Simon Foucart

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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)
Article history: Received 14 December 2007 Revised 9 September 2008 Accepted 11 September 2008 Available online 25 September 2008 Communicated by Naoki Saito 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(More)
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)
We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x ∈ CN from the mere knowledge of linear measurements y = Ax ∈ Cm, 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)
Abstract We investigate the recovery of almost s-sparse vectors x ∈ C from undersampled and inaccurate data y = Ax + e ∈ C 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 ∈ Rm×N are used, this equality-constrained `1-minimization is known to be stable with respect to sparsity defects(More)
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)
This letter demonstrates that sparse recovery can be achieved by an L<sub>1</sub>-minimization ersatz easily implemented using a conventional nonnegative least squares algorithm. A connection with orthogonal matching pursuit is also highlighted. The preliminary results call for more investigations on the potential of the method and on its relations to(More)
Binary measurements arise naturally in a variety of statistics and engineering applications. They may be inherent to the problem&#x2014;for example, in determining the relationship between genetics and the presence or absence of a disease&#x2014;or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior(More)