Simon Foucart

Hard Thresholding Pursuit: An Algorithm for Compressive Sensing

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 the Compressive Sampling Matching Pursuit algorithm, is called Hard Thresholding Pursuit. We study its general convergence and notice in particular that only a finite number of iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the linear system, the Hard Thresholding Pursuit algorithm indeed finds all

Sparse Recovery Algorithms: Sufficient Conditions in Terms of RestrictedIsometry Constants

s

The Gelfand widths of l p -balls for 0<p≤1

-sparse solutions. This condition, which reads

Sparse Recovery by Means of Nonnegative Least Squares

\delta_{3 s} < 1/\sqrt{3}

Quikr: a Method for Rapid Reconstruction of Bacterial Communities via Compressive Sensing

, is heuristically better than the sufficient conditions currently available for other compressive sensing algorithms. It applies to fast versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed under the condition

Stability and Robustness of Weak Orthogonal Matching Pursuits

\delta_{3 s} < 1/\sqrt{3}

WGSQuikr: Fast Whole-Genome Shotgun Metagenomic Classification

. We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.

An Invitation to Compressive Sensing

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 the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These conditions are δ2s <0.4652 for basis pursuit, δ3s <0.5 and δ2s <0.25 for iterative hard thresholding, and δ4s <0.3843 for compressive sampling matching pursuit. The arguments also applies to almost sparse vectors and corrupted measurements. The analysis of iterative hard thresholding is surprisingly simple. The analysis of basis pursuit features a new inequality that encompasses several inequalities encountered in Compressive Sensing.

Restricted Isometry Property

We provide sharp lower and upper bounds for the Gelfand widths of

Random Sampling in Bounded Orthonormal Systems

\ell_p