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A Mathematical Introduction to Compressive Sensing
A Mathematical Introduction to Compressive Sensing uses a mathematical perspective to present the core of the theory underlying compressive sensing and provides a detailed account of the core theory upon which the field is build. Expand
Hard Thresholding Pursuit: An Algorithm for Compressive Sensing
  • S. Foucart
  • Mathematics, Computer Science
  • SIAM J. Numer. Anal.
  • 1 November 2011
We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems using the Iterative Hard Thresholding algorithm and the Compressive Sampling Matching Pursuit algorithm. Expand
Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0
Abstract We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal l q -quasinorm is also the sparsest one. ThisExpand
A note on guaranteed sparse recovery via ℓ1-minimization
Abstract It is proved that every s-sparse vector x ∈ C N can be recovered from the measurement vector y = A x ∈ C m via l 1 -minimization as soon as the 2s-th restricted isometry constant of theExpand
Sparse Recovery Algorithms: Sufficient Conditions in Terms of RestrictedIsometry Constants
We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x∈ℂ N from the mere knowledge of linear measurements y=A x∈ℂ m , m<N. For each of theExpand
Stability and Robustness of Weak Orthogonal Matching Pursuits
A recent result establishing, under restricted isometry conditions, the success of sparse recovery via orthogonal matching pursuit using a number of iterations proportional to the sparsity level isExpand
Stability and robustness of ℓ1-minimizations with Weibull matrices and redundant dictionaries
Abstract We investigate the recovery of almost s-sparse vectors x ∈ C N from undersampled and inaccurate data y = A x + e ∈ C m by means of minimizing ‖ z ‖ 1 subject to the equality constraints A zExpand
Hard thresholding pursuit algorithms: Number of iterations ☆
Abstract The Hard Thresholding Pursuit algorithm for sparse recovery is revisited using a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered fromExpand
Exponential Decay of Reconstruction Error From Binary Measurements of Sparse Signals
We show that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by <inline-formula> <tex-math notation="LaTeX">$\Omega (\lambda ^{-1})$ . Expand
Sparse Recovery by Means of Nonnegative Least Squares
This letter demonstrates that sparse recovery can be achieved by an L1-minimization ersatz easily implemented using a conventional nonnegative least squares algorithm. Expand