Foundations of Computational Mathematics

, Volume 16, Issue 2, pp 457–492 | Cite as

A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

  • Éric Schost
  • Pierre-Jean Spaenlehauer


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 \(\left\| M-M' \right\| \) 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).


Structured Low-Rank Approximation Newton iteration  Quadratic convergence Approximate GCD Matrix completion 

Mathematics Subject Classification

65B99 65Y20 15A83 



We are grateful to Erich Kaltofen, Giorgio Ottaviani, Olivier Ruatta, Bruno Salvy, Bernd Sturmfels and Agnes Szanto for useful discussions and for pointing out important references. We also wish to thank an anonymous referee for his useful comments which led to the second variant of the algorithm. We acknowledge the financial support of NSERC and of the Canada Research Chairs program.


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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.ORCCA and Department of Computer Science, Middlesex CollegeWestern UniversityLondonCanada
  2. 2.Inria Nancy Grand-Est, Université de Lorraine, CNRS, CARAMEL ProjectVillers-lès-Nancy CedexFrance
  3. 3.Max Planck Institute for MathematicsBonnGermany

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