Compressive sensing predicts that sufficiently sparse vectors can be recovered from highly incomplete information. Efficient recovery methods such as $\ell_1$-minimization find the sparsest solution to certain systems of equations. Random matrices have become a popular choice for the measurement matrix. Indeed, near-optimal uniform recovery results have been shown for such matrices. In this note we focus on nonuniform recovery using Gaussian random matrices and $\ell_1$-minimization. We provide a condition on the number of samples in terms of the sparsity and the signal length which guarantees that a fixed sparse signal can be recovered with a random draw of the matrix using $\ell_1$-minimization. The constant 2 in the condition is optimal, and the proof is rather short compared to a similar result due to Donoho and Tanner.
We investigate non-negative least squares (NNLS) for the recovery of sparse non-negative vectors from noisy linear and biased measurements. We build upon recent results from [1] showing that for matrices whose row-span intersects the positive orthant, the nullspace property (NSP) implies compressed sensing recovery guarantees for NNLS. Such results are as good as for $\ell_1$-regularized estimators but do not require tuning parameters that depend on the noise level. A bias in the sensing matrix improves this auto-regularization feature of NNLS and the NSP then determines the sparse recovery pe…
Data samples from $\mathbb{R}^{d}$ with a common support of size $k$ are accessed through $m$ random linear projections (measurements) per sample. It is well-known that roughly $k$ measurements from a single sample are sufficient to recover the support. In the multiple sample setting, do $k$ overall measurements still suffice when only $m$ measurements per sample are allowed, with $m<k$? We answer this question in the negative by considering a generative model setting with independent samples drawn from a subgaussian prior. We show that $n=Θ((k^2/m^2)\cdot\log k(d-k))$ samples are necessary an…
We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using a-priori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a significantly reduced number of vector-valued (sub-)Gaussian measurements via mixed l^1/l^2-minimization. We prove this by establishing an appropriate version of the restricted isometry property. Our result complements pre…
Generative neural networks (GNNs) have gained renown for efficaciously capturing intrinsic low-dimensional structure in natural images. Here, we investigate the subgaussian demixing problem for two Lipschitz signals, with GNN demixing as a special case. In demixing, one seeks identification of two signals given their sum and prior structural information. Here, we assume each signal lies in the range of a Lipschitz function, which includes many popular GNNs as a special case. We prove a sample complexity bound for nearly optimal recovery error that extends a recent result of Bora, et al. (2017)…
We analyze the asymptotic performance of sparse signal recovery from noisy measurements. In particular, we generalize some of the existing results for the Gaussian case to subgaussian and other ensembles. An achievable result is presented for the linear sparsity regime. A converse on the number of required measurements in the sub-linear regime is also presented, which cover many of the widely used measurement ensembles. Our converse idea makes use of a correspondence between compressed sensing ideas and compound channels in information theory.
We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter. Numerical…
In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted $\ell_1$ minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted $\ell_1$ minimization recovery algorithm and analyze its performance using a Grassmann angle approach. We compute explicitly the relationship between the system parameters-the weights, the number of measurements, the size of the two sets, the probabiliti…
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 …
We study the recovery of sparse vectors from subsampled random convolutions via $\ell_1$-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 $s$ is small enough, i.e., $s \lesssim \sqrt{n/\log(n)}$, we show that $m \gtrsim s \log(en/s)$ measurements are sufficient to recover $s$-sparse vectors in dimension $n$ with high probability, matching the well-known condition for recovery from standard Gaussian measure…
This paper provides novel results for the recovery of signals from undersampled measurements based on analysis $\ell_1$-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 mea…
This chapter gives an overview over recovery guarantees for total variation minimization in compressed sensing for different measurement scenarios. In addition to summarizing the results in the area, we illustrate why an approach that is common for synthesis sparse signals fails and different techniques are necessary. Lastly, we discuss a generalizations of recent results for Gaussian measurements to the subgaussian case.
We proposed a weighted l1 minimization to recover a sparse signal vector and the corrupted noise vector from a linear measurement when the sensing matrix A is an m by n row i.i.d subgaussian matrix. We obtain both uniform and nonuniform recovery guarantees when the corrupted observations occupy a constant fraction of the total measurement, provided that the signal vector is sparse enough. In the uniform recovery guarantee, the upper-bound of the cardinality of the signal vector required in this paper is asymptotically optimal. While in the non-uniform recovery guarantee, we allow the proportio…
The stability of sparse signal reconstruction is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique $\ell_1$ sparse recovery. One of our algorithm produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared with performance…
We analyze a multiple-input multiple-output (MIMO) radar model and provide recovery results for a compressed sensing (CS) approach. In MIMO radar different pulses are emitted by several transmitters and the echoes are recorded at several receiver nodes. Under reasonable assumptions the transformation from emitted pulses to the received echoes can approximately be regarded as linear. For the considered model, and many radar tasks in general, sparsity of targets within the considered angle-range-Doppler domain is a natural assumption. Therefore, it is possible to apply methods from CS in order t…
Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. The current paper discusses the recently introduced Restricted Eigenvalue (RE) condition, which is among the most general assumptions on the matrix, guaranteeing recovery. We prove a reduction principle showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces. This principle allows us to establish the RE condition for several broad classes of random matrices with dependent entrie…
This paper presents a novel power spectral density estimation technique for band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The technique employs multi-coset sampling and incorporates the advantages of compressed sensing (CS) when the power spectrum is sparse, but applies to sparse and nonsparse power spectra alike. The estimates are consistent piecewise constant approximations whose resolutions (width of the piecewise constant segments) are controlled by the periodicity of the multi-coset sampling. We show that compressive estimates exhibit better tradeoffs among th…
We consider a structured estimation problem where an observed matrix is assumed to be generated as an $s$-sparse linear combination of $N$ given $n\times n$ positive-semidefinite matrices. Recovering the unknown $N$-dimensional and $s$-sparse weights from noisy observations is an important problem in various fields of signal processing and also a relevant pre-processing step in covariance estimation. We will present related recovery guarantees and focus on the case of nonnegative weights. The problem is formulated as a convex program and can be solved without further tuning. Such robust, non-B…
Group-sparsity is a common low-complexity signal model with widespread application across various domains of science and engineering. The recovery of such signal ensembles from compressive measurements has been extensively studied in the literature under the assumption that measurement operators are modeled as densely populated random matrices. In this paper, we turn our attention to an acquisition model intended to ease the energy consumption of sensing devices by splitting the measurements up into distinct signal blocks. More precisely, we present uniform guarantees for group-sparse signal r…
In this paper we consider the problem of recovering a high dimensional data matrix from a set of incomplete and noisy linear measurements. We introduce a new model that can efficiently restrict the degrees of freedom of the problem and is generic enough to find a lot of applications, for instance in multichannel signal compressed sensing (e.g. sensor networks, hyperspectral imaging) and compressive sparse principal component analysis (s-PCA). We assume data matrices have a simultaneous low-rank and joint sparse structure, and we propose a novel approach for efficient compressed sensing (CS) of…
Sparse signals can be recovered from a reduced set of samples by using compressive sensing algorithms. In common methods the signal is recovered in the sparse domain. A method for the reconstruction of sparse signal which reconstructs the remaining missing samples/measurements is recently proposed. The available samples are fixed, while the missing samples are considered as minimization variables. Recovery of missing samples/measurements is done using an adaptive gradient-based algorithm in the time domain. A new criterion for the parameter adaptation in this algorithm, based on the gradient d…
We address the problem of recovering a sparse signal observed by a resource constrained wireless sensor network under channel fading. Sparse random matrices are exploited to reduce the communication cost in forwarding information to a fusion center. The presence of channel fading leads to inhomogeneity and non Gaussian statistics in the effective measurement matrix that relates the measurements collected at the fusion center and the sparse signal being observed. We analyze the impact of channel fading on nonuniform recovery of a given sparse signal by leveraging the properties of heavy-tailed …
Recently, the 1-bit compressive sensing (1-bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. In this paper, we first show that a necessary assumption (that has been overlooked in the literature) should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This …
Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the "lower tail" of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the one-dimensional marginals of the random vectors. A similar result holds for more general sums of random positive semidefinite matrices, and the (relatively simple) proof uses a variant of the so-called PAC-Bayesian method for bounding empirical processes. We give two applications of the main result. In the first one we obtain a new finite-sample bound for o…
