Applied Numerical Mathematics
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
Model-based compressive sensing
IEEE Transactions on Information Theory
SIAM Journal on Scientific Computing
Analysis and Generalizations of the Linearized Bregman Method
SIAM Journal on Imaging Sciences
Uncertainty principles and ideal atomic decomposition
IEEE Transactions on Information Theory
On sparse representations in arbitrary redundant bases
IEEE Transactions on Information Theory
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
IEEE Transactions on Information Theory
Fast Solution of -Norm Minimization Problems When the Solution May Be Sparse
IEEE Transactions on Information Theory
A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration
IEEE Transactions on Image Processing
IEEE Transactions on Information Theory
An Efficient Algorithm for l0 Minimization in Wavelet Frame Based Image Restoration
Journal of Scientific Computing
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We present a novel sparse signal reconstruction method, iterative support detection (ISD), aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical $\ell_1$ minimization approach. ISD addresses failed reconstructions of $\ell_1$ minimization due to insufficient measurements. It estimates a support set $I$ from a current reconstruction and obtains a new reconstruction by solving the minimization problem $\min\{\sum_{i\notin I}|x_i|:Ax=b\}$, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit method, as well as its variants, because (i) the index set $I$ in ISD is not necessarily nested or increasing, and (ii) the minimization problem above updates all the components of $x$ at the same time. We generalize the null space property to the truncated null space property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold-ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold-ISD has significant advantages over the classical $\ell_1$ minimization approach, as well as two state-of-the-art algorithms: the iterative reweighted $\ell_1$ minimization algorithm (IRL1) and the iterative reweighted least-squares algorithm (IRLS). MATLAB code is available for download from http://www.caam.rice.edu/ optimization/L1/ISD/.