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 @math , @math measurements suffice to reconstruct the direction of any @math -sparse vector up to accuracy @math via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an @math RIP-property. Under a slightly worse dependence on @math , we establish stability with respect to approximate sparsity, as well as full vector recovery results.
Standard compressive sensing with partial circulant matrices. In standard (unquantized) compressive sensing, the task is to recover an (approximately) sparse vector @math from measurements @math , where @math with @math . A number of reconstruction algorithms have been introduced, most notably @math -minimization which computes the minimizer of [ z ^N |z |_1 subject to A z = Ax. ] The ( @math -)restricted isometry property is a classical way of analyzing the performance of compressive sensing @cite_9 . The restricted isometry constant @math is defined as the smallest constant @math such that If @math then all @math -sparse signals can be reconstructed via @math -minimization exactly, see e.g. @cite_17 @cite_9 . Stability under noise and sparsity defect can be shown as well and similar guarantees also hold for other reconstruction algorithms @cite_9 . It is well-known that Gaussian random matrices satisfy @math with probability at least @math if @math [Chapter 9] fora13 .
@cite_9 @cite_17
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We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. 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 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 1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a log-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality. Abstract It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as “compressed sensing” or “compressive sampling” relies on properties of the sensing matrix such as the restricted isometry property . In this Note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant. To cite this article: E.J. Candes, C. R. Acad. Sci. Paris, Ser. I 346 (2008). The goal of sparse recovery is to recover a k-sparse signal x e Rn from (possibly noisy) linear measurements of the form y = Ax, where A e Rmxn describes the measurement process. Standard results in compressive sensing show that it is possible to recover the signal x from m = O(k log(n k)) measurements, and that this bound is tight. The framework of model-based compressive sensing [BCDH10] overcomes the lower bound and reduces the number of measurements further to O(k) by limiting the supports of x to a subset M of the (nk) possible supports. This has led to many measurement-efficient algorithms for a wide variety of signal models, including block-sparsity and tree-sparsity. Unfortunately, extending the framework to other, more general models has been stymied by the following obstacle: for the framework to apply, one needs an algorithm that, given a signal x, solves the following optimization problem exactly: [EQUATION] (here x[n] denotes the projection of x on coordinates not in Ω). However, an approximation algorithm for this optimization task is not sufficient. Since many problems of this form are not known to have exact polynomial-time algorithms, this requirement poses an obstacle for extending the framework to a richer class of models. In this paper, we remove this obstacle and show how to extend the model-based compressive sensing framework so that it requires only approximate solutions to the aforementioned optimization problems. Interestingly, our extension requires the existence of approximation algorithms for both the maximization and the minimization variants of the optimization problem. Further, we apply our framework to the Constrained Earth Mover's Distance (CEMD) model introduced in [SHI13], obtaining a sparse recovery scheme that uses significantly less than O(k log(n k)) measurements. This is the first non-trivial theoretical bound for this model, since the validation of the approach presented in [SHI13] was purely empirical. The result is obtained by designing a novel approximation algorithm for the maximization version of the problem and proving approximation guarantees for the minimization algorithm described in [SHI13]. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods from such measurements typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math within a Euclidean ball of known radius. Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candes, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size ź is used to quantize m measurements y=źx of a k-sparse signal xźźN, where ź satisfies the restricted isometry property, then the approximate recovery x# via l1-minimization is within O(ź) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma---Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order ź(k m)(rź1 2)ź for any 0<ź<1, if mźr,źk(logN)1 (1źź). The result holds with high probability on the initial draw of the measurement matrix ź from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius. This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements. The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices describe measurement mappings achieving, with overwhelming probability, nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary @math -Stable Embedding (B @math SE) property, which characterizes the robustness measurement process to sign changes. We show the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance. Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary @math . This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the @math -restricted isometry property ( @math -RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in @math . While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the @math -RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary $ ... Abstract We investigate the recovery of almost s -sparse vectors x ∈ C N from undersampled and inaccurate data y = A x + e ∈ C m by means of minimizing ‖ z ‖ 1 subject to the equality constraints A z = y . If m ≍ s ln ( N s ) and if Gaussian random matrices A ∈ R m × N are used, this equality-constrained l 1 -minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m ≍ s ln ( N s ) and if Weibull random matrices are used, we prove here that the equality-constrained l 1 -minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the l 1 -norm and the outer norm is replaced by a norm comparable to the l 2 -norm. For the l 1 -minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables. It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as sensing" or sampling" relies on properties of the sensing matrix such as the restricted isometry property. In this note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant. R This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all standard models-e.g., Gaussian, frequency measurements-discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) to hold near the sparsity level in question, nor a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s logn Fourier coefficients that are contaminated with noise. Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed with an underdetermined linear measurement model. However, in reality there is a mismatch between the assumed and the actual bases due to factors such as discretization of the parameter space defining basis components, sampling jitter in A D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes significant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this paper presents a novel perturbed orthogonal matching pursuit (POMP) algorithm that performs controlled perturbation of selected support vectors to decrease the orthogonal residual at each iteration. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of perturbed bases can be obtained as compared to standard sparse reconstruction techniques. 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 @math , @math measurements suffice to reconstruct the direction of any @math -sparse vector up to accuracy @math via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an @math RIP-property. Under a slightly worse dependence on @math , we establish stability with respect to approximate sparsity, as well as full vector recovery results. We study the recovery of sparse vectors from subsampled random convolutions via @math -minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity @math is small enough, i.e., @math , we show that @math measurements are sufficient to recover @math -sparse vectors in dimension @math with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If @math is larger, then essentially @math measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques which should be of independent interest. The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally believed necessary. In [1], it was shown empirically that using lscrp minimization with p < 1 can do so with fewer measurements than with p = 1. In this paper we consider the use of iteratively reweighted algorithms for computing local minima of the nonconvex problem. In particular, a particular regularization strategy is found to greatly improve the ability of a reweighted least-squares algorithm to recover sparse signals, with exact recovery being observed for signals that are much less sparse than required by an unregularized version (such as FOCUSS, [2]). Improvements are also observed for the reweighted-lscr1 approach of [3]. In this paper, we improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. (1) In compressed sensing, we show that if the m×n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable l 1 minimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m (log(n m)+1)). (2) In the very general sensing model introduced in Candes and Plan (IEEE Trans. Inf. Theory 57(11):7235–7254, 2011) and assuming a positive fraction of corrupted measurements, exact recovery still holds if the signal now has O(m (log2 n)) nonzero entries. (3) Finally, we prove that one can recover an n×n low-rank matrix from m corrupted sampled entries by tractable optimization provided the rank is on the order of O(m (nlog2 n)); again, this holds when there is a positive fraction of corrupted samples. It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math .