On the Theorem of Uniform Recovery of Random Sampling Matrices

Andersson, Johan; Strömberg, Jan-Olov

doi:10.1109/tit.2014.2300092

Cited by 57 publications

(57 citation statements)

References 19 publications

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“…To prove that a sufficiently small RIP in levels constant implies an equation of the form (3.2), it is natural to adapt the steps used in [5] to prove Theorem 1.3. This adaptation yields a sufficient condition for recovery even in the noisy case.…”

Section: Resultsmentioning

confidence: 99%

On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels

Bastounis¹,

Hansen²

2017

SIAM J. Imaging Sci.

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The purpose of this paper is twofold. The first is to point out that the Restricted Isometry Property (RIP) does not hold in many applications where compressed sensing is successfully used. This includes fields like Magnetic Resonance Imaging (MRI), Computerized Tomography, Electron Microscopy, Radio Interferometry and Fluorescence Microscopy. We demonstrate that for natural compressed sensing matrices involving a level based reconstruction basis (e.g. wavelets), the number of measurements required to recover all s-sparse signals for reasonable s is excessive. In particular, uniform recovery of all s-sparse signals is quite unrealistic. This realisation shows that the RIP is insufficient for explaining the success of compressed sensing in various practical applications. The second purpose of the paper is to introduce a new framework based on a generalised RIP-like definition that fits the applications where compressed sensing is used. We show that the shortcomings that show that uniform recovery is unreasonable no longer apply if we instead ask for structured recovery that is uniform only within each of the levels. To examine this phenomenon, a new tool, termed the 'Restricted Isometry Property in Levels' is described and analysed. Furthermore, we show that with certain conditions on the Restricted Isometry Property in Levels, a form of uniform recovery within each level is possible. Finally, we conclude the paper by providing examples that demonstrate the optimality of the results obtained.We will thoroughly document the lack of RIP of order s in this paper, and explain why it does not hold for reasonable s. It is then natural to ask whether there might be an alternative to the RIP that may be more suitable for the actual real world CS applications. With this in mind, we shall introduce the RIP in levels which generalises the classical RIP and is much better suited for the actual applications where CS is used.

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Section: Resultsmentioning

confidence: 99%

On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels

Bastounis¹,

Hansen²

2017

SIAM J. Imaging Sci.

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“…2 We show that the tail result based on H P (s) provides a tighter upper bound on the largest eigenvalue of a matrix i.d. series 2 In general, the tail inequality P{ξ > t} describes the probability characteristics of the event in which the value of a random variable ξ is greater than a given positive constant t. Consequently, the tail inequality provides more useful information in the case of Rt ρ(σ 2 +V ) > 0.8831 than in the case of Rt ρ(σ 2 +V ) ≤ 0.8831. than is possible with the Bernstein-type result when the matrix dimension is high. The results regarding Q(s) and H P (s) are applicable for any Bennett-type concentration inequality that involves the function Q(s).…”

Section: A Overview Of the Main Resultsmentioning

confidence: 90%

Matrix Infinitely Divisible Series: Tail Inequalities and Their Applications

Zhang

Gao

Hsieh

et al. 2020

IEEE Trans. Inform. Theory

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In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for Q(s) = (s + 1) log(s + 1) − s that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function Q(s). The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work [1] on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints.

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“…if ρ = 1 and α > 1/2). Note also that the classical condition on the RIP constant has been improved several times [9,1,7,8,22,23,33]; although a version of Theorem 1 and the main results of this paper can likely be extended to the theory of these works, we do not pursue such refinements here.…”

Section: Existing Results For ℓ 1 -Minimization and Weighted ℓ 1 -Minmentioning

confidence: 99%

Weighted ${\ell}_{{1}}$-minimization for sparse recovery under arbitrary prior information

Needell¹,

Saab²,

Woolf

2017

Information and Inference

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Weighted ℓ 1 -minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted ℓ 1 -minimization when arbitrarily many distinct weights are permitted. For example, such a setup might be used when one has multiple estimates for the support of a signal, and these estimates have varying degrees of accuracy. Our analysis yields an extension to existing works that assume only a single support estimate set upon which a constant weight is applied. We include numerical experiments, with both synthetic signals and real video data, that demonstrate the benefits of allowing non-uniform weights in the reconstruction procedure.

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On the Theorem of Uniform Recovery of Random Sampling Matrices

Cited by 57 publications

References 19 publications

On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels

On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels

Matrix Infinitely Divisible Series: Tail Inequalities and Their Applications

Weighted ${\ell}_{{1}}$-minimization for sparse recovery under arbitrary prior information

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