Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core question is, by exploiting the intrinsic low dimensional structure of the signal, how to design the sensing process and decoder in a way that the number of measurements is essentially close to the complexity of the signal set. This chapter provides a survey of important results in answering this question, with an emphasis on a basis pursuit like convex optimization decoder that admits a wide range of random sensing matrices. The results are quite established in the case signals are sparse in an orthonormal basis, while the case with frame sparse signals is much less explored. In addition to presenting the latest results on recovery guarantee and how few random heavier-tailed measurements fulfill these recovery guarantees, this chapter also aims to provide some insights in proof techniques. We also take the opportunity of this book chapter to publish an interesting result (Theorem 3.10) about a restricted isometry like property related to a frame.
Compressive Sensing (CS) is a new paradigm for the efficient acquisition of signals that have sparse representation in a certain domain. Traditionally, CS has provided numerous methods for signal recovery over an orthonormal basis. However, modern applications have sparked the emergence of related methods for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary, particularly due to their potential to generate different kinds of sparse representation of signals. To this end, we apply a signal space greedy method, which relies on the ability t…
The recovery of sparsest overcomplete representation has recently attracted intensive research activities owe to its important potential in the many applied fields such as signal processing, medical imaging, communication, and so on. This problem can be stated in the following, i.e., to seek for the sparse coefficient vector of the given noisy observation over a redundant dictionary such that, where is the corrupted error. Elad et al. made the worst-case result, which shows the condition of stable recovery of sparest overcomplete representation over is where . Although it's of easy operation f…
This paper presents the first theoretical results showing that stable identification of overcomplete $μ$-coherent dictionaries $Φ\in \mathbb{R}^{d\times K}$ is locally possible from training signals with sparsity levels $S$ up to the order $O(μ^{-2})$ and signal to noise ratios up to $O(\sqrt{d})$. In particular the dictionary is recoverable as the local maximum of a new maximisation criterion that generalises the K-means criterion. For this maximisation criterion results for asymptotic exact recovery for sparsity levels up to $O(μ^{-1})$ and stable recovery for sparsity levels up to $O(μ^{-2}…
The task of finding a sparse signal decomposition in an overcomplete dictionary is made more complicated when the signal undergoes an unknown modulation (or convolution in the complementary Fourier domain). Such simultaneous sparse recovery and blind demodulation problems appear in many applications including medical imaging, super resolution, self-calibration, etc. In this paper, we consider a more general sparse recovery and blind demodulation problem in which each atom comprising the signal undergoes a distinct modulation process. Under the assumption that the modulating waveforms live in a…
In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally s…
Let x be a signal to be sparsely decomposed over a redundant dictionary A, i.e., a sparse coefficient vector s has to be found such that x=As. It is known that this problem is inherently unstable against noise, and to overcome this instability, the authors of [Stable Recovery; Donoho et.al., 2006] have proposed to use an "approximate" decomposition, that is, a decomposition satisfying ||x - A s|| < δ, rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ||s||_0 < (1+M^{-1})/2, where M denotes the coherence of the dictionary, this decompo…
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.
“Gheorghe Asachi” Technical University of Iasi（雅西“格奥尔基·阿萨奇”技术大学）
Optimizing the acquisition matrix is useful for compressed sensing of signals that are sparse in overcomplete dictionaries, because the acquisition matrix can be adapted to the particular correlations of the dictionary atoms. In this paper a novel formulation of the optimization problem is proposed, in the form of a rank-constrained nearest correlation matrix problem. Furthermore, improvements for three existing optimization algorithms are introduced, which are shown to be particular instances of the proposed formulation. Simulation results show notable improvements and superior robustness in …
This letter is on the performance of the turbo signal recovery (TSR) algorithm for partial discrete Fourier transform (DFT) matrices based compressed sensing. Based on state evolution analysis, we prove that TSR with a partial DFT sensing matrix outperforms the well-known approximate message passing (AMP) algorithm with an independent identically distributed (IID) sensing matrix.
Prior support constrained compressed sensing has of late become popular due to its potential for applications. The existing results on recovery guarantees provide global recovery bounds in the sense that they deal with full support. However, in some applications, one might be interested in the recovery guarantees limited to the given prior support, such bounds may be termed as local recovery bounds. The present work proposes the local recovery guarantees and analyzes the conditions on associated parameters that make recovery error small.
Compressive sensing (CS) is a promising technology for realizing energy-efficient wireless sensors for long-term health monitoring. In this paper, we propose a data-driven CS framework that learns signal characteristics and individual variability from patients' data to significantly enhance CS performance and noise resilience. This is accomplished by a co-training approach that optimizes both the sensing matrix and dictionary towards improved restricted isometry property (RIP) and signal sparsity, respectively. Experimental results upon ECG signals show that our framework is able to achieve be…
We address the recovery of sparse vectors in an overcomplete, linear and noisy multiple measurement framework, where the measurement matrix is known upto a permutation of its rows. We derive sparse Bayesian learning (SBL) based updates for joint recovery of the unknown sparse vector and the sensing order, represented using a permutation matrix. We model the sparse matrix using multiple uncorrelated and correlated vectors, and in particular, we use the first order AR model for the correlated sparse vectors. We propose the Permutation-MSBL and a Kalman filtering based Permutation-KSBL algorithm …
The physics of compressive sensing (CS) and the gradient-based recovery algorithms are presented. First, the different forms for CS are summarized. Second, the physical meanings of coherence and measurement are given. Third, the gradient-based recovery algorithms and their geometry explanations are provided. Finally, we conclude the report and give some suggestion for future work.
Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be mode…
We study the recovery results of $\ell_p$-constrained compressive sensing (CS) with $p\geq 1$ via robust width property and determine conditions on the number of measurements for standard Gaussian matrices under which the property holds with high probability. Our paper extends the existing results in Cahill and Mixon (2014) from $\ell_2$-constrained CS to $\ell_p$-constrained case with $p\geq 1$ and complements the recovery analysis for robust CS with $\ell_p$ loss function.
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 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.
A combinatorial approach to compressive sensing based on a deterministic column replacement technique is proposed. Informally, it takes as input a pattern matrix and ingredient measurement matrices, and results in a larger measurement matrix by replacing elements of the pattern matrix with columns from the ingredient matrices. This hierarchical technique yields great flexibility in sparse signal recovery. Specifically, recovery for the resulting measurement matrix does not depend on any fixed algorithm but rather on the recovery scheme of each ingredient matrix. In this paper, we investigate c…
Approximate message passing (AMP) is an efficient iterative signal recovery algorithm for compressed sensing (CS). For sensing matrices with independent and identically distributed (i.i.d.) Gaussian entries, the behavior of AMP can be asymptotically described by a scaler recursion called state evolution. Orthogonal AMP (OAMP) is a variant of AMP that imposes a divergence-free constraint on the denoiser. In this paper, we extend OAMP to incorporate generic denoisers, hence the name D-OAMP. Our numerical results show that state evolution predicts the performance of D-OAMP well for generic denois…
In this study, we propose a generalized turbo signal recovery algorithm to estimate a signal from quantized measurements, in which the sensing matrix is a row-orthogonal matrix, such as the partial discrete Fourier transform matrix. The state evolution of the proposed algorithm is derived and is shown to be consistent with that obtained with the replica method. Numerical experiments illustrate the excellent agreement of the proposed algorithm with theoretical state evolution.
In this paper, we propose a general framework for the asymptotic analysis of node-based verification-based algorithms. In our analysis we tend the signal length $n$ to infinity. We also let the number of non-zero elements of the signal $k$ scale linearly with $n$. Using the proposed framework, we study the asymptotic behavior of the recovery algorithms over random sparse matrices (graphs) in the context of compressive sensing. Our analysis shows that there exists a success threshold on the density ratio $k/n$, before which the recovery algorithms are successful, and beyond which they fail. Thi…
One-bit compressive sensing has extended the scope of sparse recovery by showing that sparse signals can be accurately reconstructed even when their linear measurements are subject to the extreme quantization scenario of binary samples---only the sign of each linear measurement is maintained. Existing results in one-bit compressive sensing rely on the assumption that the signals of interest are sparse in some fixed orthonormal basis. However, in most practical applications, signals are sparse with respect to an overcomplete dictionary, rather than a basis. There has already been a surge of act…
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…
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…
In traditional compressed sensing theory, the dictionary matrix is given a priori, whereas in real applications this matrix suffers from random noise and fluctuations. In this paper we consider a signal model where each column in the dictionary matrix is affected by a structured noise. This formulation is common in direction-of-arrival (DOA) estimation of off-grid targets, encountered in both radar systems and array processing. We propose to use joint sparse signal recovery to solve the compressed sensing problem with structured dictionary mismatches and also give an analytical performance bou…
In this article, we discuss a novel greedy algorithm for the recovery of compressive sampled signals under noisy conditions. Most of the greedy recovery algorithms proposed in the literature require sparsity of the signal to be known or they estimate sparsity, for a known representation basis, from the number of measurements. These algorithms recover signals when noise level is significantly low. We propose Entropy minimization based Matching Pursuit (EMP) which has the capability to reject noise even when noise level is comparable to that of signal level. The proposed algorithm can cater to c…
In oversampled adaptive sensing (OAS), noisy measurements are collected in multiple subframes. The sensing basis in each subframe is adapted according to some posterior information exploited from previous measurements. The framework is shown to significantly outperform the classic non-adaptive compressive sensing approach. This paper extends the notion of OAS to signals with structured sparsity. We develop a low-complexity OAS algorithm based on structured orthogonal sensing. Our investigations depict that the proposed algorithm outperforms the conventional non-adaptive compressive sensing fra…
We demonstrate that a sparse signal can be estimated from the phase of complex random measurements, in a "phase-only compressive sensing" (PO-CS) scenario. With high probability and up to a global unknown amplitude, we can perfectly recover such a signal if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the signal sparsity. Our approach consists in recasting the (non-linear) PO-CS scheme as a linear compressive sensing model. We built it from a signal normalization constraint and a phase-consistency constraint. Practically, we achieve…
We study the recovery conditions of weighted $\ell_1$ minimization for real-valued signal reconstruction from phaseless compressive sensing measurements when partial support information is available. A strong restricted isometry property condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success of $k$-sparse phaseless recovery via weighted $\ell_1$ minimization. Numerical experiments are conducted to illustrate our results.
Compressive Sensing, as an emerging technique in signal processing is reviewed in this paper together with its common applications. As an alternative to the traditional signal sampling, Compressive Sensing allows a new acquisition strategy with significantly reduced number of samples needed for accurate signal reconstruction. The basic ideas and motivation behind this approach are provided in the theoretical part of the paper. The commonly used algorithms for missing data reconstruction are presented. The Compressive Sensing applications have gained significant attention leading to an intensiv…
In this paper, we present a probabilistic analysis of iterative node-based verification-based (NB-VB) recovery algorithms over irregular graphs in the context of compressed sensing. Verification-based algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The analysis predicts the average fraction of unverified signal elements at each iteration $\ell$ where the average is taken over the ensembles of input signals and sensing matrices. The analysis is asymptotic ($n \rightarrow \infty$) and is similar in nature to the well-known density evoluti…
In this paper, {the goal is to design deterministic sampling patterns on the sphere and the rotation group} and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing matrices, which consists of random samples of Wigner D-functions, satisfy the Restricted Isometry Property (RIP) with proper preconditioning and can be used for sparse recovery on the rotation group. The mutual coherence, however, is used to assess the performance of deterministic and regular sensing matrices. We show that many of widely used regular sampling patt…
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 study the problem of jointly sparse support recovery with 1-bit compressive measurements in a sensor network. Sensors are assumed to observe sparse signals having the same but unknown sparse support. Each sensor quantizes its measurement vector element-wise to 1-bit and transmits the quantized observations to a fusion center. We develop a computationally tractable support recovery algorithm which minimizes a cost function defined in terms of the likelihood function and the $l_{1,\infty}$ norm. We observe that even with noisy 1-bit measurements, jointly sparse support can be recovered accura…
We introduce q-ary compressive sensing, an extension of 1-bit compressive sensing. We propose a novel sensing mechanism and a corresponding recovery procedure. The recovery properties of the proposed approach are analyzed both theoretically and empirically. Results in 1-bit compressive sensing are recovered as a special case. Our theoretical results suggest a tradeoff between the quantization parameter q, and the number of measurements m in the control of the error of the resulting recovery algorithm, as well its robustness to noise.
Random sampling in compressive sensing (CS) enables the compression of large amounts of input signals in an efficient manner, which is useful for many applications. CS reconstructs the compressed signals exactly with overwhelming probability when incoming data can be sparsely represented with a few components. However, the theory of CS framework including random sampling has been focused on exact recovery of signal; impreciseness in signal recovery has been neglected. This can be problematic when there is uncertainty in the number of sparse components such as signal sparsity in dynamic systems…
Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. T…
Compressed Sensing (CS) is an effective approach to reduce the required number of samples for reconstructing a sparse signal in an a priori basis, but may suffer severely from the issue of basis mismatch. In this paper we study the problem of simultaneously recovering multiple spectrally-sparse signals that are supported on the same frequencies lying arbitrarily on the unit circle. We propose an atomic norm minimization problem, which can be regarded as a continuous counterpart of the discrete CS formulation and be solved efficiently via semidefinite programming. Through numerical experiments,…
Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-opti…
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 …
In this paper, we propose majority voting neural networks for sparse signal recovery in binary compressed sensing. The majority voting neural network is composed of several independently trained feedforward neural networks employing the sigmoid function as an activation function. Our empirical study shows that a choice of a loss function used in training processes for the network is of prime importance. We found a loss function suitable for sparse signal recovery, which includes a cross entropy-like term and an $L_1$ regularized term. From the experimental results, we observed that the majorit…
In this paper we propose a novel form of clipping mitigation in OFDM using compressive sensing that completely avoids tone reservation and hence rate loss for this purpose. The method builds on selecting the most reliable perturbations from the constellation lattice upon decoding at the receiver, and performs compressive sensing over these observations in order to completely recover the temporally sparse nonlinear distortion. As such, the method provides a unique practical solution to the problem of initial erroneous decoding decisions in iterative ML methods, offering both the ability to augm…
Distributed compressed sensing is concerned with representing an ensemble of jointly sparse signals using as few linear measurements as possible. Two novel joint reconstruction algorithms for distributed compressed sensing are presented in this paper. These algorithms are based on the idea of using one of the signals as side information; this allows to exploit joint sparsity in a more effective way with respect to existing schemes. They provide gains in reconstruction quality, especially when the nodes acquire few measurements, so that the system is able to operate with fewer measurements than…
We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex $\ell_1$ penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable d…
This paper studies the asymptotic performance of maximum-a-posteriori estimation in the presence of prior information. The problem arises in several applications such as recovery of signals with non-uniform sparsity pattern from underdetermined measurements. With prior information, the maximum-a-posteriori estimator might have asymmetric penalty. We consider a generic form of this estimator and study its performance via the replica method. Our analyses demonstrate an asymmetric form of the decoupling property in the large-system limit. Employing our results, we further investigate the performa…
