Prony’s Method Under an Almost Sharp Multivariate Ingham Inequality

Kunis, Stefan; Möller, H. Michael; Peter, Thomas; Ohe, Ulrich von der

doi:10.1007/s00041-017-9571-5

Cited by 17 publications

(23 citation statements)

References 40 publications

(52 reference statements)

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“…These matrices generalize the classical discrete Fourier matrices to non-equispaced nodes and the involved polynomial degree is also called bandwidth. The condition number of those matrices has recently become important in the context of stability analysis of super-resolution algorithms like Prony's method [6,15], the matrix pencil method [12,18], the ESPRIT algorithm [20,21], and the MUSIC algorithm [17,22]. If the nodes of such a Vandermonde matrix are all well-separated, with minimal separation distance greater than the inverse bandwidth, bounds on the condition number are established for example in [2,5,14,18].…”

Section: Introductionmentioning

confidence: 99%

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On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Kunis

Nagel

2020

Numer Algor

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We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are “off the grid,” pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

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Section: Introductionmentioning

confidence: 99%

“…We need additional assumptions for Lemma 3.5 to work since results for general well-separated nodes, cf [15],. seem to be too weak.…”

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confidence: 99%

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Kunis

Nagel

2020

Numer Algor

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“…The condition on n implies that A has full rank M , cf. [8,9]. Hence, T has indeed rank M since all c 1 , .…”

Section: 1mentioning

confidence: 88%

“…Theorem 2.1, which shall enable us to identify the vectors {t j } M j=1 . In the following theorem, K d denotes an absolute constant that only depends on d and is further specified in [8,9]. We also make use of z j := e −2πitj := (e −2πitj,1 , .…”

Section: 1mentioning

confidence: 99%

A randomized multivariate matrix pencil method for superresolution microscopy

Ehler¹,

Kunis²,

Peter³

et al. 2019

etna

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The matrix pencil method is an eigenvalue based approach for the parameter identification of sparse exponential sums. We derive a reconstruction algorithm for multivariate exponential sums that is based on simultaneous diagonalization. Randomization is used and quantified to reduce the simultaneous diagonalization to the eigendecomposition of a single random matrix. To verify feasibility, the algorithm is applied to synthetic and experimental fluorescence microscopy data.2010 Mathematics Subject Classification. 65T40, 42C15, 30E05, 65F30.

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“…Finally, we like to mention that there exist other methods for super-resolution as Prony-like methods [13,30,31,37,38]. These are spectral methods which perform spike localization from low frequency measurements.…”

Section: Introductionmentioning

confidence: 99%

Super-resolution for doubly-dispersive channel estimation

Beinert

Jung

Steidl

et al. 2021

Sampl. Theory Signal Process. Data Anal.

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In this work we consider the problem of identification and reconstruction of doubly-dispersive channel operators which are given by finite linear combinations of time-frequency shifts. Such operators arise as time-varying linear systems for example in radar and wireless communications. In particular, for information transmission in highly non-stationary environments the channel needs to be estimated quickly with identification signals of short duration and for vehicular application simultaneous high-resolution radar is desired as well. We consider the time-continuous setting and prove an exact resampling reformulation of the involved channel operator when applied to a trigonometric polynomial as identifier in terms of sparse linear combinations of real-valued atoms. Motivated by recent works of Heckel et al. we present an exact approach for off-the-grid super-resolution which allows to perform the identification with realizable signals having compact support. Then we show how an alternating descent conditional gradient algorithm can be adapted to solve the reformulated problem. Numerical examples demonstrate the performance of this algorithm, in particular in comparison with a simple adaptive grid refinement strategy and an orthogonal matching pursuit algorithm.

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Prony’s Method Under an Almost Sharp Multivariate Ingham Inequality

Cited by 17 publications

References 40 publications

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

A randomized multivariate matrix pencil method for superresolution microscopy

Super-resolution for doubly-dispersive channel estimation

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