Certifying the Restricted Isometry Property is Hard

Bandeira, Afonso S.; Dobriban, Edgar; Mixon, Dustin G.; Sawin, Will

doi:10.1109/tit.2013.2248414

Cited by 186 publications

(156 citation statements)

References 19 publications

Supporting

Mentioning

153

Contrasting

Unclassified

Order By: Relevance

“…Similarly, the expected number of j ∈ Q that do not satisfy (4) is at most γ|Q|/4, so by Markov's inequality, with probability at least 3/4 it holds that the number of j ∈ Q that do not satisfy (4) is at most γ|Q|. It follows that there exists a vector g ∈ G i for which (4) holds for all but at most γ fraction of j ∈ [N] and for all but at most γ fraction of j ∈ Q, as required. (1) , .…”

Section: Proof: Observe That For Everymentioning

confidence: 98%

“…This notion, due to Candès and Tao [11], was intensively studied during the last decade and found various applications and connections to several areas of theoretical computer science, including sparse recovery [8,20,27], coding theory [14], norm embeddings [6,23], and computational complexity [4,31,25]. The original motivation for the restricted isometry property comes from the area of compressed sensing.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Restricted Isometry Property of Subsampled Fourier Matrices

Haviv¹,

Regev²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

Section: Proof: Observe That For Everymentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

The Restricted Isometry Property of Subsampled Fourier Matrices

Haviv¹,

Regev²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…With the exception of random matrices, constructing fast matrices satisfying the RIP is known to be non-trivial -although recent attempts towards addressing this are presented in Nelson et al (2014). Also, verifying the RIP for deterministic matrices is NP-hard, as shown in Bandeira et al (2012). So the idea is to devise an embedding operator that reduces the dimensionality of the measurements while preserving the NSP of the original measurement operator, thus maintaining the same compressed sensing-based guarantees on recovering the image.…”

Section: Preliminary Studies and Testsmentioning

confidence: 99%

A Fourier dimensionality reduction model for big data interferometric imaging

Kartik

Carrillo

Thiran

et al. 2017

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

Data dimensionality reduction in radio interferometry can provide savings of computational resources for image reconstruction through reduced memory footprints and lighter computations per iteration, which is important for the scalability of imaging methods to the big data setting of the next-generation telescopes. This article sheds new light on dimensionality reduction from the perspective of the compressed sensing theory and studies its interplay with imaging algorithms designed in the context of convex optimization. We propose a post-gridding linear data embedding to the space spanned by the left singular vectors of the measurement operator, providing a dimensionality reduction below image size. This embedding preserves the null space of the measurement operator and hence its sampling properties are also preserved in light of the compressed sensing theory. We show that this can be approximated by first computing the dirty image and then applying a weighted subsampled discrete Fourier transform to obtain the final reduced data vector. This Fourier dimensionality reduction model ensures a fast implementation of the full measurement operator, essential for any iterative image reconstruction method. The proposed reduction also preserves the i.i.d. Gaussian properties of the original measurement noise. For convex optimization-based imaging algorithms, this is key to justify the use of the standard 2 -norm as the data fidelity term. Our simulations confirm that this dimensionality reduction approach can be leveraged by convex optimization algorithms with no loss in imaging quality relative to reconstructing the image from the complete visibility data set. Reconstruction results in simulation settings with no direction dependent effects or calibration errors show promising performance of the proposed dimensionality reduction. Further tests on real data are planned as an extension of the current work. matlab code implementing the proposed reduction method is available on GitHub.

show abstract

“…However, determining whether an arbitrary matrix satisfies the RIP condition has recently been proved to be NP-hard [2]. Nonetheless, it is known that sensing matrices can be formed by a sampling process such that they obey the RIP with overwhelming probability under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of reweightedl1greedy algorithm for noisy reconstruction

Zhu

Arroyo

et al. 2015

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tSparse solutions for an underdetermined system of linear equations Φx = u can be found more accurately by l 1 -minimization type algorithms, such as the reweighted l 1 -minimization and l 1 greedy algorithms, than with analytical methods, in particular in the presence of noisy data. Recently, a generalized l 1 greedy algorithm was introduced and applied to signal and image recovery. Numerical experiments have demonstrated the convergence of the new algorithm and the superiority of the algorithm over the reweighted l 1 -minimization and l 1 greedy algorithms although the convergence has not yet been proven theoretically. In this paper, we provide an error bound for the reweighted l 1 greedy algorithm, a type of the generalized l 1 greedy algorithm, in the noisy case and show its improvement over the reweighted l 1 -minimization.

show abstract

Certifying the Restricted Isometry Property is Hard

Cited by 186 publications

References 19 publications

The Restricted Isometry Property of Subsampled Fourier Matrices

The Restricted Isometry Property of Subsampled Fourier Matrices

A Fourier dimensionality reduction model for big data interferometric imaging

Error analysis of reweightedl1greedy algorithm for noisy reconstruction

Contact Info

Product

Resources

About