On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

Hansen, Anders C.

doi:10.1090/s0894-0347-2010-00676-5

Cited by 127 publications

(132 citation statements)

References 45 publications

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“…The proof of the propositions relies on an idea that originated in a paper by Gross [43], namely, the golfing scheme. The variant we are using here is based on an idea from [3] as well as uneven section techniques from [47,48], see also [42]. However, the informed reader will recognize that the setup here differs substantially from both [43] and [3].…”

Section: )mentioning

confidence: 99%

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Adcock

Hansen

Poon³

et al. 2017

Forum of Mathematics, Sigma

274

461

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This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.2010 Mathematics Subject Classification: 94A20, 94A08 (primary); 42C40, 65R32, 92C55 (secondary)

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Section: )mentioning

confidence: 99%

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Adcock

Hansen

Poon³

et al. 2017

Forum of Mathematics, Sigma

274

461

View full text Add to dashboard Cite

show abstract

“…Step V: We claim that for γ > 0, then 36) when N ∈ N and θ > 0 are chosen according to (6.3), (6.4) and 37) for some universal constant C > 0. Also, if θ = 1 then the left hand side of (11.36) is equal to zero.…”

Section: (116)mentioning

confidence: 99%

Generalized Sampling and Infinite-Dimensional Compressed Sensing

2015

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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are ill-suited for solving a number of important problems.This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.

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“…The use of uneven sections (as we do in this paper) of infinite matrices seems to be the best way to combat these problems. This approach stems from [20] where the technique was used to solve a long standing open problem in computational spectral theory. The reader may consult [17,21] for other examples of uneven section techniques.…”

Section: Remark 46mentioning

confidence: 99%

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

Adcock

Hansen

2012

J Fourier Anal Appl

108

144

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We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the well-known consistent reconstruction technique (Eldar et al). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable.Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.

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On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

Cited by 127 publications

References 45 publications

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

Generalized Sampling and Infinite-Dimensional Compressed Sensing

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

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