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, relative to the general iteration steps of ADMM, the proposed algorithms have no splitting or dual variables in iterations and thus the dependence of the current approximation on past iterations is direct. Secondly, provable theoretical guarantees are provided in terms of restricted isometry property, which is the first theoretical guarantee of ADMM for $\ell_0$-LS to the best of our knowledge. Finally, they outperform the corresponding IHT, NIHT and HTP greatly when reconstructing both constant amplitude signals with random signs (CARS signals) and Gaussian signals.
Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial $\ell_1$-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context \cite{CRT,DonohoPol} proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization succeeds in solving the system. In thi…
Compressed sensing with sparse frame representations is seen to have much greater range of practical applications than that with orthonormal bases. In such settings, one approach to recover the signal is known as $\ell_1$-analysis. We expand in this article the performance analysis of this approach by providing a weaker recovery condition than existing results in the literature. Our analysis is also broadly based on general frames and alternative dual frames (as analysis operators). As one application to such a general-dual-based approach and performance analysis, an optimal-dual-based techniq…
In this paper we address the recovery conditions of weighted $\ell_p$ minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that weighted $\ell_p$ minimization with $0<p<1$ is stable and robust under weaker sufficient conditions compared to weighted $\ell_1$ minimization. Moreover, the sufficient recovery conditions of weighted $\ell_p$ are weaker than those of regular $\ell_p$ minimization if at least $50%$ of the support estimate is accurate. We also review some algorithms which exist to solve the non-convex $\ell_p…
An algorithmic framework, based on the difference of convex functions algorithm (DCA), is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence of $\ell_1$ minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit ($\ell_1$ minimization) and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literatu…
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 …
We present reconstruction algorithms for smooth signals with block sparsity from their compressed measurements. We tackle the issue of varying group size via group-sparse least absolute shrinkage selection operator (LASSO) as well as via latent group LASSO regularizations. We achieve smoothness in the signal via fusion. We develop low-complexity solvers for our proposed formulations through the alternating direction method of multipliers.
Compressed sensing (CS) involves sampling signals at rates less than their Nyquist rates and attempting to reconstruct them after sample acquisition. Most such algorithms have parameters, for example the regularization parameter in LASSO, which need to be chosen carefully for optimal performance. These parameters can be chosen based on assumptions on the noise level or signal sparsity, but this knowledge may often be unavailable. In such cases, cross validation (CV) can be used to choose these parameters in a purely data-driven fashion. Previous work analysing the use of CV in CS has been base…
One of the most basic problems in compressed sensing is solving an under-determined system of linear equations. Although this problem seems rather hard certain $\ell_1$-optimization algorithm appears to be very successful in solving it. The recent work of \cite{CRT,DonohoPol} rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the len…
Reliable and energy-efficient wireless data transmission remains a major challenge in resource-constrained wireless neural recording tasks, where data compression is generally adopted to relax the burdens on the wireless data link. Recently, Compressed Sensing (CS) theory has successfully demonstrated its potential in neural recording application. The main limitation of CS, however, is that the neural signals have no good sparse representation with commonly used dictionaries and learning a reliable dictionary is often data dependent and computationally demanding. In this paper, a novel CS appr…
This paper follows the recent discussion on the sparse solution recovery with quasi-norms $\ell_q,~q\in(0,1)$ when the sensing matrix possesses a Restricted Isometry Constant $δ_{2k}$ (RIC). Our key tool is an improvement on a version of "the converse of a generalized Cauchy-Schwarz inequality" extended to the setting of quasi-norm. We show that, if $δ_{2k}\le 1/2$, any minimizer of the $l_q$ minimization, at least for those $q\in(0,0.9181]$, is the sparse solution of the corresponding underdetermined linear system. Moreover, if $δ_{2k}\le0.4931$, the sparse solution can be recovered by any $l…
In this paper we look at a well known linear inverse problem that is one of the mathematical cornerstones of the compressed sensing field. In seminal works \cite{CRT,DOnoho06CS} $\ell_1$ optimization and its success when used for recovering sparse solutions of linear inverse problems was considered. Moreover, \cite{CRT,DOnoho06CS} established for the first time in a statistical context that an unknown vector of linear sparsity can be recovered as a known existing solution of an under-determined linear system through $\ell_1$ optimization. In \cite{DonohoPol,DonohoUnsigned} (and later in \cite{…
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…
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 seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by $\ell_1$-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by $\ell_1$-minimization with random partial circulant…
We extend the techniques of Hügel, Rauhut and Strohmer (Found. Comput. Math., 2014) to show that for every $δ\in(0,1]$, there exists an explicit random $m\times N$ partial Fourier matrix $A$ with $m=s\operatorname{polylog}(N/ε)$ and entropy $s^δ\operatorname{polylog}(N/ε)$ such that for every $s$-sparse signal $x\in\mathbb{C}^N$, there exists an event of probability at least $1-ε$ over which $x$ is the unique minimizer of $\|z\|_1$ subject to $Az=Ax$. The bulk of our analysis uses tools from decoupling to estimate the extreme singular values of the submatrix of $A$ whose columns correspond to …
The null space condition for $\ell_1$ minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via $\ell_1$ minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on…
In this paper we revisit one of the classical problems of compressed sensing. Namely, we consider linear under-determined systems with sparse solutions. A substantial success in mathematical characterization of an $\ell_1$ optimization technique typically used for solving such systems has been achieved during the last decade. Seminal works \cite{CRT,DOnoho06CS} showed that the $\ell_1$ can recover a so-called linear sparsity (i.e. solve systems even when the solution has a sparsity linearly proportional to the length of the unknown vector). Later considerations \cite{DonohoPol,DonohoUnsigned} …
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.
Compressed sensing of simultaneously sparse and low-rank matrices enables recovery of sparse signals from a few linear measurements of their bilinear form. One important question is how many measurements are needed for a stable reconstruction in the presence of measurement noise. Unlike conventional compressed sensing for sparse vectors, where convex relaxation via the $\ell_1$-norm achieves near optimal performance, for compressed sensing of sparse low-rank matrices, it has been shown recently Oymak et al. that convex programmings using the nuclear norm and the mixed norm are highly suboptima…
The sparse signal recovery in the standard compressed sensing (CS) problem requires that the sensing matrix be known a priori. Such an ideal assumption may not be met in practical applications where various errors and fluctuations exist in the sensing instruments. This paper considers the problem of compressed sensing subject to a structured perturbation in the sensing matrix. Under mild conditions, it is shown that a sparse signal can be recovered by $\ell_1$ minimization and the recovery error is at most proportional to the measurement noise level, which is similar to the standard CS result.…
This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ell_1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on `Compressed Sensing' edited by Yonina Eldar and Gitta Kutyn…
Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. This line of work shows that $\ell_1$-regularized least squares regression can accurately estimate a sparse linear model from $n$ noisy examples in $p$ dimensions, even if $p$ is much larger than $n$. In this paper we study a variant of this problem where the original $n$ input variables are compressed by a random linear transformation to $m \ll n$ examples in $p$ dimensions, and establish conditions under whic…
The achievable and converse regions for sparse representation of white Gaussian noise based on an overcomplete dictionary are derived in the limit of large systems. Furthermore, the marginal distribution of such sparse representations is also inferred. The results are obtained via the Replica method which stems from statistical mechanics. A direct outcome of these results is the introduction of sharp threshold for $\ell_{0}$-norm decoding in noisy compressed sensing, and its mean-square error for underdetermined Gaussian vector channels.
Goal: This paper deals with the problems that some EEG signals have no good sparse representation and single channel processing is not computationally efficient in compressed sensing of multi-channel EEG signals. Methods: An optimization model with L0 norm and Schatten-0 norm is proposed to enforce cosparsity and low rank structures in the reconstructed multi-channel EEG signals. Both convex relaxation and global consensus optimization with alternating direction method of multipliers are used to compute the optimization model. Results: The performance of multi-channel EEG signal reconstruction…
We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block $k$-sparse signals in no more than $k$ steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed $\ell_2/\ell_1$-optimization approach. This complements previous recove…
In compressed sensing sparse solutions are usually obtained by solving an $\ell^1$-minimization problem. Furthermore, the sparsity of the signal does need not be directly given. In fact, it is sufficient to have a signal that is sparse after an application of a suitable transform. In this paper we consider the stability of solutions obtained from $\ell^p$-minimization for arbitrary $0<p \leq1$. Further we suppose that the signals are sparse with respect to general redundant transforms associated to not necessarily tight frames. Since we are considering general frames the role of the dual frame…
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 compressed sensing, one wishes to acquire an approximately sparse high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear measurements, then later approximately recover $x$ given only those measurement outcomes. Various guarantees have been studied in terms of the notion of approximation in recovery, and some isolated folklore results are known stating that some forms of recovery are stronger than others, via black-box reductions. In this note we provide a general theorem concerning the hierarchy of strengths of various recovery guarantees. As a corollary of this theorem, by re…
In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\ell_1$ minimization. \cite{Icassp reweighted l_1}, \cite{Isit reweighted l_1}, \cite{XuScaingLaw} and \cite{iterativereweightedjournal} have shown that a two-stage re-weighted $\ell_1$ minimization algorithm can boost the phase transition performance for signals whose nonzero elem…
In the problem of compressive phase retrieval, one wants to recover an approximately $k$-sparse signal $x \in \mathbb{C}^n$, given the magnitudes of the entries of $Φx$, where $Φ\in \mathbb{C}^{m \times n}$. This problem has received a fair amount of attention, with sublinear time algorithms appearing in \cite{cai2014super,pedarsani2014phasecode,yin2015fast}. In this paper we further investigate the direction of sublinear decoding for real signals by giving a recovery scheme under the $\ell_2 / \ell_2$ guarantee, with almost optimal, $\Oh(k \log n )$, number of measurements. Our result outperf…
The weak-$\ell^p$ norm can be used to define a measure $s$ of sparsity. When we compute $s$ for the discrete cosine transform coefficients of a signal, the value of $s$ is related to the information content of said signal. We use this value of $s$ to define a reference-free index $\mathcal{E}$, called the sparsity index, that we can use to predict with high accuracy the quality of signal reconstruction in the setting of compressed sensing imaging. That way, when compressed sensing is framed in the context of sampling theory, we can use $\mathcal{E}$ to decide when to further partition the samp…
We consider the compressed sensing problem, where the object $x_0 \in \bR^N$ is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$. A popular method of sparsity-promoting reconstruction is $\ell^1$-penalized least-squares reconstruction (aka LASSO, Basis Pursuit). It is currently popular to consider the strict sparsity model, where the object $x_0$ is nonzero in only a small fraction of entries. In this paper, we instead consider the much more broadly applicable $\ell_p$-sparsity model,…
Compressed sensing is a developing field aiming at reconstruction of sparse signals acquired in reduced dimensions, which make the recovery process under-determined. The required solution is the one with minimum $\ell_0$ norm due to sparsity, however it is not practical to solve the $\ell_0$ minimization problem. Commonly used techniques include $\ell_1$ minimization, such as Basis Pursuit (BP) and greedy pursuit algorithms such as Orthogonal Matching Pursuit (OMP) and Subspace Pursuit (SP). This manuscript proposes a novel semi-greedy recovery approach, namely A* Orthogonal Matching Pursuit (…
Model-based compressed sensing refers to compressed sensing with extra structure about the underlying sparse signal known a priori. Recent work has demonstrated that both for deterministic and probabilistic models imposed on the signal, this extra information can be successfully exploited to enhance recovery performance. In particular, weighted $\ell_1$-minimization with suitable choice of weights has been shown to improve performance in the so called non-uniform sparse model of signals. In this paper, we consider a full generalization of the non-uniform sparse model with very mild assumptions…
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. The reconstruction method is based on a convex minimization program coined "innovative Basis Pursuit DeNoise" (or iBPDN). Under the common $\ell_2$-fidelity constraint made on the available measurements, this optimization promotes the ($\ell_1$) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results …
We consider the problem of recovering a single or multiple frequency-sparse signals, which share the same frequency components, from a subset of regularly spaced samples. The problem is referred to as continuous compressed sensing (CCS) in which the frequencies can take any values in the normalized domain [0,1). In this paper, a link between CCS and low rank matrix completion (LRMC) is established based on an $\ell_0$-pseudo-norm-like formulation, and theoretical guarantees for exact recovery are analyzed. Practically efficient algorithms are proposed based on the link and convex and nonconvex…
Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modelled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling…
The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. Our main theorem is that, if $G$ is $L$-Lipschitz, then roughly $O(k …
In the paper, we introduce an unconstrained analysis model based on the $\ell_{1}-α\ell_{2}$ $(0< α\leq1)$ minimization for the signal and image reconstruction. We develop some new technology lemmas for tight frame, and the recovery guarantees based on the restricted isometry property adapted to frames. The effective algorithm is established for the proposed nonconvex analysis model. We illustrate the performance of the proposed model and algorithm for the signal and compressed sensing MRI reconstruction via extensive numerical experiments. And their performance is better than that of the exis…
We introduce a new class of measurement matrices for compressed sensing, using low order summaries over binary sequences of a given length. We prove recovery guarantees for three reconstruction algorithms using the proposed measurements, including $\ell_1$ minimization and two combinatorial methods. In particular, one of the algorithms recovers $k$-sparse vectors of length $N$ in sublinear time $\text{poly}(k\log{N})$, and requires at most $Ω(k\log{N}\log\log{N})$ measurements. The empirical oversampling constant of the algorithm is significantly better than existing sublinear recovery algorit…
We propose a compressed sampling and dictionary learning framework for fiber-optic sensing using wavelength-tunable lasers. A redundant dictionary is generated from a model for the reflected sensor signal. Imperfect prior knowledge is considered in terms of uncertain local and global parameters. To estimate a sparse representation and the dictionary parameters, we present an alternating minimization algorithm that is equipped with a pre-processing routine to handle dictionary coherence. The support of the obtained sparse signal indicates the reflection delays, which can be used to measure impa…
Compressed sensing of sparse sources can be improved by incorporating prior knowledge of the source. In this paper we demonstrate a method for optimal selection of weights in weighted $L_1$ norm minimization for a noiseless reconstruction model, and show the improvements in compression that can be achieved.
Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and $\ell_1$ analysis minimization becomes po…
The phase transition is a performance measure of the sparsity-undersampling tradeoff in compressed sensing (CS). This letter reports our first observation and evaluation of an empirical phase transition of the $\ell_1$ minimization approach to the complex valued CS (CVCS), which is positioned well above the known phase transition of the real valued CS in the phase plane. This result can be considered as an extension of the existing phase transition theory of the block-sparse CS (BSCS) based on the universality argument, since the CVCS problem does not meet the condition required by the phase t…
This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, s…
In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies $\{u_i\}_{i=1}^M$, where $M$ is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies $\{u_i + δ_i\}_{i=1}^M$ that are different from the base frequencies and where $\{δ_i\}_{i=1}^M$ are unknown. Ignoring perturbations of this nature can lead to major errors in signal recovery. In this paper, we present a simple but effective altern…
Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and recoverable by $\ell_1$-minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax = b. Specifically, suppose A is of full spark with m rows, and suppose $\frac{m}{2}$ < s < m. Then the solution to Ax = b is unique for x with $\|x\|_0 \leq s$ up to a set …
Solving under-determined systems of linear equations with sparse solutions attracted enormous amount of attention in recent years, above all, due to work of \cite{CRT,CanRomTao06,DonohoPol}. In \cite{CRT,CanRomTao06,DonohoPol} it was rigorously shown for the first time that in a statistical and large dimensional context a linear sparsity can be recovered from an under-determined system via a simple polynomial $\ell_1$-optimization algorithm. \cite{CanRomTao06} went even further and established that in \emph{noisy} systems for any linear level of under-determinedness there is again a linear spa…
In this paper, we propose a method to predict the asymptotic performance of the alternating direction method of multipliers (ADMM) for compressed sensing, where we reconstruct an unknown structured signal from its underdetermined linear measurements. The derivation of the proposed method is based on the recently developed convex Gaussian min-max theorem (CGMT), which can be applied to various convex optimization problems to obtain its asymptotic error performance. Our main idea is to analyze the convex subproblem in the update of ADMM iteratively and characterize the asymptotic distribution of…
