S-sparse Signal Research Articles

Sparse recovery using expanders via hard thresholding algorithm

Expanders play an important role in combinatorial compressed sensing. Via expanders measurements, we propose the expander normalized heavy ball hard thresholding algorithm (ENHB-HT) based on expander iterative hard thresholding (E-IHT) algorithm. We provide convergence analysis of ENHB-HT, and it turns out that ENHB-HT can recover an s-sparse signal if the measurement matrix A∈{0,1}m×n satisfies some mild conditions. Numerical experiments are simulated to support our two main theorems which describe the convergence rate and the accuracy of the proposed algorithm. Simulations are also performed to compare the performance of ENHB-HT and several existing algorithms under different types of noise, the empirical results demonstrate that our algorithm outperform a few existing ones in the presence of outliers.

High-order block RIP for nonconvex block-sparse compressed sensing

Abstract This paper concentrates on the recovery of block-sparse signals, which are not only sparse but also nonzero elements are arrayed into some blocks (clusters) rather than being arbitrary distributed all over the vector, from linear measurements. We establish high-order sufficient conditions based on block RIP, which could ensure the exact recovery of every block s-sparse signal in the noiseless case via mixed l 2 / l p {l_{2}/l_{p}} minimization method, and the stable and robust recovery in the case that signals are not accurately block-sparse in the presence of noise. Additionally, a lower bound on necessary number of random Gaussian measurements is gained for the condition to be true with overwhelming probability. Furthermore, a series of numerical experiments are conducted to demonstrate the performance of the mixed l 2 / l p {l_{2}/l_{p}} minimization. To the best of the authors’ knowledge, the recovery guarantees established in this paper are superior to those currently available.

Performance bounds of the intensity-based estimators for noisy phase retrieval

The aim of noisy phase retrieval is to estimate a signal x0∈Cd from m noisy intensity measurements bj=|〈aj,x0〉|2+ηj,j=1,…,m, where aj∈Cd are known measurement vectors and η=(η1,…,ηm)⊤∈Rm is a noise vector. A commonly used estimator for x0 is to minimize the intensity-based loss function, i.e., xˆ:=argminx∈Cd∑j=1m(|〈aj,x〉|2−bj)2. Although many algorithms for solving the intensity-based estimator have been developed, there are very few results about its estimation performance. In this paper, we focus on the performance of the intensity-based estimator and prove that the error bound satisfies minθ∈R⁡‖xˆ−eiθx0‖2≲min⁡{‖η‖2m1/4,‖η‖2‖x0‖2⋅m} under the assumption of m≳d and aj∈Cd,j=1,…,m, being complex Gaussian random vectors. We also show that the error bound is rate optimal when m≳dlog⁡m. In the case where x0 is an s-sparse signal, we present a similar result under the assumption of m≳slog⁡(ed/s). To the best of our knowledge, our results provide the first theoretical guarantees for both the intensity-based estimator and its sparse version.

Improved Sufficient Conditions Based on RIC of Order 2s for IHT and HTP Algorithms

Reconstructing an s-sparse signal x from an un- derdetermined noisy linear model arises in many applications. The iterative hard thresholding (IHT) and hard thresholding pursuit (HTP) algorithms are two popular sparse signal recovery algorithms. Since their recovery performances can be theoreti- cally characterized by their sufficient conditions of stable sparse signal recovery, it is essential to establish less restrictive sufficient conditions. This work develops δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2s</sub> -based sufficient conditions for stable recovery with IHT and HTP. The two new sufficient conditions are significantly less restrictive than the state-of-the- art ones for IHT and HTP.

Analysis of the ratio of ℓ1 and ℓ2 norms in compressed sensing

We study the ratio of ℓ1 and ℓ2 norms (ℓ1/ℓ2) as a sparsity-promoting objective in compressed sensing. We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the ℓ1/ℓ2 objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare ℓ1/ℓ2 with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach support selection, and we demonstrate that it empirically improves the performance of existing ℓ1/ℓ2 algorithms.

Efficient iterative thresholding algorithms with functional feedbacks and null space tuning

An accelerated class of adaptive scheme of iterative thresholding algorithms is studied analytically and empirically. They are based on the feedback mechanism of the null space tuning techniques. The main contribution of this article is the accelerated convergence analysis and proofs with a variable/adaptive index selection and different feedback principles at each iteration. The convergence analysis requires no longer a priori sparsity information s of a signal. It is shown that uniform recovery of all s-sparse signals from given linear measurements can be achieved under reasonable (preconditioned) restricted isometry conditions. Accelerated convergence rate and improved convergence conditions are obtained by selecting an appropriate size of the index support per iteration. The theoretical findings are sufficiently demonstrated and confirmed by extensive numerical experiments. It is also observed that the proposed algorithms have a clearly advantageous balance of efficiency, adaptivity and accuracy compared with all other state-of-the-art greedy iterative algorithms.

A new upper bound of p for lp-minimization in compressed sensing

Signal recovery through l1-minimization may fail for restricted isometry constant δ2s>2/2, therefore lp-minimization is an alternative method. In this paper, we consider the use of lp-minimization with 0 < p ≤ 1 to recover sparse signals from an under-determined linear measurement Ax=y. We prove that if A satisfies a restricted isometry property with δ2s>2/2, an upper bound p¯=(2+2)(1−δ2s) exists such that any s-sparse signal can be recovered through a nonconvex lp-minimization with any p≤p¯. To the best of our knowledge, this upper bound p¯ is a significant improvement compared to the best existing results proposed by Wen et al. (2015) [19], i.e.,p¯={5031(1−δ2s),δ2s∈[2/2,0.7183),0.4541,δ2s∈[0.7183,0.7729),2(1−δ2s),δ2s∈[0.7729,1).Numerical experiment indicates that lp-minimization with a larger p has better recovery effect than with a smaller p. Our result expands the upper bound of p.

One-bit compressed sensing via ℓp(0 < p < 1)-minimization method

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓp(0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓp effectively s-sparse signals which contains genuinely s-sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of ℓp effectively s-sparse signals with error order . We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any ℓp effectively s-sparse bounded signal x can be estimated using augmented ℓp minimization model and empirical distribution function method respectively. Especially, to recover ℓp effectively s-sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm.

Sparsity and incoherence in orthogonal matching pursuit

Recovery of sparse signals via approximation methods has been extensively studied in recently years. We consider the nonuniform recovery of orthogonal matching pursuit (OMP) from fewer noisy random measurements. Rows of sensing matrices are assumed to be drawn independently from a probability distribution obeying the isotropy property and the incoherence property. Our models not only include the standard sensing matrices in compressed sensing context, but also cover other new sensing matrices such as random convolutions, subsampled tight or continuous frames. Given m admissible random measurements of a fixed s-sparse signal $$\varvec{x}\in \mathbb {R}^n$$ , we show that OMP can recover the support of $$\varvec{x}$$ exactly after s iterations with overwhelming probability provided that $$\begin{aligned} m=O( s(s+\log (n-s))). \end{aligned}$$ It follows that the approximation order of OMP is $$\begin{aligned} \Vert \varvec{x}- \varvec{x}_j\Vert =O(\eta ^j) \end{aligned}$$ where $$0<\eta <1$$ and $$\varvec{x}_j$$ denotes the recovered signal at j-th iteration. As a byproduct of the proof, the necessary number of measurements to ensure sparse recovery by $$l_1$$ -minimization with random partial circulant or Toeplitz matrices is proved to be optimal.

Estimating linear functionals of a sparse family of Poisson means

Assume that we observe a sample of size n composed of p-dimensional signals, each signal having independent entries drawn from a scaled Poisson distribution with an unknown intensity. We are interested in estimating the sum of the n unknown intensity vectors, under the assumption that most of them coincide with a given “background” signal. The number s of p-dimensional signals different from the background signal plays the role of sparsity and the goal is to leverage this sparsity assumption in order to improve the quality of estimation as compared to the naive estimator that computes the sum of the observed signals. We first introduce the group hard thresholding estimator and analyze its mean squared error measured by the squared Euclidean norm. We establish a nonasymptotic upper bound showing that the risk is at most of the order of $$\sigma ^2(sp+s^2\sqrt{p}\log ^{3/2}(np))$$ . We then establish lower bounds on the minimax risk over a properly defined class of collections of s-sparse signals. These lower bounds match with the upper bound, up to logarithmic terms, when the dimension p is fixed or of larger order than $$s^2$$ . In the case where the dimension p increases but remains of smaller order than $$s^2$$ , our results show a gap between the lower and the upper bounds, which can be up to order $$\sqrt{p}$$ .

New Conditions on Stable Recovery of Weighted Sparse Signals via Weighted $$l_1$$ l 1 Minimization

A problem of recovering weighted sparse signals via weighted $$l_1$$ minimization has recently drawn considerable attention with application to function interpolation. The weighted robust null space property (NSP) of order s and the weighted restricted isometry property (RIP) with the weighted 3s-RIP constant $$\delta _{\mathbf {w},3s}$$ have been proposed and proved to be sufficient conditions for guaranteeing stable recovery of weighted s-sparse signals. In this paper, we propose two new sufficient conditions, i.e., the weighted $$l_q$$ -robust NSP of order s and the weighted RIP with $$\delta _{\mathbf {w},2s}$$ . Different from the existing results, the weighted $$l_q$$ -robust NSP of order s is more general and weaker than the weighted robust NSP of order s, and the weighted RIP is characterized by $$\delta _{\mathbf {w},2s}$$ instead of $$\delta _{\mathbf {w},3s}$$ . Accordingly, the reconstruction error estimations based on the newly proposed recovery conditions are also derived, respectively. Moreover, we demonstrate that the weighted RIP with small $$\delta _{\mathbf {w},2s}$$ implies the weighted $$l_1$$ -robust NSP of order s.

A new bound on the block restricted isometry constant in compressed sensing

This paper focuses on the sufficient condition of block sparse recovery with the l_{2}/l_{1}-minimization. We show that if the measurement matrix satisfies the block restricted isometry property with delta_{2s|mathcal{I}}< 0.6246, then every block s-sparse signal can be exactly recovered via the l_{2}/l_{1}-minimization approach in the noiseless case and is stably recovered in the noisy measurement case. The result improves the bound on the block restricted isometry constant delta_{2s|mathcal {I}} of Lin and Li (Acta Math. Sin. Engl. Ser. 29(7):1401-1412, 2013).

Alternating projection for sparse recovery

Reconstructing the sparse signal from a few linear measurements has attracted increasing attentions in recent years. In this study, the authors propose the alternating projection (AP) method for sparse signal recovery with learning the sparsity of the original signal. Different with classical hard thresholding algorithms, the AP method regards the signal recovery problem as finding an intersect point of two sets. Theoretically, the authors prove that the proposed algorithm can reconstruct the s-sparse original signal provided the sensing matrix satisfies several assumptions when the noise is absent. They also prove that AP method is a noise-robust algorithm, i.e. a tolerable reconstruction can be obtained by AP if the noise is small. In numerical experiments, the authors compare AP with several existing algorithms when being applied to sparse signals recovery and images reconstruction. The results demonstrate the efficiency of the proposed algorithm.

The Performance of Orthogonal Multi-Matching Pursuit under the Restricted Isometry Property

The orthogonal multi-matching pursuit (OMMP) is a natural extension of the orthogonal matching pursuit (OMP). We denote the OMMP with the parameter M as OMMP(M) where M ≥ 1 is an integer. The main difference between OMP and OMMP(M) is that OMMP(M) selects M atoms per iteration, while OMP only adds one atom to the optimal atom set. In this paper, we study the performance of orthogonal multi-matching pursuit under RIP. In particular, we show that, when the measurement matrix A satisfies (25s, 1/10)-RIP, OMMP(M0) with M0 = 12 can recover s-sparse signals within s iterations. We furthermore prove that OMMP(M) can recover s-sparse signals within O(s/M) iterations for a large class of M . Mathematics subject classification: 94A12, 65H99, 65D15.

On the Theorem of Uniform Recovery of Random Sampling Matrices

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m ≳ s(ln s) 3 lnN. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m × N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, δ s , ensuring sparse recovery via l 1 -minimization. We show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type .

A new approach of conditions on δ 2s (Φ) for s-sparse recovery

In this paper, we provide a unified expression to obtain the conditions on the restricted isometry constant δ 2s (Φ). These conditions cover the important results proposed by Candes et al. and each of them is a sufficient condition for sparse signal recovery. In the noiseless case, when δ 2s (Φ) satisfies any one of these conditions, the s-sparse signal can be exactly recovered via (l 1) constrained minimization.

Block sparse recovery via mixed l 2/l 1 minimization

We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s < 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2/l1-minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali (in IEEE Trans. Inform. Theory 55: 5302–5316, 2009). We also give another sufficient condition on block RIP for such recovery method: δs < 0.307.

Nonuniform support recovery from noisy random measurements by Orthogonal Matching Pursuit

This article considers nonuniform support recovery via Orthogonal Matching Pursuit (OMP) from noisy random measurements. Given m admissible random measurements (of which Subgaussian measurements is a special case) of a fixed s-sparse signal x in Rn corrupted with additive noise, we show that under a condition on the minimum magnitude of the nonzero components of x, OMP can recover the support of x exactly after s iterations with overwhelming probability provided that m=O(slogn). This extends the results of Tropp and Gilbert (2007) [53] to the case with noise. It is a real improvement over previous results in the noisy case, which are based on mutual incoherence property or restricted isometry property analysis and require O(s2logn) random measurements. In addition, this article also considers sparse recovery from noisy random frequency measurements via OMP. Similar results can be obtained for the partial random Fourier matrix via OMP provided that m=O(s(s+log(n−s))). Thus, for some special cases, this answers the open question raised by Kunis and Rauhut (2008) [34], and Tropp and Gilbert (2007) [53].

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

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

Restricted p–isometry property and its application for nonconvex compressive sensing

Compressed sensing is a new scheme which shows the ability to recover sparse signal from fewer measurements, using l 1 minimization. Recently, Chartrand and Staneva showed in Chartrand and Staneva (Inverse Problems 24:1–14, 2009) that the l p minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does the l 1 minimization. They proved that l p minimization with 0 < p < 1 recovers S-sparse signals x ∈ ℝ N from fewer Gaussian random measurements for some smaller p with probability exceeding $$ 1 - 1 \Bigg/ {N\choose S}. $$ The first aim of this paper is to show that above result is right for the case of Gaussian random measurements with probability exceeding 1 − 2e − c(p)M , where M is the numbers of rows of Gaussian random measurements and c(p) is a positive constant that guarantees $1-2e^{-c(p)M}>1 - 1 / {N\choose S}$ for p smaller. The second purpose of the paper is to show that under certain weaker conditions, decoders Δ p are stable in the sense that they are (2,p) instance optimal for a large class of encoder for 0 < p < 1.