Abstract. This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is faster and easier to implement, which makes it an attractive alternative to BP for signal recovery problems. 1.

We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.

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We consider the problem of recovering a matrix from its action on a known vector in the setting where the matrix can be represented efficiently in a known matrix dictionary. Connections with sparse signal recovery allows for the use of efficient reconstruction techniques such as Basis Pursuit (BP). Of particular interest is the dictionary of time-frequency shift matrices and its role for channel estimation and identification in communications engineering. We present recovery results for BP with the time-frequency shift dictionary and various dictionaries of random matrices.

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix n −1 U ∗ U and its expectation where U is a complex random n × D matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned from const · D ln D random samples.

Abstract. The problem of imaging extended targets (sources or scatterers) is formulated in the framework of compressed sensing with emphasis on subwavelength resolution. The proposed formulation of the problems of inverse source/scattering is essentially exact and leads to the random partial Fourier measurement matrix. In the case of square-integrable targets, the proposed sampling scheme in the Littlewood-Paley wavelet basis block-diagonalizes the scattering matrix with each block in the form of random partial Fourier matrix corresponding to each dyadic scale of the target. The resolution issue is analyzed from two perspectives: stability and the signal-to-noise ratio (SNR). The subwavelength modes are shown to be typically unstable. The stability in the subwavelength modes requires additional techniques such as near-field measurement or illumination. The number of the stable modes typically increases as the negative d-th (the dimension of the target) power of the distance between the target and the sensors/source. The resolution limit is shown to be inversely proportional to the SNR in the high SNR limit. Numerical simulations are provided to validate the theoretical predictions. 1.

We consider the estimation of doubly selective wireless channels within pulseshaping multicarrier systems (which include OFDM systems as a special case). A pilot-assisted channel estimation technique using the methodology of compressed sensing (CS) is proposed. By exploiting a channel’s delay-Doppler sparsity, CS-based channel estimation allows an increase in spectral efficiency through a reduction of the number of pilot symbols that have to be transmitted. We also present an extension of our basic channel estimator that employs a sparsity-improving basis expansion. We propose a framework for optimizing the basis and an iterative approximate basis optimization algorithm. Simulation results using three different CS recovery algorithms demonstrate significant performance gains (in terms of improved estimation accuracy or reduction of the number of pilots) relative to conventional least-squares estimation, as well as substantial advantages of using an optimized basis.

Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The prediction of theorems are confirmed by numerical simulations. 1.

Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1-minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.