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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, Ming-Jun Lai
- 2009

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a r t i c l e i n f o a b s t r a c t We present a condition on the matrix of an… (More)

- Simon Foucart, Ming-Jun Lai
- 2008

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
- 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)

We provide sharp lower and upper bounds for the Gelfand widths of p-balls in the N-dimensional 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.

- Jitse Niesen, Trinity Hall, Sergio Blanes, Chris Budd, Stig Faltinsen, Tony Humphries +19 others
- 2004

Discretization methods for ordinary differential equations are usually not exact; they commit an error at every step of the algorithm. All these errors combine to form the global error, which is the error in the final result. The global error is the subject of this thesis. In the first half of the thesis, accurate a priori estimates of the global error are… (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)

- Simon Foucart, Rick Chartrand, Ron Devore, Anna Gilbert, Justin Romberg, Roman Vershynin
- 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)

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)