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- Simon Foucart, Holger Rauhut
- Applied and Numerical Harmonic Analysis
- 2013

We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal q-quasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the 1-norm. We then introduce a simple numerical scheme to compute solutions with minimal q-quasinorm, and we study its… (More)

- Simon Foucart
- SIAM J. Numerical Analysis
- 2011

We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and of the Compressive Sampling Matching Pursuit or Subspace Pursuit algorithms, is called Hard Thresholding Pursuit. We study its general convergence, and notice in particular… (More)

- Simon Foucart
- 2010

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 = Ax ∈ C m , m < N. For each of the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These… (More)

- Simon Foucart
- 2009

Foreword I have started to put these notes together in December 2008. They are intended for a graduate course on Compressed Sensing in the Department of Mathematics at Vanderbilt University. I will update them during the Spring semester of 2009 to produce the draft of a coherent manuscript by May 2009. I expect to continue improving it afterwards. You… (More)

- Simon Foucart, Alain Pajor, Holger Rauhut, Tino Ullrich
- J. Complexity
- 2010

We provide sharp lower and upper bounds for the Gelfand widths of ℓ p-balls in the N-dimensional ℓ N q-space for 0 < p ≤ 1 and p < q ≤ 2. Such estimates are highly relevant to the novel theory of compressive sensing, and our proofs rely on methods from this area.

Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem—e.g., in determining the relationship between genetics and the presence or absence of a disease—or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can… (More)

- Simon Foucart
- 2012

We investigate the recovery of almost s-sparse vectors x ∈ C N from undersampled and inaccurate data y = Ax + e ∈ C m by means of minimizing z 1 subject to the equality constraints Az = y. If m s ln(N/s) and if Gaussian random matrices A ∈ R m×N are used, this equality-constrained 1-minimization is known to be stable with respect to sparsity defects and… (More)

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 from compressive linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a certain restricted isometry… (More)

We identify and solve an overlooked problem about the characterization of underdeter-mined systems of linear equations for which sparse solutions have minimal 1-norm. This characterization is known as the null space property. When the system has real coefficients , sparse solutions can be considered either as real or complex vectors, leading to two… (More)