The restricted isometry property and its implications for compressed sensing

2008 ◽  
Vol 346 (9-10) ◽  
pp. 589-592 ◽  
Author(s):  
Emmanuel J. Candès
2019 ◽  
Vol 9 (1) ◽  
pp. 157-193 ◽  
Author(s):  
Marius Junge ◽  
Kiryung Lee

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.


2013 ◽  
Vol 718-720 ◽  
pp. 669-674 ◽  
Author(s):  
Rui Wu ◽  
Wei Huang

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.


2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yao Wang ◽  
Jianjun Wang

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.


2019 ◽  
Vol 9 (3) ◽  
pp. 601-626 ◽  
Author(s):  
Sjoerd Dirksen ◽  
Hans Christian Jung ◽  
Holger Rauhut

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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