This paper presents a simple yet very effective timedomain sparse representation and the associated sparse recovery techniques that can robustly capture raw ultra-wideband (UWB) synthetic aperture radar (SAR) records. Unlike previous approaches in compressed sensing for SAR, we take advantage of the sparsity and the correlation directly in the raw received pulses before even attempting image fo...

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, ...

We introduce a new signal model, called (K,C)-sparse, to capture K-sparse signals in N dimensions whose nonzero coefficients are contained within at most C clusters, with C < K ≪ N . In contrast to the existing work in the sparse approximation and compressive sensing literature on block sparsity, no prior knowledge of the locations and sizes of the clusters is assumed. We prove that O (K + C lo...

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which we call Bregman ISS and Linearized Bregman ISS. We show that under proper conditions, there exists a bias-free and sign-consistent point on their solution paths, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the...

Definition 1 (Restricted isometry and orthogonality). The S-restricted isometry constant δS of a matrix F ∈ Rn×m is the smallest quantity such that (1− δS)‖x‖2 ≤ ‖FTx‖2 ≤ (1 + δS)‖x‖2 for all T ⊆ [m] with |T | ≤ S and all x ∈ R|T |. The (S, S′)-restricted orthogonality constant θS,S′ of F is the smallest quantity such that |FTx · FT ′x′| ≤ θS,S′‖c‖‖c‖ for all disjoint T, T ′ ⊆ [m] with |T | ≤ S...

A novel theory of sparse recovery is presented in order to bridge the standard compressive sensing framework and the one-bit compressive sensing framework. In the former setting, sparse vectors observed via few linear measurements can be reconstructed exactly. In the latter setting, the linear measurements are only available through their signs, so exact reconstruction of sparse vectors is repl...

In traditional compressed sensing theory, the dictionary matrix is given a priori, whereas in real applications this matrix suffers from random noise and fluctuations. In this paper we consider a signal model where each column in the dictionary matrix is affected by a structured noise. This formulation is common in problems such as radar signal processing and direction-of-arrival (DOA) estimati...

We discuss a general notion of “sparsity structure” and associated recoveries of a sparse signal from its linear image of reduced dimension possibly corrupted with noise. Our approach allows for unified treatment of (a) the “usual sparsity” and “usual l1 recovery,” (b) block-sparsity with possibly overlapping blocks and associated block-l1 recovery, and (c) low-rank-oriented recovery by nuclear...

Linear sketching and recovery of sparse vectors with randomly constructed sparse matrices has numerous applications in several areas, including compressive sensing, data stream computing, graph sketching, and combinatorial group testing. This paper considers the same problem with the added twist that the sparse coefficients of the unknown vector exhibit further correlations as determined by a k...

Compressive sensing (sparse recovery) is a new area in mathematical image and signal processing that predicts that sparse signals can be recovered from what was previously believed to be highly incomplete measurement [3, 5, 7, 12]. Recently, the ideas of this field have been extended to the recovery of low rank matrices from undersampled information [6, 8]; most notably to the matrix completion...