Learning algebraic decompositions using Prony structures

Kunis, Stefan; Römer, Tim; Ohe, Ulrich von der

doi:10.1016/j.aam.2020.102044

Cited by 7 publications

(7 citation statements)

References 39 publications

(62 reference statements)

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“…Moreover, one can work with much more general filtrations of the polynomial ring; see the statements in [vdOhe17, Chapter 2]. See also [KRvdO20] for an approach relating Toeplitz and Hankel matrices in this context. ♦…”

Section: Prony's Methodsmentioning

confidence: 99%

Truncated moment problems on positive-dimensional algebraic varieties

Wageringel¹

2022

Preprint

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Section: Prony's Methodsmentioning

confidence: 99%

Truncated moment problems on positive-dimensional algebraic varieties

Wageringel¹

2022

Preprint

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“…The proof of this statement follows a completely similar course as that for the cosine atom g(φ i ; t) given in the Appendix. The factorization in Theorem 3 allows to write down a spread polynomial analogue of (15). Corollary 3.…”

Section: Polynomial Functionsmentioning

confidence: 99%

Parametric spectral analysis: scale and shift

Cuyt¹,

Lee²

2020

Preprint

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We introduce the paradigm of dilation and translation for use in the spectral analysis of complex-valued univariate or multivariate data. The new procedure stems from a search on how to solve ambiguity problems in this analysis, such as aliasing because of too coarsely sampled data (see [8]), or collisions in projected data (see [7]), which may be solved by a translation of the sampling locations.In Section 2 both dilation and translation are first presented for the classical one-dimensional exponential analysis. In the subsequent Sections 3-7 the paradigm is extended to more functions, among which the trigonometric functions cosine, sine, the hyperbolic cosine and sine functions, the Chebyshev and spread polynomials, the sinc, gamma and Gaussian function, and several multivariate versions of all of the above.Each of these function classes needs a tailored approach, making optimal use of the properties of the base function used in the considered sparse interpolation problem. With each of the extensions a structured linear matrix pencil is associated, immediately leading to a computational scheme for the spectral analysis, involving a generalized eigenvalue problem and several structured linear systems.In Section 8 we illustrate the new methods in several examples: fixed width Gaussian distribution fitting, sparse cardinal sine or sinc interpolation, and lacunary or supersparse Chebyshev polynomial interpolation.

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“…Exact measure recovery in the classical Fourier setting, i.e. when the underlying space is the torus T d , has a very long history starting with the initial work by G. R. de Prony in 1795 [1] in the univariate case, and then moving on to different one-dimensional [7,6,21] and multi-dimensional Prony-based techniques [31,37,41,47,48] that have stabilized and generalized the method in various directions. Recently, the Prony's method has also been extend to the ddimensional sphere S d .…”

Section: Introductionmentioning

confidence: 99%

“…To analyze the convergence of the discretization process we build on results stated in [29]. We also would like to mention the for the nonnegative total variation minimization problem an alternative way of construction of a dual certificate that involves some algebraic techniques has been proposed in [48], and in pure compressed sensing setting a recovery of sparse signal on the two-dimensional sphere has been considered in [23] The outline of this paper is as follows. In Section 2, briefly the necessary analytical tools on the sphere including spherical harmonics are introduced, and the problem of super-resolution is stated.…”

Section: Introductionmentioning

confidence: 99%