The restricted isometry property and its implications for compressed sensing

2008 ◽  
Vol 346 (9-10) ◽  
pp. 589-592 ◽  
Author(s):  
Emmanuel J. Candès
Keyword(s):  
Compressed Sensing ◽  
Restricted Isometry Property

scholarly journals Generalized notions of sparsity and restricted isometry property. Part I: a unified framework

2019 ◽  
Vol 9 (1) ◽  
pp. 157-193 ◽  
Author(s):  
Marius Junge ◽  
Kiryung Lee
Keyword(s):  
Banach Space ◽  
Inverse Problems ◽  
Compressed Sensing ◽  
Dimensionality Reduction ◽  
Group Actions ◽  
Function Spaces ◽  
Restricted Isometry Property ◽  
Low Rank ◽  
Unified Framework ◽  
Order Tensor

Abstract The restricted isometry property (RIP) is an integral tool in the analysis of various inverse problems with sparsity models. Motivated by the applications of compressed sensing and dimensionality reduction of low-rank tensors, we propose generalized notions of sparsity and provide a unified framework for the corresponding RIP, in particular when combined with isotropic group actions. Our results extend an approach by Rudelson and Vershynin to a much broader context including commutative and non-commutative function spaces. Moreover, our Banach space notion of sparsity applies to affine group actions. The generalized approach in particular applies to high-order tensor products.

Hard Thresholding Pursuit with Partially Known Support for Compressed Sensing

2013 ◽  
Vol 718-720 ◽  
pp. 669-674 ◽  
Author(s):  
Rui Wu ◽  
Wei Huang
Keyword(s):  
Theoretical Analysis ◽  
Compressed Sensing ◽  
Prior Information ◽  
Recovery Process ◽  
Restricted Isometry Property ◽  
Hard Thresholding ◽  
Simulation Experiments ◽  
Recovery Performance ◽  
Support Information

Compressed sensing has attracted lots of interest in recent years. Recent works in modified compressed sensing exploited the prior information about the signal to reduce the number of measurements. In this paper, we propose a hard thresholding pursuit algorithm with partially known support (HTP-PKS), which incorporates the prior support information into the recovery process. Theoretical analysis shows that by using prior information of partially known support, the HTP-PKS algorithm presents stable and robust recovery performance under a relaxed restricted isometry property (RIP) condition. To illustrate, simulation experiments are given.

scholarly journals Improved RIP Conditions for Compressed Sensing with Coherent Tight Frames

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yao Wang ◽  
Jianjun Wang
Keyword(s):  
Compressed Sensing ◽  
Sufficient Conditions ◽  
Tight Frame ◽  
Restricted Isometry Property ◽  
Tight Frames ◽  
Analysis Method

This paper establishes new sufficient conditions on the restricted isometry property (RIP) for compressed sensing with coherent tight frames. One of our main results shows that the RIP (adapted to D) condition δk+θk,k<1 guarantees the stable recovery of all signals that are nearly k-sparse in terms of a coherent tight frame D via the l1-analysis method, which improves the existing ones in the literature.

scholarly journals One-bit compressed sensing with partial Gaussian circulant matrices

2019 ◽  
Vol 9 (3) ◽  
pp. 601-626 ◽  
Author(s):  
Sjoerd Dirksen ◽  
Hans Christian Jung ◽  
Holger Rauhut
Keyword(s):  
Compressed Sensing ◽  
Restricted Isometry Property ◽  
Circulant Matrices ◽  
Vector Norm ◽  
Sparse Vector ◽  
Efficient Program ◽  
Full Vector

Abstract In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\simeq \delta ^{-4} s\log (N/s\delta )$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell _1/\ell _2$ restricted isometry property property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results, i.e., estimation of both vector norm and direction.

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