On the approximation of spectra of linear operators on Hilbert spaces

Hansen, Anders C.

doi:10.1016/j.jfa.2008.01.006

Cited by 111 publications

(71 citation statements)

References 27 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

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273

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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

Self Cite

273

461

View full text Add to dashboard Cite

show abstract

“…the finite section) of U . Finite sections are extremely widely used in practice [11,12,36]. However, for general operators U there is no guarantee thatα…”

Section: Finite Sections: a Warning From Spectral Theorymentioning

confidence: 99%

“…Finite sections have been studied extensively from the viewpoint of computational spectral theory. Therein one typically wishes to gain information about the spectrum of U by considering discretizations of the form P N U P N [9,36]. The main conclusion is that, unless U satisfies some very stringent restrictions (such as positive self-adjointness), its finite sections P N U P N may have wildly different (spectral) properties.…”

Section: Finite Sections: a Warning From Spectral Theorymentioning

confidence: 99%

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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 n-pseudospectrum was first introduced in Hansen (2008) as a tool for the general spectral problem. It was further investigated in Hansen (submitted a), and in this section, we will discuss some of the nice properties of these sets.…”

Section: Properties Of the N-pseudospectra Of Bounded Operatorsmentioning

confidence: 99%