IEEE 802.16e LDPC codes as measurement matrices are given to show the effect of stopping sets in the performance of the IPA.

Concentration of S-largest mutilated vectors with [formula omitted]-quasinorm for [formula omitted] and its applications

Suppose Φ∈RM×N is a known sensing matrix. For any nonzero vector h in the null space of Φ and any given positive integer S<N, we further introduce a definition of S-largest mutilated vector of h and develop some of its important properties associated with the concept of concentration in lp space (0<p⩽1) based on the concept of concentration proposed by Donoho in compressive sensing. As application of these important properties, we assert that any S-sparse signal can be exactly recovered via (lp) minimization in the noiseless case as soon as the concentration of S-largest mutilated vectors with lp-quasinorm is less than 1/2. This is a complement and generalization for a similar result proposed by Donoho. Moreover, the upper bound condition of δS(Φ) proposed by Cai for exact recovery of sparse signals via (l1) minimization is also derived from this result simply.