Non-uniform spline recovery from small degree polynomial approximation

Castro, Yohann de; Mijoule, Guillaume

doi:10.1016/j.jmaa.2015.05.034

Cited by 10 publications

(9 citation statements)

References 20 publications

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“…A consecutive paper [12] showed that the recovery is robust to noisy measurements. Similar results are given for support detection from low Fourier coefficients [4], [23], recovery of non-uniform splines from their projection onto spaces of algebraic polynomials [8], [18] and recovery of streams of pulses [6], [9]. (see also [17]).…”

Section: Introductionsupporting

confidence: 62%

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Super-Resolution on the Sphere Using Convex Optimization

Bendory

Dekel²,

Feuer

2015

IEEE Trans. Signal Process.

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This paper considers the problem of recovering an ensemble of Diracs on a sphere from its low resolution measurements. The Diracs can be located at any location on the sphere, not necessarily on a grid. We show that under a separation condition, one can recover the ensemble with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting. The algorithm's computation time depends solely on the number of measurements, and not on the required solution accuracy. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for recovery. Furthermore, in the discrete setting, we estimate the recovery error in the presence of noise as a function of the noise level and the super-resolution factor.

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Section: Introductionsupporting

confidence: 62%

“…Up to now, these method were applied to projections onto trigonometric [10], [12], [13], [15], [52], [53], [55] and algebraic polynomial spaces [8], [18]. This work showed that it can be applied to the sphere as well.…”

Section: Discussionmentioning

confidence: 98%

Super-Resolution on the Sphere Using Convex Optimization

Bendory

Dekel²,

Feuer

2015

IEEE Trans. Signal Process.

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“…For h : C n × R → R defined by h = persp( · 2 /2), one can write the primal in the form P λ (µ, σ) = τ z (h)(F n (µ), nσ)+σ/2+f (µ, σ), where z = (y, 0) ∈ C n × R and f (µ, σ) = λ µ TV + I R++ . Then, we can apply [4,Proposition,19.18], with g : C n × R → R by g(·, σ) = τ z (h)(·, nσ) + σ/2 and L = F n . This leads to the Lagrangian formulation L(µ, σ, c, t) := f (µ, σ) + F n (µ), c + σt − g * (c, t) .…”

Section: E2 Proof Of Propositionmentioning

confidence: 99%

Adapting to unknown noise level in sparse deconvolution

Boyer

Castro

Salmon

2017

Information and Inference

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In this paper, we study sparse spike deconvolution over the space of complex-valued measures when the input measure is a finite sum of Dirac masses. We introduce a modified version of the Beurling Lasso (BLasso), a semi-definite program that we refer to as the Concomitant Beurling Lasso (CBLasso). This new procedure estimates the target measure and the unknown noise level simultaneously. Contrary to previous estimators in the literature, theory holds for a tuning parameter that depends only on the sample size, so that it can be used for unknown noise level problems. Consistent noise level estimation is standardly proved. As for Radon measure estimation, theoretical guarantees match the previous state-of-the-art results in Super-Resolution regarding minimax prediction and localization. The proofs are based on a bound on the noise level given by a new tail estimate of the supremum of a stationary non-Gaussian process through the Rice method.More precisely, pioneering works were proposed in [12] treating inverse problems on the space of Borel measures and in [13], where the Super-Resolution problem was investigated via Semi-Definite Programming and a groundbreaking construction of a "dual certificate". Exact recovery (in the noiseless case), minimax prediction and localization (in the noisy case) have been performed using the Beurling Lasso (BLasso) estimator [2,38,25,37] which minimizes the total variation norm over complex-valued Borel measures. Noise robustness (as the noise level tends to zero) has been thoroughly investigated in [22]; the reader may also consult [23,20,24] for more details. Change point detection and grid-less spline decomposition are studied in [7,19]. Several interesting extensions, such as deconvolution over spheres, have been also recently provided in [6,9,8]. For more general settings, we refer to the work of [30]. Concomitant Beurling Lasso: adapting to the noiseOur proposed estimator is an adaptation to the Super-Resolution framework of a methodology first developed for sparse high dimensional regression. In the latter case, the joint estimation of the parameter and of the noise level has first been considered in [33,1], though without any theory. It was based on concomitant estimation ideas that could be traced back to the work of Huber [29]. The formulation we consider in this work appeared first in [33, 1] with a statistical point of view, as well as in [40] with a game theory flavor. Note that interestingly, both approaches rely on the notion of robustness. An equivalent definition of this estimator was proposed and extensively studied independently in [5] under the name Square-root Lasso. The formulation we investigate is also closer to the one analyzed in [36] under the name Scaled-Lasso. Yet, we adopt the terminology of "Concomitant Beurling Lasso" in reference to the seminal paper [33]. Last but not least, our contribution borrows some ideas from the stimulating lecture notes [39].Remark that an alternative formulation was investigated in [35] with a particular aim at Gaussian m...

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“…Particularly, it was suggested to recast Total-Variation (TV) and atomic norm minimization as semi-definite programs (SDP) in order to recover point sources from low-resolution data on the line [13,12], and on the sphere [7,8,9], or for line spectral estimation [10,47,46]. Similar approach was applied to the recovery of non-uniform splines from their projection onto algebraic polynomial spaces [6,20] (see also [18,1]).…”

Section: Introductionmentioning

confidence: 99%

Robust recovery of stream of pulses using convex optimization

Bendory

Dekel

Feuer

2016

Journal of Mathematical Analysis and Applications

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This paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses. This problem is motivated by a variety of applications, such as ultrasound and radar. We show that for univariate and bivariate stream of pulses, one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem, provided that a pulse-dependent separation condition is satisfied. The main result of this paper states that the recovery is robust to additive noise or model mismatch.

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Non-uniform spline recovery from small degree polynomial approximation

Cited by 10 publications

References 20 publications

Super-Resolution on the Sphere Using Convex Optimization

Super-Resolution on the Sphere Using Convex Optimization

Adapting to unknown noise level in sparse deconvolution

Robust recovery of stream of pulses using convex optimization

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