A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

作者：Schost Eric; Spaenlehauer Pierre Jean^*

来源：Foundations of Computational Mathematics, 2016, 16(2): 457-492.

DOI：10.1007/s10208-015-9256-x

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

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix , the goal is to compute a matrix of given rank in a linear or affine subspace of matrices (usually encoding a specific structure) such that the Frobenius distance 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 and the linear/affine subspace . 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 matrices in . 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).

• 出版日期2016-4
• 单位INRIA