Jeffrey D. Blanchard

Compressed Sensing: How Sharp Is the Restricted Isometry Property?

Compressed sensing (CS) seeks to recover an unknown vector with

GPU accelerated greedy algorithms for compressed sensing

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Fast k -selection algorithms for graphics processing units

entries by making far fewer than

Performance comparisons of greedy algorithms in compressed sensing

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Conjugate Gradient Iterative Hard Thresholding: Observed Noise Stability for Compressed Sensing

measurements; it posits that the number of CS measurements should be comparable to the information content of the vector, not simply

Matricial filters and crystallographic composite dilation wavelets

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Recovery Guarantees for Rank Aware Pursuits

. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004 there have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS—exact reconstruction from seemingly undersampled measurements—it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry.

Crystallographic Haar-Type Composite Dilation Wavelets

For appropriate matrix ensembles, greedy algorithms have proven to be an efficient means of solving the combinatorial optimization problem associated with compressed sensing. This paper describes an implementation for graphics processing units (GPU) of hard thresholding, iterative hard thresholding, normalized iterative hard thresholding, hard thresholding pursuit, and a two-stage thresholding algorithm based on compressive sampling matching pursuit and subspace pursuit. The GPU acceleration of the former bottleneck, namely the matrix–vector multiplications, transfers a significant portion of the computational burden to the identification of the support set. The software solves high-dimensional problems in fractions of a second which permits large-scale testing at dimensions currently unavailable in the literature. The GPU implementations exhibit up to 70

Large Scale Iterative Hard Thresholding on a Graphical Processing Unit

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