A multivariate generalization of Prony's method

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

doi:10.1016/j.laa.2015.10.023

Cited by 66 publications

(80 citation statements)

References 31 publications

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“…As mentioned above, one advantage of the Hankel Prony structure H over the Toeplitz Prony structure T is that H works with exponential sums with arbitrary bases in K n while T needs bases in (K \ {0}) n . On the other hand, some relevant results in this context are known only for Toeplitz matrices; see, e.g., [18,Theorem 3.7].…”

Section: Remark 45mentioning

confidence: 99%

Learning algebraic decompositions using Prony structures

Kunis

Römer

Ohe

2020

Advances in Applied Mathematics

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Section: Remark 45mentioning

confidence: 99%

Learning algebraic decompositions using Prony structures

Kunis

Römer

Ohe

2020

Advances in Applied Mathematics

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“…Next, it is natural to extend the techniques developed here to the multi-dimensional setting as multivariate signals are of high importance in many applications such as DNA sequencing and Mass Spectrometry. This could be investigated using for instance some extension of Prony's method to several dimensions such as in [37], [28] and [22].…”

Section: Future Directionsmentioning

confidence: 99%

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Chrétien

Tyagi

2020

J Fourier Anal Appl

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Consider L groups of point sources or spike trains, with the l th group represented by x l (t). For a function g : R → R, let g l (t) = g(t/µ l ) denote a point spread function with scale µ l > 0, and with µ 1 < · · · < µ L . With y(t) = L l=1 (g l x l )(t), our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein L = 1; we call this the multi-kernel unmixing superresolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage 1 ≤ l ≤ L involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).

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“…Under some natural assumptions, solutions have been proposed for the univariate case, for the multivariate case with a univariate resolution (projection method) and with a multivariate approach …”

Section: The Univariate Case: Gaussian Quadraturesmentioning

confidence: 99%

Algorithms for computing cubatures based on moment theory

Collowald

Hubert

2018

Stud Appl Math

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Quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomials of degree 2r−1 or less by a suitable choice of r nodes and weights. Cubature is a generalization of quadrature in higher dimension. In this article, we elaborate algorithms to compute all minimal cubatures for a given domain and a given degree. We propose first algorithm in symbolic computation to characterize all cubatures of a given degree with a fixed number of nodes. The determination of the nodes and weights is then left to the computation of the eigenvectors of the matrix identified at the characterization stage and can be performed numerically. The characterization of cubatures on which our algorithms are based stems from moment theory. We formulate the results there in a basis‐independent way: Rather than considering the moment matrix, the central object in moment problems, we introduce the underlying linear map from the polynomial ring to its dual, the Hankel operator. This makes natural the use of bases of polynomials other than the monomial basis, and proves to be computationally relevant, either for numerical properties or to exploit symmetry.

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A multivariate generalization of Prony's method

Cited by 66 publications

References 31 publications

Learning algebraic decompositions using Prony structures

Learning algebraic decompositions using Prony structures

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Algorithms for computing cubatures based on moment theory

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