Certifying the Restricted Isometry Property is Hard
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It is demonstrated that testing whether a matrix satisfies RIP is NP-hard, which means it is impossible to efficiently test for RIP provided P ≠ NP.Abstract:
This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P ≠ NP.read more
Citations
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Best subset selection via a modern optimization lens
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Minimization of $\ell_{1-2}$ for Compressed Sensing
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The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
TL;DR: It is confirmed by showing that for a given matrix A and positive integer k, computing the best constants for which the RIP or NSP hold is, in general, NP-hard.
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The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
TL;DR: In this article, it was shown that for a given matrix A and positive integer k, computing the best constants for which the restricted isometry property (RIP) or nullspace property (NSP) hold is NP-hard.
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Optimal detection of sparse principal components in high dimension
TL;DR: In this paper, a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix is performed, based on a sparse eigenvalue statistic.
References
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Decoding by linear programming
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: F can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program) and numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted.
Posted Content
Decoding by Linear Programming
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: In this paper, it was shown that under suitable conditions on the coding matrix, the input vector can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program).
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CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
Deanna Needell,Joel A. Tropp +1 more
TL;DR: A new iterative recovery algorithm called CoSaMP is described that delivers the same guarantees as the best optimization-based approaches and offers rigorous bounds on computational cost and storage.
Journal ArticleDOI
The restricted isometry property and its implications for compressed sensing
TL;DR: Candes et al. as discussed by the authors established new results about the accuracy of the reconstruction from undersampled measurements, which improved on earlier estimates, and have the advantage of being more elegant. But they did not consider the restricted isometry property of the sensing matrix.
Journal ArticleDOI
CoSaMP: iterative signal recovery from incomplete and inaccurate samples
Deanna Needell,Joel A. Tropp +1 more
TL;DR: This extended abstract describes a recent algorithm, called, CoSaMP, that accomplishes the data recovery task and was the first known method to offer near-optimal guarantees on resource usage.