Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

Bernstein, Brett; Fernandez‐Granda, Carlos

doi:10.1002/cpa.21805

Cited by 20 publications

(32 citation statements)

References 68 publications

(137 reference statements)

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“…As a result, different sparse combinations of features may yield essentially the same data. For a detailed analysis of this issue in the context of super-resolution and deconvolution of point sources we refer the reader to Section 3.2 in [13] (see also [59] and [78]) and Section 2.1 in [3], respectively.…”

Section: Beyond Sparsity and Randomness: Separation And Correlation Dmentioning

confidence: 99%

“…where ξ > 0 is a parameter that must be tuned according to the noise level. Previous works have established robustness guarantees for TV-norm minimization applied to specific SNL problems such as super-resolution [14,34] and deconvolution [3] at small noise levels. These proofs are based on dual certificates.…”

Section: Robustness To Noisementioning

confidence: 99%

“…These proofs are based on dual certificates. Combining the arguments in [3,34] with our dual-certificate construction in Section 3 yields robustness guarantees for general SNL problems in terms of support recovery at high signal-to-noise regimes. We omit the details, since the proof would essentially mimic the ones in [3,34].…”

Section: Robustness To Noisementioning

confidence: 99%

“…Of course, the true parameters may not lie on the grid used to solve the 1 -norm minimization problem. The proof techniques used to derive robustness guarantees for super-resolution and deconvolution in [3,34] can be leveraged to provide some control over the discretization error. Performing a more accurate analysis of discretization error for SNL problems is an interesting direction for future research.…”

Section: )mentioning

confidence: 99%

“…Subsequent works build upon these results to study the robustness of this methodology to robustness [14,32,34,83], missing data [81], and outliers [36]. A similar analysis is carried out in [3] for deconvolution. The authors establish a sampling theorem for Gaussian and Ricker-wavelet convolution kernels, which characterizes what sampling patterns yield exact recovery under a minimum-separation condition.…”

Section: Convex Programming Applied To Specific Snl Problemsmentioning

confidence: 99%

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Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Bernstein

Liu

Papadaniil

et al. 2020

IEEE Trans. Inform. Theory

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Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. This approach has had empirical success in a variety of domains, from geophysics to medical imaging, but lacks a theoretical justification. In particular, compressed-sensing theory does not apply, because the measurement operators are deterministic and violate incoherence conditions such as the restricted-isometry property. Our main contribution is a theory for sparse recovery adapted to deterministic settings. We show that convex programming succeeds in recovering the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurement operator. The theoretical results are illustrated through numerical experiments for two applications: heat-source localization and estimation of brain activity from electroencephalography data.

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Section: Beyond Sparsity and Randomness: Separation And Correlation Dmentioning

confidence: 99%

Section: Robustness To Noisementioning

confidence: 99%

Section: Robustness To Noisementioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Convex Programming Applied To Specific Snl Problemsmentioning

confidence: 99%

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Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Bernstein

Liu

Papadaniil

et al. 2020

IEEE Trans. Inform. Theory

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Sparse non-negative super-resolution — simplified and stabilised

Eftekhari¹,

Tanner

Thompson

et al. 2021

Applied and Computational Harmonic Analysis

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The convolution of a discrete measure, x " ř k i"1 aiδt i , with a local window function, φps´tq, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources tai, tiu k i"1 with an accuracy beyond the essential support of φps´tq, typically from m samples ypsjq " ř k i"1 aiφpsj´tiq`δj , where δj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. δj " 0, m ě 2k`1 samples are available, and φps´tq generates a Chebyshev system. This is independent of how close the sample locations are and does not rely on any regulariser beyond non-negativity; as such, it extends and clarifies the work by Schiebinger et al. in [1] and De Castro et al. in [2], who achieve the same results but require a total variation regulariser, which we show is unnecessary.Moreover, we characterise non-negative solutionsx consistent with the samples within the bound ř m j"1 δ 2 j ď δ 2 . Any such non-negative measure is within Opδ 1{7 q of the discrete measure x generating the samples in the generalised Wasserstein distance. Similarly, we show using somewhat different techniques that the integrals ofx and x over pti´ǫ, ti`ǫq are similarly close, converging to one another as ǫ and δ approach zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of φps´tq being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution and that, while regularisers such as total variation might be particularly effective, they are not required in the non-negative setting.

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A mathematical theory of the computational resolution limit in one dimension

Liu

Zhang

2022

Applied and Computational Harmonic Analysis

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Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

Cited by 20 publications

References 68 publications

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Sparse non-negative super-resolution — simplified and stabilised

A mathematical theory of the computational resolution limit in one dimension

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