Prony’s method in several variables

Sauer, Tomas

doi:10.1007/s00211-016-0844-8

Cited by 30 publications

(44 citation statements)

References 32 publications

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“…Though determining Ω is a nonlinear problem, it can be approached by methods from Numerical Linear Algebra. As pointed out in [7,8], the Hankel matrices…”

Section: Prony's Problem In Several Variablesmentioning

confidence: 93%

“…By increasing B in a proper way, one can so construct Gröbner or H-bases for I Ω with which the computation of the frequencies is reduced to an eigenvalue problem. The main observation from [7,8] is now as follows. 2. if rank F A,Γ n = rank F A,Γ n+1 , then ker F A,Γ n (x) is a basis of I Ω , where Γ n := {α ∈ N s 0 : |α| ≤ n}.…”

Section: Prony's Problem In Several Variablesmentioning

confidence: 97%

“…In the latter, the main numerical problem in the algorithms presented in [7,8] is to determine reliably the nullspaces of sequences of matrices that are generated by successively adding blocks of columns. Adding these columns corresponds to extending a symmetric H-basis for an ideal, and treating them in a symmetric way and not attaching them in some order is of great importance for the numerical performance of such algorithms.…”

Section: Introductionmentioning

confidence: 99%

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SVD update methods for large matrices and applications

Peña

Sauer

2019

Linear Algebra and its Applications

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We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix A ∈ R m×n with m n. Supposing that an SVD of A is already known, and given a matrix B ∈ R m×n , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [AB] ∈ R m×(n+n ) . This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted.

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“…Though determining Ω is a nonlinear problem, it can be approached by methods from Numerical Linear Algebra. As pointed out in [7,8], the Hankel matrices…”

Section: Prony's Problem In Several Variablesmentioning

confidence: 93%

Section: Prony's Problem In Several Variablesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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SVD update methods for large matrices and applications

Peña

Sauer

2019

Linear Algebra and its Applications

Self Cite

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“…is a bijection. For an exponential sum f ∈ Exp and d ∈ N, consider the Hankel matrix [30], and Mourrain [21]).…”

Section: Common Options For the Sequencementioning

confidence: 99%

Learning algebraic decompositions using Prony structures

Kunis

Römer

Ohe

2020

Advances in Applied Mathematics

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“…In [28], a projection based method is used to compute univariate polynomials which roots determine the coordinates of the terms in the sparse representation. In [38], an H-basis of the kernel ideal of a moment matrix is computed and used to find the roots which determine the sparse decomposition. Direct decomposition methods which compute the algebraic structure of the Artinian Gorenstein algebra associated to the moment matrix and deduce the sparse representation from eigenvectors of multiplication operators are developed in [31] and [22].…”

Section: Pronymentioning

confidence: 99%

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

Josz¹,

Lasserre²,

Mourrain³

2019

Adv Comput Math

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We show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the n-dimensional torus. Therefore the semidefinite programming approach initiated by in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of ℓ 1 -norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive sparse recovery LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).

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Prony’s method in several variables

Cited by 30 publications

References 32 publications

SVD update methods for large matrices and applications

SVD update methods for large matrices and applications

Learning algebraic decompositions using Prony structures

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

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