Deterministic performance analysis of subspace methods for cisoid parameter estimation

Aubel, Céline; Bölcskei, Helmut

doi:10.1109/isit.2016.7541559

Cited by 13 publications

(14 citation statements)

References 31 publications

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“…Regarding the stability analysis of subspace methods, there have been works on bounding the error in terms of the minimum singular value of Vandermonde matrices. Such inequalities can be found in [28,25] for MUSIC and in [15,1] for ESPRIT, as well as in [30] for the matrix pencil method. One major roadblock for this approach is that accurate bounds for the smallest singular value in the ∆ ≤ 1/M regime were not readily available.…”

Section: Related Workmentioning

confidence: 95%

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Super-Resolution Limit of the ESPRIT Algorithm

Liao

Fannjiang

2020

IEEE Trans. Inform. Theory

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The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M + 1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT (Estimation of Signal Parameters via Rotation Invariance Techniques) is an efficient method that does not depend on the sign of the measure and this paper provides a stability analysis: we prove an explicit upper bound on the recovery error of ESPRIT in terms of the minimum singular value of Vandermonde matrices. Using prior estimates for the minimum singular value explains its resolution limit -when the support of µ consists of multiple well-separated clumps, the noise level that ESPRIT can tolerate scales like SRF −(2λ−2) , where the super-resolution factor SRF governs the difficulty of the problem and λ is the cardinality of the largest clump. Our theory is validated by numerical experiments.

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Section: Related Workmentioning

confidence: 95%

“…This difficulty was addressed in [25], which provided the first accurate analysis of MUSIC in the ∆ ≤ 1/M regime. As for ESPRIT, the bounds in [15,1] do not capture the exact dependence of the error on the minimum singular value, and consequently, are inaccurate when ∆ ≤ 1/M (see (3.6) and (3.7) and the discussion there).…”

Section: Related Workmentioning

confidence: 99%

Super-Resolution Limit of the ESPRIT Algorithm

Liao

Fannjiang

2020

IEEE Trans. Inform. Theory

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“…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%

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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“…1 for an illustration. 1 If we use 2M moments, we see directly by (2) that the moments are entire functions of the M nodes and M weights such that the inverse mapping theorem gives local smoothness of the Prony-map around every point where the Jacobian of the mapping from the moments to the parameters has non-vanishing determinant. As the determinant is analytic, we have local smoothness of the Prony-map almost everywhere.…”

Section: Preliminaries and Known Stability Of Subspace Methodsmentioning

confidence: 99%

“…µ(X) = 1, and denote the set of probability-like measures on T by M(T). 2 For some specified minimal weight c min > 0 and minimal separation q > 0, we consider the set of measures…”

Section: Preliminaries and Known Stability Of Subspace Methodsmentioning

confidence: 99%

Sparse super resolution is Lipschitz continuous

Hockmann,

Kunis

2021

Preprint

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Motivated by the application of neural networks in super resolution microscopy, this paper considers super resolution as the mapping of trigonometric moments of a discrete measure on [0, 1) d to its support and weights. We prove that this map satisfies a local Lipschitz property where we give explicit estimates for the Lipschitz constant depending on the dimension d and the sampling effort. Moreover, this local Lipschitz estimate allows to conclude that super resolution with the Wasserstein distance as the metric on the parameter space is even globally Lipschitz continuous. As a byproduct, we improve an estimate for the smallest singular value of multivariate Vandermonde matrices having pairwise clustering nodes.

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Deterministic performance analysis of subspace methods for cisoid parameter estimation

Cited by 13 publications

References 31 publications

Super-Resolution Limit of the ESPRIT Algorithm

Super-Resolution Limit of the ESPRIT Algorithm

Compressive Sampling Using Annihilating Filter-Based Low-Rank Interpolation

Sparse super resolution is Lipschitz continuous

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