A simple hard-thresholding operation is shown to be able to recover $L$ signals $\mathbf{x}_1,...,\mathbf{x}_L \in \mathbb{R}^n$ that share a common support of size $s$ from $m = \mathcal{O}(s)$ one-bit measurements per signal if $L \ge \log(en/s)$. This result improves the single signal recovery bounds with $m = \mathcal{O}(s\log(en/s))$ measurements in the sense that asymptotically fewer measurements per non-zero entry are needed. Numerical evidence supports the theoretical considerations.
Compressed sensing (CS) demonstrates that a sparse, or compressible signal can be acquired using a low rate acquisition process below the Nyquist rate, which projects the signal onto a small set of vectors incoherent with the sparsity basis. In this paper, we propose a new framework for compressed sensing recovery problem using iterative approximation method via L0 minimization. Instead of directly solving the unconstrained L0 norm optimization problem, we use the linearization and proximal points techniques to approximate the penalty function at each iteration. The proposed algorithm is very …
Optimal $k$-thresholding algorithms are a class of $k$-sparse signal recovery algorithms that overcome the shortcomings of traditional hard thresholding algorithms caused by the oscillation of the residual function. In this paper, a novel convergence analysis for optimal $k$-thresholding algorithms is established, which reveals the data-time tradeoffs of these algorithms. Both the analysis and numerical results demonstrate that when the number of measurements is small, the algorithms cannot converge; when the number of measurements is suitably large, the number of iterations required for succe…
We propose a new iterative greedy algorithm for reconstructions of sparse signals with or without noisy perturbations in compressed sensing. The proposed algorithm, called \emph{subspace thresholding pursuit} (STP) in this paper, is a simple combination of subspace pursuit and iterative hard thresholding. Firstly, STP has the theoretical guarantee comparable to that of $\ell_1$ minimization in terms of restricted isometry property. Secondly, with a tuned parameter, on the one hand, when reconstructing Gaussian signals, it can outperform other state-of-the-art reconstruction algorithms greatly;…
In this work, we show that reconstructing a sparse signal from quantized compressive measurement can be achieved in an unified formalism whatever the (scalar) quantization resolution, i.e., from 1-bit to high resolution assumption. This is achieved by generalizing the iterative hard thresholding (IHT) algorithm and its binary variant (BIHT) introduced in previous works to enforce the consistency of the reconstructed signal with respect to the quantization model. The performance of this algorithm, simply called quantized IHT (QIHT), is evaluated in comparison with other approaches (e.g., IHT, b…
The optimal $k$-thresholding (OT) and optimal $k$-thresholding pursuit (OTP) are newly introduced frameworks of thresholding techniques for compressed sensing and signal approximation. Such frameworks motivate the practical and efficient algorithms called relaxed optimal $k$-thresholding ($\textrm{ROT}ω$) and relaxed optimal $k$-thresholding pursuit ($\textrm{ROTP}ω$) which are developed through the tightest convex relaxations of OT and OTP, where $ω$ is a prescribed integer number. The preliminary numerical results demonstrated in \cite{Z19} indicate that these approaches can stably reconstru…
We consider the problem of recovering a set of correlated signals (e.g., images from different viewpoints) from a few linear measurements per signal. We assume that each sensor in a network acquires a compressed signal in the form of linear measurements and sends it to a joint decoder for reconstruction. We propose a novel joint reconstruction algorithm that exploits correlation among underlying signals. Our correlation model considers geometrical transformations between the supports of the different signals. The proposed joint decoder estimates the correlation and reconstructs the signals usi…
Compressed sensing (CS) or sparse signal reconstruction (SSR) is a signal processing technique that exploits the fact that acquired data can have a sparse representation in some basis. One popular technique to reconstruct or approximate the unknown sparse signal is the iterative hard thresholding (IHT) which however performs very poorly under non-Gaussian noise conditions or in the face of outliers (gross errors). In this paper, we propose a robust IHT method based on ideas from $M$-estimation that estimates the sparse signal and the scale of the error distribution simultaneously. The method h…
The one-bit compressed sensing framework aims to reconstruct a sparse signal by only using the sign information of its linear measurements. To compensate for the loss of scale information, past studies in the area have proposed recovering the signal by imposing an additional constraint on the L2-norm of the signal. Recently, an alternative strategy that captures scale information by introducing a threshold parameter to the quantization process was advanced. In this paper, we analyze the typical behavior of the thresholding 1-bit compressed sensing utilizing the replica method of statistical me…
In this paper we propose three iterative greedy algorithms for compressed sensing, called \emph{iterative alternating direction} (IAD), \emph{normalized iterative alternating direction} (NIAD) and \emph{alternating direction pursuit} (ADP), which stem from the iteration steps of alternating direction method of multiplier (ADMM) for $\ell_0$-regularized least squares ($\ell_0$-LS) and can be considered as the alternating direction versions of the well-known iterative hard thresholding (IHT), normalized iterative hard thresholding (NIHT) and hard thresholding pursuit (HTP) respectively. Firstly,…
Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) - It gives near-optimal error guarantees. - It is robust to observation noise. - It succeeds with a minimum number of observations. - It can be used with any sampling operator for which the …
The $\ell^0$ minimization of compressed sensing is often relaxed to $\ell^1$, which yields easy computation using the shrinkage mapping known as soft thresholding, and can be shown to recover the original solution under certain hypotheses. Recent work has derived a general class of shrinkages and associated nonconvex penalties that better approximate the original $\ell^0$ penalty and empirically can recover the original solution from fewer measurements. We specifically examine p-shrinkage and firm thresholding. In this work, we prove that given data and a measurement matrix from a broad class …
Commonly employed reconstruction algorithms in compressed sensing (CS) use the $L_2$ norm as the metric for the residual error. However, it is well-known that least squares (LS) based estimators are highly sensitive to outliers present in the measurement vector leading to a poor performance when the noise no longer follows the Gaussian assumption but, instead, is better characterized by heavier-than-Gaussian tailed distributions. In this paper, we propose a robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse signals in the presence of impulsive noise. To address this p…
We provide another framework of iterative algorithms based on thresholding, feedback and null space tuning for sparse signal recovery arising in sparse representations and compressed sensing. Several thresholding algorithms with various feedbacks are derived, which are seen as exceedingly effective and fast. Convergence results are also provided. The core algorithm is shown to converge in finite many steps under a (preconditioned) restricted isometry condition. The algorithms are seen as particularly effective for large scale problems. Numerical studies about the effectiveness and the speed of…
In this letter, a binary sparse Bayesian learning (BSBL) algorithm is proposed to slove the one-bit compressed sensing (CS) problem in both single measurement vector (SMV) and multiple measurement vectors (MMVs). By utilising the Bussgang-like decomposition, the one-bit CS problem can be approximated as a standard linear model. Consequently, the standard SBL algorithm can be naturally incorporated. Numerical results demonstrate the effectiveness of the BSBL algorithm.
In one-bit compressed sensing, previous results state that sparse signals may be robustly recovered when the measurements are taken using Gaussian random vectors. In contrast to standard compressed sensing, these results are not extendable to natural non-Gaussian distributions without further assumptions, as can be demonstrated by simple counter-examples. We show that approximately sparse signals that are not extremely sparse can be accurately reconstructed from single-bit measurements sampled according to a sub-gaussian distribution, and the reconstruction comes as the solution to a convex pr…
Sign truncated matching pursuit (STrMP) algorithm is presented in this paper. STrMP is a new greedy algorithm for the recovery of sparse signals from the sign measurement, which combines the principle of consistent reconstruction with orthogonal matching pursuit (OMP). The main part of STrMP is as concise as OMP and hence STrMP is simple to implement. In contrast to previous greedy algorithms for one-bit compressed sensing, STrMP only need to solve a convex and unconstraint subproblem at each iteration. Numerical experiments show that STrMP is fast and accurate for one-bit compressed sensing c…
In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization s…
Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how "compressed" compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signa…
We examine the use of a structured thresholding algorithm for sparse underwater channel estimation using compressed sensing. This method shows some improvements over standard algorithms for sparse channel estimation such as matching pursuit, iterative detection and least squares.
In this paper, we analyze the information theoretic lower bound on the necessary number of samples needed for recovering a sparse signal under different compressed sensing settings. We focus on the weighted graph model, a model-based framework proposed by Hegde et al. (2015), for standard compressed sensing as well as for one-bit compressed sensing. We study both the noisy and noiseless regimes. Our analysis is general in the sense that it applies to any algorithm used to recover the signal. We carefully construct restricted ensembles for different settings and then apply Fano's inequality to …
This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method is the first tractable algorithm with such guarantees for this setting. The results are complemented by numerical experiments confirming the validity of our approach.
We consider the problem of sparse signal recovery from 1-bit measurements. Due to the noise present in the acquisition and transmission process, some quantized bits may be flipped to their opposite states. These sign flips may result in severe performance degradation. In this study, a novel algorithm, termed HISTORY, is proposed. It consists of Hamming support detection and coefficients recovery. The HISTORY algorithm has high recovery accuracy and is robust to strong measurement noise. Numerical results are provided to demonstrate the effectiveness and superiority of the proposed algorithm.
Compressed sensing (CS) and 1-bit CS cannot directly recover quantized signals and require time consuming recovery. In this paper, we introduce \textit{Hamming compressed sensing} (HCS) that directly recovers a k-bit quantized signal of dimensional $n$ from its 1-bit measurements via invoking $n$ times of Kullback-Leibler divergence based nearest neighbor search. Compared with CS and 1-bit CS, HCS allows the signal to be dense, takes considerably less (linear) recovery time and requires substantially less measurements ($\mathcal O(\log n)$). Moreover, HCS recovery can accelerate the subsequent…
The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that…
There have been a number of studies on sparse signal recovery from one-bit quantized measurements. Nevertheless, little attention has been paid to the choice of the quantization thresholds and its impact on the signal recovery performance. This paper examines the problem of one-bit quantizer design for sparse signal recovery. Our analysis shows that the magnitude ambiguity that ever plagues conventional one-bit compressed sensing methods can be resolved, and an arbitrarily small reconstruction error can be achieved by setting the quantization thresholds close enough to the original data sample…
This paper studies the stability of some reconstruction algorithms for compressed sensing in terms of the bit precision. Considering the fact that practical digital systems deal with discretized signals, we motivate the importance of the total number of accurate bits needed from the measurement outcomes in addition to the number of measurements. It is shown that if one uses a $2k \times n$ Vandermonde matrix with roots on the unit circle as the measurement matrix, $O(\ell + k \log(n/k))$ bits of precision per measurement are sufficient to reconstruct a $k$-sparse signal $x \in \R^n$ with dynam…
Is it possible to obliviously construct a set of hyperplanes H such that you can approximate a unit vector x when you are given the side on which the vector lies with respect to every h in H? In the sparse recovery literature, where x is approximately k-sparse, this problem is called one-bit compressed sensing and has received a fair amount of attention the last decade. In this paper we obtain the first scheme that achieves almost optimal measurements and sublinear decoding time for one-bit compressed sensing in the non-uniform case. For a large range of parameters, we improve the state of the…
The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm ($p \in (0,1)$). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all paramete…
Wireless OFDM channels can be approximated by a time varying filter with sparse time domain taps. Recent achievements in sparse signal processing such as compressed sensing have facilitated the use of sparsity in estimation, which improves the performance significantly. The problem of these sparse-based methods is the need for a stable transformation matrix which is not fulfilled in the current transmission setups. To assist the analog filtering at the receiver, the transmitter leaves some of the subcarriers at both edges of the bandwidth unused which results in an ill-conditioned DFT submatri…
We study the problem of approximately recovering signals on a manifold from one-bit linear measurements drawn from either a Gaussian ensemble, partial circulant ensemble, or bounded orthonormal ensemble and quantized using Sigma-Delta or distributed noise shaping schemes. We assume we are given a Geometric Multi-Resolution Analysis, which approximates the manifold, and we propose a convex optimization algorithm for signal recovery. We prove an upper bound on the recovery error which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the cl…
Compressed sensing aims to undersample certain high-dimensional signals, yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization -- which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offe…
We investigate the sign-linear embeddings of 1-bit compressed sensing given by Gaussian measurements. One can give short arguments concerning a Restricted Isometry Property of such maps using Vapnik-Chervonenkis dimension of sparse hemispheres. This approach has a natural extension to the presence of additive white noise prior to quantization. Noisy one-bit mappings are shown to satisfy an RIP when the metric on the sphere is given by the noise.
This letter proposes a dictionary learning algorithm for blind one bit compressed sensing. In the blind one bit compressed sensing framework, the original signal to be reconstructed from one bit linear random measurements is sparse in an unknown domain. In this context, the multiplication of measurement matrix $\Ab$ and sparse domain matrix $Φ$, \ie $\Db=\AbΦ$, should be learned. Hence, we use dictionary learning to train this matrix. Towards that end, an appropriate continuous convex cost function is suggested for one bit compressed sensing and a simple steepest-descent method is exploited to…
The replica method is a non-rigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an n-dimensional vector "decouples" as n scalar postulated MAP estimators. The result is based on applyi…
Compressed sensing (CS) is on recovery of high dimensional signals from their low dimensional linear measurements under a sparsity prior and digital quantization of the measurement data is inevitable in practical implementation of CS algorithms. In the existing literature, the quantization error is modeled typically as additive noise and the multi-bit and 1-bit quantized CS problems are dealt with separately using different treatments and procedures. In this paper, a novel variational Bayesian inference based CS algorithm is presented, which unifies the multi- and 1-bit CS processing and is ap…
This letter proposes a low-computational Bayesian algorithm for noisy sparse recovery in the context of one bit compressed sensing with sensing matrix perturbation. The proposed algorithm which is called BHT-MLE comprises a sparse support detector and an amplitude estimator. The support detector utilizes Bayesian hypothesis test, while the amplitude estimator uses an ML estimator which is obtained by solving a convex optimization problem. Simulation results show that BHT-MLE algorithm offers more reconstruction accuracy than that of an ML estimator (MLE) at a low computational cost.
In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $δ$, $m\sim δ^{-4} s\log(N/sδ)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $δ$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $δ$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.
Qom University of Technology（库姆理工大学）；Swinburne University of Technology（斯威本科技大学）；Sharif University of Technology（谢里夫理工大学）
This letter proposes a sparse diffusion steepest-descent algorithm for one bit compressed sensing in wireless sensor networks. The approach exploits the diffusion strategy from distributed learning in the one bit compressed sensing framework. To estimate a common sparse vector cooperatively from only the sign of measurements, steepest-descent is used to minimize the suitable global and local convex cost functions. A diffusion strategy is suggested for distributive learning of the sparse vector. Simulation results show the effectiveness of the proposed distributed algorithm compared to the stat…
The 1-bit compressed sensing framework enables the recovery of a sparse vector x from the sign information of each entry of its linear transformation. Discarding the amplitude information can significantly reduce the amount of data, which is highly beneficial in practical applications. In this paper, we present a Bayesian approach to signal reconstruction for 1-bit compressed sensing, and analyze its typical performance using statistical mechanics. Utilizing the replica method, we show that the Bayesian approach enables better reconstruction than the L1-norm minimization approach, asymptotical…
In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in…
The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components for generating a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate m…
We consider iterative (`turbo') algorithms for compressed sensing. First, a unified exposition of the different approaches available in the literature is given, thereby enlightening the general principles and main differences. In particular we discuss i) the estimation step (matched filter vs. optimum MMSE estimator), ii) the unbiasing operation (implicitly or explicitly done and equivalent to the calculation of extrinsic information), and iii) thresholding vs. the calculation of soft values. Based on these insights we propose a low-complexity but well-performing variant utilizing a Krylov spa…
We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellati…
We present optimal sample complexity estimates for one-bit compressed sensing problems in a realistic scenario: the procedure uses a structured matrix (a randomly sub-sampled circulant matrix) and is robust to analog pre-quantization noise as well as to adversarial bit corruptions in the quantization process. Our results imply that quantization is not a statistically expensive procedure in the presence of nontrivial analog noise: recovery requires the same sample size one would have needed had the measurement matrix been Gaussian and the noisy analog measurements been given as data.
We address the problem of sparse recovery using greedy compressed sensing recovery algorithms, without explicit knowledge of the sparsity. Estimating the sparsity order is a crucial problem in many practical scenarios, e.g., wireless communications, where exact value of the sparsity order of the unknown channel may be unavailable a priori. In this paper we have proposed a new greedy algorithm, referred to as the Multiple Choice Hard Thresholding Pursuit (MCHTP), which modifies the popular hard thresholding pursuit (HTP) suitably to iteratively recover the unknown sparse vector along with the s…
In this letter, the problem of sparse signal reconstruction from one bit compressed sensing measurements is investigated. To solve the problem, a variational Bayes framework with a new statistical multivariate model is used. The dependency of the wavelet decomposition coefficients is modeled with a multivariate Gaussian copula. This model can separate marginal structure of coefficients from their intra scale dependency. In particular, the drawable Gaussian vine copula multivariate double Lomax model is suggested. The reconstructed signal is derived by variational Bayes algorithm which can calc…
In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the second of two conference papers describing the derivation of these algorithms, connection with related literature, extensions of original framework, and new empirical evidence. This paper describes the state evolution formalism for analyzing these algorithms, and some of the conclusions that can be drawn from thi…
In the problem of one-bit compressed sensing, the goal is to find a $δ$-close estimation of a $k$-sparse vector $x \in \mathbb{R}^n$ given the signs of the entries of $y = Φx$, where $Φ$ is called the measurement matrix. For the one-bit compressed sensing problem, previous work \cite{Plan-robust,support} achieved $Θ(δ^{-2} k \log(n/k))$ and $\tilde{ \Oh} ( \frac{1}{ δ} k \log (n/k))$ measurements, respectively, but the decoding time was $Ω( n k \log (n / k ))$. \ In this paper, using tools and techniques developed in the context of two-stage group testing and streaming algorithms, we contribut…
We address the issue of applying quantized compressed sensing (CS) on low-energy telemonitoring. So far, few works studied this problem in applications where signals were only approximately sparse. We propose a two-stage data compressor based on quantized CS, where signals are compressed by compressed sensing and then the compressed measurements are quantized with only 2 bits per measurement. This compressor can greatly reduce the transmission bit-budget. To recover signals from underdetermined, quantized measurements, we develop a Bayesian De-quantization algorithm. It can exploit both the mo…
In this paper, we address the problem of distributed sparse recovery of signals acquired via compressed measurements in a sensor network. We propose a new class of distributed algorithms to solve Lasso regression problems, when the communication to a fusion center is not possible, e.g., due to communication cost or privacy reasons. More precisely, we introduce a distributed iterative soft thresholding algorithm (DISTA) that consists of three steps: an averaging step, a gradient step, and a soft thresholding operation. We prove the convergence of DISTA in networks represented by regular graphs,…
We consider the variational free energy approach for compressed sensing. We first show that the naïve mean field approach performs remarkably well when coupled with a noise learning procedure. We also notice that it leads to the same equations as those used for iterative thresholding. We then discuss the Bethe free energy and how it corresponds to the fixed points of the approximate message passing algorithm. In both cases, we test numerically the direct optimization of the free energies as a converging sparse-estimationalgorithm.
In frequency-division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems, 1-bit compressed sensing (CS)-based superimposed channel state information (CSI) feedback has shown many advantages, while still faces many challenges, such as low accuracy of the downlink CSI recovery and large processing delays. To overcome these drawbacks, this paper proposes a deep learning (DL) scheme to improve the 1-bit compressed sensing-based superimposed CSI feedback. On the user side, the downlink CSI is compressed with the 1-bit CS technique, superimposed on the uplink user data sequences (…
This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high pro…
