Stability and super-resolution of generalized spike recovery

Batenkov, Dmitry

doi:10.1016/j.acha.2016.09.004

Cited by 34 publications

(51 citation statements)

References 54 publications

(134 reference statements)

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“…Accordingly, our robustness guarantees on the low-rank matrix entries can be translated in terms of the actual signal that is recovered (for example, on the support or amplitudes of the spike in the case of recovery of spike superpositions). In fact, this has been also an active area of researches [29]- [33], [38], and we again exploit these findings for our second step of signal recovery. For example, see Moitra [32] for more details on the error bound for the case of modified matrix pencil approach for recovery of Diracs.…”

Section: F Recovery Of Continuous Domain Fri Signals After Interpolamentioning

confidence: 80%

“…The common findings are that the estimation error for the location parameter {t j } p−1 j=0 and the magnitude a j,l are bounded by the condition number of the confluent Vandermonde matrix as well as the minimum separation distance ∆. Moreover, matrix pencil approaches such as Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) method [39] is shown to stably recovery the locations [33], [38].…”

Section: F Recovery Of Continuous Domain Fri Signals After Interpolamentioning

confidence: 99%

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Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

Kim

Jin

et al. 2017

IEEE Trans. Inform. Theory

115

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Abstract-While the recent theory of compressed sensing provides an opportunity to overcome the Nyquist limit in recovering sparse signals, a solution approach usually takes the form of an inverse problem of an unknown signal, which is crucially dependent on specific signal representation. In this paper, we propose a drastically different two-step Fourier compressive sampling framework in a continuous domain that can be implemented via measurement domain interpolation, after which signal reconstruction can be done using classical analytic reconstruction methods. The main idea originates from the fundamental duality between the sparsity in the primary space and the low-rankness of a structured matrix in the spectral domain, showing that a low-rank interpolator in the spectral domain can enjoy all of the benefits of sparse recovery with performance guarantees. Most notably, the proposed low-rank interpolation approach can be regarded as a generalization of recent spectral compressed sensing to recover large classes of finite rate of innovations (FRI) signals at a near-optimal sampling rate. Moreover, for the case of cardinal representation, we can show that the proposed low-rank interpolation scheme will benefit from inherent regularization and an optimal incoherence parameter. Using a powerful dual certificate and the golfing scheme, we show that the new framework still achieves a near-optimal sampling rate for a general class of FRI signal recovery, while the sampling rate can be further reduced for a class of cardinal splines. Numerical results using various types of FRI signals confirm that the proposed low-rank interpolation approach offers significantly better phase transitions than conventional compressive sampling approaches.Index Terms-Compressed sensing, signals of finite rate of innovations, spectral compressed sensing, low rank matrix completion, dual certificates, golfing scheme.

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Section: F Recovery Of Continuous Domain Fri Signals After Interpolamentioning

confidence: 80%

Section: F Recovery Of Continuous Domain Fri Signals After Interpolamentioning

confidence: 99%

Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

Kim

Jin

et al. 2017

IEEE Trans. Inform. Theory

115

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show abstract

“…The smallest known choice for A that works unconditionally without any further assumptions on Ω has cardinality #Ω log #Ω s−1 which still grows quite fast for large s. Moreover, it is known to be beneficial to oversample, i.e., to choose A larger than needed, cf. [19]. Thus, Algorithm 1 addresses the two main issues here: how to handle large columns in a still efficient way and how to ensure that the rank is controlled well.…”

Section: Prony's Problem In Several Variablesmentioning

confidence: 99%

SVD update methods for large matrices and applications

Peña

Sauer

2019

Linear Algebra and its Applications

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We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix A ∈ R m×n with m n. Supposing that an SVD of A is already known, and given a matrix B ∈ R m×n , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [AB] ∈ R m×(n+n ) . This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted.

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“…The oldest we are aware of is the paper [8] of Prony, where the problem is considered without noise. We note also that there is some effort [31,3,2] in the direction of overcoming this barrier in the case of univariate trigonometric setting, where the information is in the form K k=1 R r=0 a k,r (−ij) r exp(−ijt k ), |j| < N, (1.5) so that each t k appears with multiplicity R + 1.…”

mentioning

confidence: 99%

Super-Resolution Meets Machine Learning: Approximation of Measures

Mhaskar

2019

J Fourier Anal Appl

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The problem of super-resolution in general terms is to recuperate a finitely supported measure µ given finitely many of its coefficientsμ(k) with respect to some orthonormal system. The interesting case concerns situations, where the number of coefficients required is substantially smaller than a power of the reciprocal of the minimal separation among the points in the support of µ.In this paper, we consider the more severe problem of recuperating µ approximately without any assumption on µ beyond having a finite total variation. In particular, µ may be supported on a continuum, so that the minimal separation among the points in the support of µ is 0. A variant of this problem is also of interest in machine learning as well as the inverse problem of de-convolution.We define an appropriate notion of a distance between the target measure and its recuperated version, give an explicit expression for the recuperation operator, and estimate the distance between µ and its approximation. We show that these estimates are the best possible in many different ways.We also explain why for a finitely supported measure the approximation quality of its recuperation is bounded from below if the amount of information is smaller than what is demanded in the super-resolution problem. This paper is motivated by two apparently disjoint areas; super-resolution and machine learning. A problem of interest in both of these areas is the approximation of a measure using a finite amount of information on the measure. Thus we wish to develop a theory of (weak-star) approximation of measures. We will describe our motivation and the connections of this work to the problem of super-resolution and the problem of machine learning in Sections 1.1 and 1.2 respectively. The aims and contributions of this paper, and its outline is given in Section 1.3. This section and the next being introductory in nature, the notation used in these two sections may not be the same as the one used in the remainder of the paper. Super-resolutionThe problem of super-resolution is stated by Donoho [12] as follows. Given observations of the form k∈Z a k exp(−iωk∆) + z(ω), |ω| ≤ Ω, (1.1)where {a k } is sequence of complex numbers, ∆, Ω > 0, and z represents a perturbation subject to the condition that Ω −Ω |z(ω)| 2 dω ≤ ǫ, recuperate the sequence {a k } up to an accuracy O(ǫ) in the sense of optimal recovery, when Ω∆ < π. It is shown in [12] that this is not possible in general, but it is possible under some sparsity assumptions.Relevant to the current paper is a generalization stated by Candés and Fernandez-Granda in [6, Theorem 1.2]. We denote the quotient space R/(2πZ) by T. Let µ(t) = K k=1 a k δ(t − t k ) + Z(t),(1.2)where δ denotes the Dirac delta measure at 0. We assume that the moments K k=1 a k exp(−ijt k ) + T Z(t) exp(−ijt)dt,

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Stability and super-resolution of generalized spike recovery

Cited by 34 publications

References 54 publications

Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

SVD update methods for large matrices and applications

Super-Resolution Meets Machine Learning: Approximation of Measures

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