Recovery of Exact Sparse Representations in the Presence of Bounded Noise

Fuchs, Jean–Jacques

doi:10.1109/tit.2005.855614

Cited by 258 publications

(251 citation statements)

References 19 publications

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“…Our measurement rate-MSE tradeoff (Proposition 3) is related to the results in [13]- [15], [29], [32], [33]. However, we observe the following significant differences.…”

Section: B Relation To Previous Worksupporting

confidence: 55%

“…• As opposed to the stability results in [13]- [15], [29], [32], [33], our approach guarantees reliable recovery of (parts of) the support set, which is a highly desirable property.…”

Section: B Relation To Previous Workmentioning

confidence: 99%

“…Conditions for stable recovery for the SP, CoSaMP, and IHT algorithms were derived in [13], [14], and [15], respectively (see also [31]). In [32], Donoho et al derived sufficient conditions for stable recovery and partial support recovery of sparse signals using a combinatorial formulation (based on the L 0 "norm") as well as for convex relaxations (see also [33]). As we will shortly see, all these works implicitly showed the existence of a tradeoff between the triple (k, m, n) and the MSE, and characterized it.…”

Section: Introductionmentioning

confidence: 99%

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A Measurement Rate-MSE Tradeoff for Compressive Sensing Through Partial Support Recovery

Blasco-Serrano

Zachariah

Sundman

et al. 2014

IEEE Trans. Signal Process.

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We study the fundamental relationship between two relevant quantities in compressive sensing: the measurement rate, which characterizes the asymptotic behavior of the dimensions of the measurement matrix in terms of the ratio m/ log n (m being the number of measurements and n the dimension of the sparse signal), and the mean square estimation error. First, we use an information-theoretic approach to derive sufficient conditions on the measurement rate to reliably recover a part of the support set that represents a certain fraction of the total signal power when the sparsity level is fixed. Second, we characterize the mean square error of an estimator that uses partial support set information. Using these two parts, we derive a tradeoff between the measurement rate and the mean square error. This tradeoff is achievable using a two-step approach: first support set recovery, then estimation of the active components. Finally, for both deterministic and random signals, we perform a numerical evaluation to verify the advantages of the methods based on partial support set recovery.

QC 20140617

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“…Our measurement rate-MSE tradeoff (Proposition 3) is related to the results in [13]- [15], [29], [32], [33]. However, we observe the following significant differences.…”

Section: B Relation To Previous Worksupporting

confidence: 55%

“…• As opposed to the stability results in [13]- [15], [29], [32], [33], our approach guarantees reliable recovery of (parts of) the support set, which is a highly desirable property.…”

Section: B Relation To Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Measurement Rate-MSE Tradeoff for Compressive Sensing Through Partial Support Recovery

Blasco-Serrano

Zachariah

Sundman

et al. 2014

IEEE Trans. Signal Process.

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QC 20140617

show abstract

“…The usual way to tackle this problem is performing a best least-square (LS) fitting of the input data. However, this criterion has many local minima for real valued spectral peaks [35][36][37], hence in principle requiring a combinatorial exploration for integer-valued mode numbers n l , and an aposteriori thresholding scheme to differentiate the "correct" from the "wrong" solutions, which is very much CPU-time consuming and cannot possibly be adapted for real-time applications on the submillisecond time scale required for the analysis of the JET measurements. The only possibility to solve this problem is to provide an estimate for the amplitudes of all possible modes numbers in the range {−K, …, K} (where |K| is much larger than the maximum mode number that can be conceivably present in the input spectrum), at the same time enforcing that most of these modes actually have a null amplitude, i.e.…”

Section: Acknowledgementsmentioning

confidence: 99%

The JET Alfvén Eigenmode Local Manager for the real-time detection and tracking of a frequency-degenerate spectrum of MHD instabilities

Testa

Carfantan

Fasoli

et al. 2011

Fusion Engineering and Design

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We present the real-time VME system used to detect and track MHD instabilities, and particularly Alfvén Eigenmodes, on the JET tokamak [J.Wesson, Tokamaks, 3 rd edition, (Oxford Science Publication, Oxford, 2003), p.617]. This system runs on a 1kHz clock cycle, and allows performing a real-time, unsupervised and blind detection, decomposition and tracking of the individual components in a frequency-degenerate, multi-harmonic spectrum, using a small number of input data which are unevenly sampled in the spatial domain. This makes it possible to follow in real-time the detected modes as the plasma background evolves, and measure in real-time their frequency, damping rate, toroidal mode-number and relative amplitude. The successful implementation of this system opens a clear path towards developing real-time control tools for electro-magnetic instabilities in future fusion devices aimed at achieving a net energy gain, such as ITER [J.Wesson, Tokamaks,3 1) Introduction.The problem of blind and unsupervised real-time detection of the different components in a multiharmonics spectrum using a small number of input data which are un-evenly sampled in the spatial domain is now becoming one of the main aspects required for machine protection and the control of plasma discharges in thermonuclear fusion experiments, and a wealth of literature is available on this subject (for some examples see Chapter3 and Chapter7 and references therein in Ref. [1] and Chapter2 and references therein in Ref. [2]). The method routinely used for this analysis involves sampling of a (relatively) small set of magnetic and non-magnetic signals, often containing some spatial periodicities so as to enhance or eliminate detection of certain components when the input signals are processed appropriately. A real-time algorithm then runs, and a global alarm is generated which may trigger a feedback control mechanism under certain specified conditions. The main drawback of this method is that it can only detect modes when they have become unstable, i.e. when they may have already had some, possibly detrimental, effect on the plasma background parameters.Conversely, an innovative method has been employed for quite some time now on the JET tokamak [3], which combines active excitation of magnetic field perturbations with a very small amplitude at the plasma edge (maximum intensity max(|B DRIVEN |)~0.1G, i.e. 10 5 times smaller than the typical value of the toroidal magnetic field in JET, B TOR~( 1-3)T) with synchronous real-time detection of the driven perturbations. This is the so-called Alfvén Eigenmodes (AEs) Active Diagnostic (AEAD) system [4,5], of which the real-time Alfvén Eigenmodes Local Manager (AELM) constitutes one essential, moreover worldwide unique component. This diagnostic system allows the real-time detection and tracking of the driven modes, usually magneto-hydrodynamic (MHD) Eigenmodes supported by the plasma, when still stable, i.e. when the modes have a positive damping >0, and have not yet caused any effect on the plas...

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“…More recent results concerning recovery conditions in the presence of noise when (Opt 2 ) is used have been published in [6,8,7,9] …”

Section: Previous Resultsmentioning

confidence: 99%

Recovery Conditions of Sparse Representations in the Presence of Noise.

Fuchs

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

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When seeking a representation of a signal on a redundant basis one generally replaces the quest for the sparsest model by an 1 minimization and solves thus a linear program. In the presence of noise one has in addition to replace the exact reconstruction constraint by an approximate one. We consider simultaneously several ways to allow for reconstruction errors and analyze precisely under which conditions exact recovery is possible in the absence of noise. These are then also the conditions that allow recovery in presence of noise in case of large signal to noise ratio. We illustrate the results on an example that shows that the chances of recovery do indeed depend upon the criterion.

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Recovery of Exact Sparse Representations in the Presence of Bounded Noise

Abstract: The purpose of this contribution is to extend some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. The type of questions addressed so far is : given a (n,m)-matrix with

Cited by 258 publications

References 19 publications

A Measurement Rate-MSE Tradeoff for Compressive Sensing Through Partial Support Recovery

A Measurement Rate-MSE Tradeoff for Compressive Sensing Through Partial Support Recovery

The JET Alfvén Eigenmode Local Manager for the real-time detection and tracking of a frequency-degenerate spectrum of MHD instabilities

Recovery Conditions of Sparse Representations in the Presence of Noise.

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