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Exact Support Recovery for Sparse Spikes Deconvolution
, 2013
"... This paper studies sparse spikes deconvolution over the space of measures. For nondegenerate sums of Diracs, we show that, when the signaltonoise ratio is large enough, total variation regularization (which the natural extension of ℓ 1 norm of vector to the setting of measures) recovers the exact ..."
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This paper studies sparse spikes deconvolution over the space of measures. For nondegenerate sums of Diracs, we show that, when the signaltonoise ratio is large enough, total variation regularization (which the natural extension of ℓ 1 norm of vector to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. Finally we draw connections between the performances of sparse recovery on a continuous domain and on a discretized grid.
The recoverability limit for superresolution via sparsity. arXiv:1502.01385,
, 2014
"... Abstract We consider the problem of robustly recovering a ksparse coefficient vector from the Fourier series that it generates, restricted to the interval [−Ω, Ω]. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of Ω) b ..."
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Abstract We consider the problem of robustly recovering a ksparse coefficient vector from the Fourier series that it generates, restricted to the interval [−Ω, Ω]. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of Ω) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm σ, we show upper and lower bounds on the minimax error rate that both scale like (SRF ) 2k−1 σ, providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity 2k, or equivalently from comparing 2k to the socalled σspark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. Acknowledgments.
Sparse Spikes Deconvolution on Thin Grids
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
"... We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve ..."
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We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed. Key words and phrases: Dirac pulses, spike train, finite rate of innovation, superresolution, sparse recovery, structured low rank approximation, alternating projections, Cadzow denoising
SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization
"... Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory w ..."
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Abstract—The mathematical theory of superresolution developed recently by Candès and FernandesGranda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of regularly spaced timespace samples. This theory was then extended to the cases with partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as offgrid/continuous compressed sensing. However, a major problem of existing atomic norm methods is that the frequencies can be recovered only if they are sufficiently separated, prohibiting commonly known high resolution. In this paper, a novel nonconvex optimization method is proposed which guarantees exact recovery under no resolution limit and hence achieves high resolution. A locally convergent iterative algorithm is implemented to solve the nonconvex problem. The algorithm iteratively carries out ANM with a sound reweighting strategy which enhances sparsity and resolution, and is termed as reweighted atomicnorm minimization (RAM). Extensive numerical simulations are carried out to demonstrate the performance of the proposed method with application to direction of arrival (DOA) estimation. Index Terms—Continuous compressed sensing (CCS), DOA estimation, frequency estimation, gridless sparse method, high resolution, reweighted atomic norm minimization (RAM). I.