The Noise Collector for sparse recovery in high dimensions
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Pennsylvania State University（宾夕法尼亚州立大学）；Stanford University（斯坦福大学）；University of California, Merced（加州大学默塞德分校）
Miguel Moscoso, Alexei Novikov, George Papanicolaou, Chrysoula Tsogka
The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. A sparse solution of the linear system $A ρ= b_0$ can be found efficiently with an $l_1$-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) matrix $C$ and solve an augmented system $A ρ+ C η= b_0 + e$, where $ e$ is the noise. We show that the $l_1$-norm minimal solution of the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $ b_0$ increases to infinity. We also obtain exact support recovery if the noise is not too large, and develop a Fast Noise Collector Algorithm which makes the computational cost of solving the augmented system comparable to that of the original one. Finally, we demonstrate the effectiveness of the method in applications to passive array imaging.
Random non-linear Fourier features have recently shown remarkable performance in a wide-range of regression and classification applications. Motivated by this success, this article focuses on a sparse non-linear Fourier feature (NFF) model. We provide a characterization of the sufficient number of data points that guarantee perfect recovery of the unknown parameters with high-probability. In particular, we show how the sufficient number of data points depends on the kernel matrix associated with the probability distribution function of the input data. We compare our results with the recoverabi…
Sharif University of Technology（谢里夫理工大学）；Qom University of Technology（库姆理工大学）；Instituto Superior Técnico（高等理工学院）
This paper investigates the problem of sparse signal recovery in the presence of additive impulsive noise. The heavytailed impulsive noise is well modelled with stable distributions. Since there is no explicit formulation for the probability density function of $SαS$ distribution, alternative approximations like Generalized Gaussian Distribution (GGD) are used which impose $\ell_p$-norm fidelity on the residual error. In this paper, we exploit a Continuous Mixed Norm (CMN) for robust sparse recovery instead of $\ell_p$-norm. We show that in blind conditions, i.e., in case where the parameters …
Iowa State University（爱荷华州立大学）；New York University（纽约大学）；Duke University（杜克大学）
In this paper, we study two important problems in the automated design of neural networks -- Hyper-parameter Optimization (HPO), and Neural Architecture Search (NAS) -- through the lens of sparse recovery methods. In the first part of this paper, we establish a novel connection between HPO and structured sparse recovery. In particular, we show that a special encoding of the hyperparameter space enables a natural group-sparse recovery formulation, which when coupled with HyperBand (a multi-armed bandit strategy), leads to improvement over existing hyperparameter optimization methods. Experiment…
Sharif University of Technology（谢里夫理工大学）；Qom University of Technology（库姆理工大学）
In this paper, we study the missing sample recovery problem using methods based on sparse approximation. In this regard, we investigate the algorithms used for solving the inverse problem associated with the restoration of missed samples of image signal. This problem is also known as inpainting in the context of image processing and for this purpose, we suggest an iterative sparse recovery algorithm based on constrained $l_1$-norm minimization with a new fidelity metric. The proposed metric called Convex SIMilarity (CSIM) index, is a simplified version of the Structural SIMilarity (SSIM) index…
The University of Adelaide（阿德莱德大学）；University of Technology, Sydney（悉尼科技大学）；Brown University（布朗大学）
Matching Pursuit LASSIn Part I \cite{TanPMLPart1}, a Matching Pursuit LASSO ({MPL}) algorithm has been presented for solving large-scale sparse recovery (SR) problems. In this paper, we present a subspace search to further improve the performance of MPL, and then continue to address another major challenge of SR -- batch SR with many signals, a consideration which is absent from most of previous $\ell_1$-norm methods. As a result, a batch-mode {MPL} is developed to vastly speed up sparse recovery of many signals simultaneously. Comprehensive numerical experiments on compressive sensing and fac…
In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based spa…
School of Electrical and Computer Engineering（电气与计算机工程学院）；Georgia Institute of Technology（佐治亚理工学院）；Department of Electrical Engineering（电气工程系）；Stanford University（斯坦福大学）
We apply the OSCAR (octagonal selection and clustering algorithms for regression) in recovering group-sparse matrices (two-dimensional---2D---arrays) from compressive measurements. We propose a 2D version of OSCAR (2OSCAR) consisting of the $\ell_1$ norm and the pair-wise $\ell_{\infty}$ norm, which is convex but non-differentiable. We show that the proximity operator of 2OSCAR can be computed based on that of OSCAR. The 2OSCAR problem can thus be efficiently solved by state-of-the-art proximal splitting algorithms. Experiments on group-sparse 2D array recovery show that 2OSCAR regularization …
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 note we compare two recently proposed semidefinite relaxations for the sparse linear regression problem by Pilanci, Wainwright and El Ghaoui (Sparse learning via boolean relaxations, 2015) and Dong, Chen and Linderoth (Relaxation vs. Regularization A conic optimization perspective of statistical variable selection, 2015). We focus on the cardinality constrained formulation, and prove that the relaxation proposed by Dong, etc. is theoretically no weaker than the one proposed by Pilanci, etc. Therefore any sufficient condition of exact recovery derived by Pilanci can be readily applied t…
In this paper we introduce a new optimization formulation for sparse regression and compressed sensing, called CLOT (Combined L-One and Two), wherein the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norms. This formulation differs from the Elastic Net (EN) formulation, in which the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norm squared. It is shown that, in the context of compressed sensing, the EN formulation does not achieve robust recovery of sparse vectors, whereas the new CLOT formulation achieves robust recovery. Also, like EN but unlike LASSO…
We formulate sparse support recovery as a salient set identification problem and use information-theoretic analyses to characterize the recovery performance and sample complexity. We consider a very general model where we are not restricted to linear models or specific distributions. We state non-asymptotic bounds on recovery probability and a tight mutual information formula for sample complexity. We evaluate our bounds for applications such as sparse linear regression and explicitly characterize effects of correlation or noisy features on recovery performance. We show improvements upon previ…
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant $t\ge {4/3}$, in compressed sensing $δ_{tk}^A < \sqrt{(t-1)/t}$ guarantees the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$…
Rutgers, The State University of New Jersey（罗格斯，新泽西州立大学）
We study the problem of recovering sparse signals from compressed linear measurements. This problem, often referred to as sparse recovery or sparse reconstruction, has generated a great deal of interest in recent years. To recover the sparse signals, we propose a new method called multiple orthogonal least squares (MOLS), which extends the well-known orthogonal least squares (OLS) algorithm by allowing multiple $L$ indices to be chosen per iteration. Owing to inclusion of multiple support indices in each selection, the MOLS algorithm converges in much fewer iterations and improves the computat…
In this paper, we propose a novel sparse recovery method based on the generalized error function. The penalty function introduced involves both the shape and the scale parameters, making it very flexible. The theoretical analysis results in terms of the null space property, the spherical section property and the restricted invertibility factor are established for both constrained and unconstrained models. The practical algorithms via both the iteratively reweighted $\ell_1$ and the difference of convex functions algorithms are presented. Numerical experiments are conducted to illustrate the im…
This paper develops a general theoretical framework to analyze structured sparse recovery problems using the notation of dual certificate. Although certain aspects of the dual certificate idea have already been used in some previous work, due to the lack of a general and coherent theory, the analysis has so far only been carried out in limited scopes for specific problems. In this context the current paper makes two contributions. First, we introduce a general definition of dual certificate, which we then use to develop a unified theory of sparse recovery analysis for convex programming. Secon…
We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of the matrix is solved using a simple least square algorithm. The results are verified in the context of recovering CINE data from undersampled measurements; we obtain good recovery when the sampling conditions are satisfied.
We consider the Orthogonal Least-Squares (OLS) algorithm for the recovery of a $m$-dimensional $k$-sparse signal from a low number of noisy linear measurements. The Exact Recovery Condition (ERC) in bounded noisy scenario is established for OLS under certain condition on nonzero elements of the signal. The new result also improves the existing guarantees for Orthogonal Matching Pursuit (OMP) algorithm. In addition, This framework is employed to provide probabilistic guarantees for the case that the coefficient matrix is drawn at random according to Gaussian or Bernoulli distribution where we e…
Sparse Principal Component Analysis (PCA) is a prevalent tool across a plethora of subfields of applied statistics. While several results have characterized the recovery error of the principal eigenvectors, these are typically in spectral or Frobenius norms. In this paper, we provide entrywise $\ell_{2,\infty}$ bounds for Sparse PCA under a general high-dimensional subgaussian design. In particular, our results hold for any algorithm that selects the correct support with high probability, those that are sparsistent. Our bound improves upon known results by providing a finer characterization of…
A regularized risk minimization procedure for regression function estimation is introduced that achieves near optimal accuracy and confidence under general conditions, including heavy-tailed predictor and response variables. The procedure is based on median-of-means tournaments, introduced by the authors in [8]. It is shown that the new procedure outperforms standard regularized empirical risk minimization procedures such as lasso or slope in heavy-tailed problems.
In this paper, we generalize Huber's criterion to multichannel sparse recovery problem of complex-valued measurements where the objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vectors. This requires careful characterization of robust complex-valued loss functions as well as Huber's criterion function for the multivariate sparse regression problem. We devise a greedy algorithm based on simultaneous normalized iterative hard thresholding (SNIHT) algorithm. …
Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability issue arises with traffic anomaly detection in backbone networks, and subsumes compressed sensing as well as the timely low-rank plus sparse matrix recovery tasks encountered in matrix decomposition problems. Leveraging the ability of $\ell_1$- and nuclear norms to recover sparse and low-rank matrice…
Compressed sensing (sparse signal recovery) often encounters nonnegative data (e.g., images). Recently we developed the methodology of using (dense) Compressed Counting for recovering nonnegative K-sparse signals. In this paper, we adopt very sparse Compressed Counting for nonnegative signal recovery. Our design matrix is sampled from a maximally-skewed p-stable distribution (0<p<1), and we sparsify the design matrix so that on average (1-g)-fraction of the entries become zero. The idea is related to very sparse stable random projections (Li et al 2006 and Li 2007), the prior work for estimati…
We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evalu…
Model selection and sparse recovery are two important problems for which many regularization methods have been proposed. We study the properties of regularization methods in both problems under the unified framework of regularized least squares with concave penalties. For model selection, we establish conditions under which a regularized least squares estimator enjoys a nonasymptotic property, called the weak oracle property, where the dimensionality can grow exponentially with sample size. For sparse recovery, we present a sufficient condition that ensures the recoverability of the sparsest s…
We derive an information-theoretic lower bound for sample complexity in sparse recovery problems where inputs can be chosen sequentially and adaptively. This lower bound is in terms of a simple mutual information expression and unifies many different linear and nonlinear observation models. Using this formula we derive bounds for adaptive compressive sensing (CS), group testing and 1-bit CS problems. We show that adaptivity cannot decrease sample complexity in group testing, 1-bit CS and CS with linear sparsity. In contrast, we show there might be mild performance gains for CS in the sublinear…
We consider the problem of recovering sparse vectors from underdetermined linear measurements via $\ell_p$-constrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that two phenomena occur if $p\neq 2$. First, one may need substantially more than $s \log(en/s)$ measurements (optimal for $p=2$) for uniform recovery of all $s$-sparse vectors. Second, the matrix that achieves recovery with the optimal number of measurements may not be Gaussian (as for $p=2$). We present a new, direct analysis which shows that in fact neither o…
University of Electronic Science and Technology of China（电子科技大学）；China Academy of Engineering Physics（中国工程物理研究院）
In this paper, we consider the block-sparse signals recovery problem in the context of multiple measurement vectors (MMV) with common row sparsity patterns. We develop a new method for recovery of common row sparsity MMV signals, where a pattern-coupled hierarchical Gaussian prior model is introduced to characterize both the block-sparsity of the coefficients and the statistical dependency between neighboring coefficients of the common row sparsity MMV signals. Unlike many other methods, the proposed method is able to automatically capture the block sparse structure of the unknown signal. Our …
In this paper, we investigate jointly sparse signal recovery and jointly sparse support recovery in Multiple Measurement Vector (MMV) models for complex signals, which arise in many applications in communications and signal processing. Recent key applications include channel estimation and device activity detection in MIMO-based grant-free random access which is proposed to support massive machine-type communications (mMTC) for Internet of Things (IoT). Utilizing techniques in compressive sensing, optimization and deep learning, we propose two model-driven approaches, based on the standard aut…
The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been recently proposed to deal with the case of finite-valued sparse signals. In this work, we focus on binary sparse signals and we propose a novel formulation, based on polynomial optimization. This approach is analyzed and compared to the state-of-the-art binary compressed sensing methods.
Spectrum sensing is an important process in cognitive radio. A number of sensing techniques that have been proposed suffer from high processing time, hardware cost and computational complexity. To address these problems, compressive sensing has been proposed to decrease the processing time and expedite the scanning process of the radio spectrum. Selection of a suitable sparse recovery algorithm is necessary to achieve this goal. A number of sparse recovery algorithms have been proposed. This paper surveys the sparse recovery algorithms, classify them into categories, and compares their perform…
Due to its self-regularizing nature and its ability to quantify uncertainty, the Bayesian approach has achieved excellent recovery performance across a wide range of sparse signal recovery applications. However, most existing methods are based on the real-value signal model, with the complex-value signal model rarely considered. Motivated by the adaptive least absolute shrinkage and selection operator (LASSO) and the sparse Bayesian learning (SBL) framework, a hierarchical model with adaptive Laplace priors is proposed in this paper for recovery of complex sparse signals. Moreover, the space a…
Sparse recovery Space-time Adaptive Processing (STAP) can reduce the requirements of clutter samples, and suppress clutter effectively using limited training samples for airborne radar. The whole angle-Doppler plane is discretized into small grid points uniformly in presently available sparse recovery STAP methods, however, the clutter ridge is not located exactly on the pre-discretized grid points in non-sidelooking STAP radar. The off-grid effect degrades the performance of STAP significantly. In this paper, a gridless sparse recovery STAP method is proposed based on reweighted atomic norm m…
Block-sparse signal recovery without knowledge of block sizes and boundaries, such as those encountered in multi-antenna mmWave channel models, is a hard problem for compressed sensing (CS) algorithms. We propose a novel Sparse Bayesian Learning (SBL) method for block-sparse recovery based on popular CS based regularizers with the function input variable related to total variation (TV). Contrary to conventional approaches that impose the regularization on the signal components, we regularize the SBL hyperparameters. This iterative TV-regularized SBL algorithm employs a majorization-minimizatio…
In this paper, we consider the recovery of the high-dimensional block-sparse signal from a compressed set of measurements, where the non-zero coefficients of the recovered signal occur in a small number of blocks. Adopting the idea of deep unfolding, we explore the block-sparse structure and put forward a block-sparse reconstruction network named Ada-BlockLISTA, which performs gradient descent on every single block followed by a block-wise shrinkage. Furthermore, we prove the linear convergence rate of our proposed network, which also theoretically guarantees exact recovery for a potentially h…
Compared with digital methods, sparse recovery based on spiking neural networks has great advantages like high computational efficiency and low power-consumption. However, current spiking algorithms cannot guarantee more accurate estimates since they are usually designed to solve the classical optimization with convex penalties, especially the $\ell_{1}$-norm. In fact, convex penalties are observed to underestimate the true solution in practice, while non-convex ones can avoid the underestimation. Inspired by this, we propose an adaptive version of spiking sparse recovery algorithm to solve th…
In this paper, we present a computationally efficient sparse signal recovery scheme using Deep Neural Networks (DNN). The architecture of the introduced neural network is inspired from sparse Bayesian learning (SBL) and named as Learned-SBL (L-SBL). We design a common architecture to recover sparse as well as block sparse vectors from single measurement vector (SMV) or multiple measurement vectors (MMV) depending on the nature of the training data. In the MMV model, the L-SBL network can be trained to learn any underlying sparsity pattern among the vectors including joint sparsity, block spars…
We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Greedy Pursuit (TGP). We demonstrate that an appropriate choice of thresholding parameter, even without the knowledge of sparsity level of the signal and strength of the noise, can result in exact recovery with no false discoveries as the dimension of the data increases to infinity.
Motivated by re-weighted $\ell_1$ approaches for sparse recovery, we propose a lifted $\ell_1$ (LL1) regularization which is a generalized form of several popular regularizations in the literature. By exploring such connections, we discover there are two types of lifting functions which can guarantee that the proposed approach is equivalent to the $\ell_0$ minimization. Computationally, we design an efficient algorithm via the alternating direction method of multiplier (ADMM) and establish the convergence for an unconstrained formulation. Experimental results are presented to demonstrate how t…
Random stepped frequency (RSF) radar, which transmits random-frequency pulses, can suppress the range ambiguity, improve convert detection, and possess excellent electronic counter-countermeasures (ECCM) ability [1]. In this paper, we apply a sparse recovery method to estimate the range and Doppler of targets. We also propose a cognitive mechanism for RSF radar to further enhance the performance of the sparse recovery method. The carrier frequencies of transmitted pulses are adaptively designed in response to the observed circumstance. We investigate the criterion to design carrier frequencies…
Classical results in sparse recovery guarantee the exact reconstruction of $s$-sparse signals under assumptions on the dictionary that are either too strong or NP-hard to check. Moreover, such results may be pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We derive necessary and sufficient conditions for sparse recovery, which depend on properties of the cycles of the graph that can be checked in polynomial time. We also derive s…
This paper proposes a federated learning technique for deep algorithm unfolding with applications to sparse signal recovery and compressed sensing. We refer to this architecture as Fed-CS. Specifically, we unfold and learn the iterative shrinkage thresholding algorithm for sparse signal recovery without transporting to a central location, the training data distributed across many clients. We propose a layer-wise federated learning technique, in which each client uses local data to train a common model. Then we transmit only the model parameters of that layer from all the clients to the server,…
Newton-Direction-Based ReLU-Thresholding Methods for Nonnegative Sparse Signal Recovery
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；Shenzhen International Center for Industrial and Applied Mathematics, SRIBD, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院深圳市工业与应用数学中心）；Department of Mathematical Sciences, University of Nottingham Ningbo China（宁波诺丁汉大学数学系）
We present a compressive sensing based defect detection by multiple input multiple output (MIMO) wireless radar. Here, defects are inside a layered material structure, therefore, due to reflections from the surface of the layered material structure the defect detection is challenging. By utilizing a low-rank nature of the reflections of the layered material structure and sparse nature of the defects, we propose a method based on rank minimization and sparse recovery. To improve the accuracy in the recovery of low-rank and sparse components, we propose a non-convex approach based on the iterati…
Intensively growing approach in signal processing and acquisition, the Compressive Sensing approach, allows sparse signals to be recovered from small number of randomly acquired signal coefficients. This paper analyses some of the commonly used threshold-based algorithms for sparse signal reconstruction. Signals satisfy the conditions required by the Compressive Sensing theory. The Orthogonal Matching Pursuit, Iterative Hard Thresholding and Single Iteration Reconstruction algorithms are observed. Comparison in terms of reconstruction error and execution time is performed within the experiment…
Sparse support recovery arises in many applications in communications and signal processing. Existing methods tackle sparse support recovery problems for a given measurement matrix, and cannot flexibly exploit the properties of sparsity patterns for improving performance. In this letter, we propose a data-driven approach to jointly design the measurement matrix and support recovery method for complex sparse signals, using auto-encoder in deep learning. The proposed architecture includes two components, an auto-encoder and a hard thresholding module. The proposed auto-encoder successfully handl…
A new family of operators, coined hierarchical measurement operators, is introduced and discussed within the well-known hierarchical sparse recovery framework. Such operator is a composition of block and mixing operations and notably contains the Kronecker product as a special case. Results on their hierarchical restricted isometry property (HiRIP) are derived, generalizing prior work on recovery of hierarchically sparse signals from Kronecker-structured linear measurements. Specifically, these results show that, very surprisingly, sparsity properties of the block and mixing part can be traded…
The goal of phaseless compressed sensing is to recover an unknown sparse or approximately sparse signal from the magnitude of its measurements. However, it does not take advantage of any support information of the original signal. Therefore, our main contribution in this paper is to extend the theoretical framework for phaseless compressed sensing to incorporate with prior knowledge of the support structure of the signal. Specifically, we investigate two conditions that guarantee stable recovery of a weighted $k$-sparse signal via weighted l1 minimization without any phase information. We firs…
