A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

Éric Schost 1, 2 Pierre-Jean Spaenlehauer 3
3 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $M$, the goal is to compute a matrix $M'$ of given rank $r$ in a linear or affine subspace $E$ of matrices (usually encoding a specific structure) such that the Frobenius distance $\lVert M-M'\rVert$ is small. We propose a Newton-like iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank $r$ and the linear/affine subspace $E$. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank $r$ matrices in $E$. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured Low-Rank Approximation: univariate approximate GCDs (Sylvester matrices), low-rank Matrix completion (coordinate spaces) and denoising procedures (Hankel matrices).
Type de document :
Article dans une revue
Foundations of Computational Mathematics, Springer Verlag, 2015, pp.1-36
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https://hal.archives-ouvertes.fr/hal-00953684
Contributeur : Pierre-Jean Spaenlehauer <>
Soumis le : vendredi 28 février 2014 - 14:51:28
Dernière modification le : jeudi 22 septembre 2016 - 14:31:10

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  • HAL Id : hal-00953684, version 1
  • ARXIV : 1312.7279

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Éric Schost, Pierre-Jean Spaenlehauer. A Quadratically Convergent Algorithm for Structured Low-Rank Approximation. Foundations of Computational Mathematics, Springer Verlag, 2015, pp.1-36. <hal-00953684>

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