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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).
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https://hal.archives-ouvertes.fr/hal-00953684
Contributor : Pierre-Jean Spaenlehauer <>
Submitted on : Friday, February 28, 2014 - 2:51:28 PM
Last modification on : Thursday, February 7, 2019 - 3:14:49 PM

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Éric Schost, Pierre-Jean Spaenlehauer. A Quadratically Convergent Algorithm for Structured Low-Rank Approximation. Foundations of Computational Mathematics, Springer Verlag, 2016, 16 (2), pp.457-492. ⟨10.1007/s10208-015-9256-x⟩. ⟨hal-00953684⟩

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